Mathematical Surveys and Monographs
Volume 62
Gaussian Measures
Vladimir I. Bogachev
American Mathematical Society
Selected Titles in This Series 62 Vladimir I. Bogachev, Gaussian measures. 1998 61 W. Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators, 1998
60 lain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C'-algebras. 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice. 1998
58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations. 1998 57 Marc Levine, Mixed motives. 1998
56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part 1, 1998
55 J. Scott Carter and Masahico Salto, Knotted surfaces and their diagrams. 1998 54 Casper Goffman, Togo Nishlura, and Daniel Waterman, Homeomorphisms in analysis. 1997
53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis. 1997 52 V. A. Kozlov, V. G. Mas'ya, and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997
51 Jan Mali and William P. Zlemer, Fine regularity of solutions of elliptic partial differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory. 1997
49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations. 1997
48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997
47 A. D. Elmendorf, I. Kris, M. A. Mandell, and J. P. May (with an appendix by M. Cole), Rings, modules, and algebras in stable homotopy theory. 1997 46 Stephen Lipscomb, Symmetric inverse semigroups. 1996
45 George M. Bergman and Adam 0. Hausknecht, Cogroups and co-rings in categories of associative rings. 1996
44 J. Amords, M. Burger, K. Corlette, D. Kotachick, and D. Toledo, Fundamental groups of compact Kiihler manifolds. 1996
43 James E. Humphreys, Conjugacy classes in semisimple algebraic groups. 1995
42 Ralph Freese, Jaroslav Jeiiek, and J. B. Nation, Free lattices. 1995 41 Hal L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. 1995
40.3 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 3; 1998
40.2 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 2. 1995 40.1
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 1. 1994
39 Sigurdur Helgason, Geometric analysis on symmetric spaces. 1994 38 Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, 1993 37 Leonard Lewin, Editor, Structural properties of polylogarithms, 1991 36 John B. Conway, The theory of subnormal operators, 1991 35 Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers. 1990 34 Victor Isakov, Inverse source problems, 1990 33 Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, 1990
32 Howard Jacobowits, An introduction to CR structures. 1990 (Continued in the back of this publication)
Gaussian Measures
Mathematical Surveys and Monographs
Volume 62
Gaussian Measures
Vladimir 1. Bogachev
American Mathematical Society
Editorial Board Tudor Stefan Ratiu, Chair Michael Renardy
Georgia M. Benkart Peter Landweber
Translated from the original Russian manuscript by Vladimir I. Bogachev. 1991 Mathematics Subject Classification. Primary 28C20, 60B11; Secondary 60015, 60H07. ABSTRACT. This book presents a systematic exposition of the modern theory of Gaussian measures. The basic properties of finite and infinite dimensional Gaussian distributions, including their linear
and nonlinear transformations. are discussed. The book is intended for graduate students and researchers in probability theory, mathematical statistics, functional analysis, and mathematical physics. It contains a lot of examples and exercises. The bibliography contains 844 items; the detailed bibliographical comments and subject index are included.
Library of Congress Cataloging-in-Publication Data Bogachev. V. 1. (Vladimir Igorevich). 1961iGaussovskie mery. English] Gaussian measures / Vladimir I. Bogachev. (Mathematical surveys and monographs. ISSN 0076.5376 ; v. 62) p. cm. Includes bibliographical references and index. ISBN 0-8218-1054-5 (he : alk. paper)
1. Gaussian measures. QA312.B64 1998
I. Title. II. Series: Mathematical surveys and monographs ; no. 62.
515'.42 -dc21
98-27239
CIP
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Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint -peraissionaass.org. © 1998 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. E) The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.
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Contents Preface
xi
Chapter 1. Finite Dimensional Gaussian Distributions 1.1. Gaussian measures on the real line 1.2. Multivariate Gaussian distributions 1.3. Hermite polynomials 1.4. The Ornstein-Uhlenbeck semigroup 1.5. I.G. 1.7. 1.8. 1.9. 1.10.
Sobolev classes Hypercontractivity Several useful estimates Convexity inequalities Characterizations of Gaussian measures Complements and problems
Chapter 2. Infinite Dimensional Gaussian Distributions 2.1. Cylindrical sets 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9. 2.10.
Basic definitions Examples
2.11.
Stochastic integrals
2.12.
Complements and problems
The Cameron-Martin space Zero-one laws Separability and oscillations Equivalence and singularity Measurable seminorms The Ornstein-Uhlenbeck semigroup Measurable linear functionals
Chapter 3. Radon Gaussian Measures 3.1. Radon measures 3.2. Basic properties of Radon Gaussian measures 3.3. Gaussian covariances 3.4. The structure of Radon Gaussian measures 3.5. Gaussian series 3.6. Supports of Gaussian measures 3.7. Measurable linear operators 3.8. Weak convergence of Gaussian measures 3.9. Abstract Wiener spaces 3.10. Conditional measures and conditional expectations Vii
97 97 100 104
109 112 119 122
129
136 140
Contents
viii 3.11.
Complements and problems
142
Chapter 4. Convexity of Gaussian Measures 4.1. Gaussian symmetrization Ehrhard's inequality 4.2. 4.3. Isoperimetric inequalities 4.4. Convex functions 4.5. H-Lipschitzian functions 4.6. Correlation inequalities 4.7. The Onsager-Machlup functions 4.8. Small ball probabilities 4.9. Large deviations 4.10. Complements and problems
157 157 162 167
Chapter 5. Sobolev Classes over Gaussian Measures 5.1. Integration by parts 5.2. The Sobolev classes Wp'' and Dr"
205 205
5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10. 5.11. 5.12.
The Sobolev classes HP2' Properties of Sobolev classes and examples The logarithmic Sobolev inequality Multipliers and Meyer's inequalities Equivalence of different definitions Divergence of vector fields Gaussian capacities Measurable polynomials Differentiability of H-Lipschitzian functions Complements and problems
171 174
177 181
187 195 197
211 215
218 226 229 234 238 243 249 261 266
Chapter 6. Nonlinear Transformations of Gaussian Measures Auxiliary results 6.1. Measurable linear automorphisms 6.2. 6.3. Linear transformations 6.4. Radon-Nikodym densities 6.5. Examples of equivalent measures and linear transformations Nonlinear transformations 6.6. 6.7. Examples of nonlinear transformations Finite dimensional mappings 6.8. Malliavin's method 6.9. 6.10. Surface measures 6.11. Complements and problems
279 279 282 285 288 295 298 308 314 316
Chapter 7. Applications 7.1. Trajectories of Gaussian processes 7.2. Infinite dimensional Wiener processes Logarithmic gradients 7.3. 7.4. Spherically symmetric measures
333 333 336 339 344 346 353
7.5. 7.6.
Infinite dimensional diffusions Complements and problems
Appendix A. Locally Convex Spaces, Operators, and Measures
321
324
361
Contents
ix
A.1.
Locally convex spaces
361
A.2.
Linear operators
365
A.3.
Measures and measurability
371
Bibliographical Comments
381
References
391
Index
427
Preface The modem theory of Gaussian measures lies at the intersection of the theory of random processes, functional analysis, and mathematical physics and is closely connected with diverse applications in quantum field theory, statistical physics, financial mathematics, and other areas of sciences. The study of Gaussian measures combines ideas and methods from probability theory, nonlinear analysis, geometry, linear operators, and topological vector spaces in a beautiful and nontrivial way. The goal of this book is to present the modern theory of Gaussian measures. Chapter 1 contains basic facts about Gaussian measures on IR". In addition to the standard probabilistic facts the reader will find here a discussion of the Hermite polynomials, the Ornstein-Uhlenbeck semigroup, the Sobolev classes with Gaussian weights, the logarithmic Sobolev and Poincark inequalities, and convexity of Gaussian measures. These analytic tools play a fundamental role in the theory. Principal results belonging to linear topological theory are discussed in Chapters 2 and 3. These include classical theorems about equivalence and singularity, zero-one laws, Cameron-Martin spaces, measurable linear functionals, and the topological properties of supports. Chapter 4 contains some inequalities and estimates related to the convexity of Gaussian measures, such as Gaussian isoperimetric inequalities, and Ehrhard's and Anderson's inequalities. These inequalities are applied to the study of the exponential integrability, probabilities of small balls, and large deviations. Nonlinear problems are discussed in Chapters 5 and 6, where, in particular, the Sobolev classes over Gaussian measures and Gaussian capacities are investigated. Chapter 6 deals with transformations of Gaussian measures. In addition, we give an introduction to the Malliavin calculus. In Chapter 7 we discuss some properties of finite and infinite dimensional Gaussian processes and certain related diffusion processes. These results, apart from the interest in their own right, provide a good illustration of the ideas and methods of the previous chapters. It is worth noting here that one of the fundamental ideas in the theory of Gaussian measures is that the various centered Radon Gaussian measures are realizations of one and the same "canonical" Gaussian measure: the countable product of the standard normal Gaussian distributions on the line. This canonical measure ry is defined on the space IR°` of all real sequences. The Cameron-Martin space of ry is the classical Hilbert space 12. The space IR" has a relatively poor collection of continuous linear functionals (consisting only of the functionals that depend on finitely many variables). However, the set of measurable linear functionals is much broader: it can be identified with 12: for any (c,) E 12 the series F,',,=, C,,x,, converges -f-almost everywhere, and every measurable linear functional admits such a representation. Although the Cameron-Martin space 12 has measure zero, every continuous linear functional on it (even every continuous linear operator) admits Xi
xii
Preface
a unique measurable linear extension to all of R". Having in mind this basic example, one can understand better the "coordinate" interpretation of the results presented in this book. Certainly, there are problems in which the reduction to R" is useless. For instance, this is the case in most of the problems concerning sample path properties of Gaussian processes. Nevertheless, readers who prefer not to dwell on topological subtleties connected with infinite dimensional spaces may assume (and the isomorphism theorem from Chapter 3 gives a full justification for this) that our discussion concerns Hilbert spaces or the space R". The "unique" Gaussian measure mentioned above is often encountered in the appearance of the Wiener measure on the space of continuous trajectories; in this case very interesting objects arise that have no natural analogues in the other isomorphic representations. All auxiliary results from functional analysis and general topology used in the texts are collected in the Appendix. Prerequisites include basics of Lebesgue integration, multivariate calculus, and probability theory.
Formulas and assertions (theorems, definitions, remarks, etc.) are numbered independently of their type within each section so that the number of an assertion or a formula is preceded by the chapter number and section number. The book contains a large number of problems (exercises). The role of the
problems (in addition to the usual one) is to present a disordered collection of interesting facts and to unburden the main text from some technical details of proofs. Many problems are provided with hints; this, however, has no relation to the level of difficulty (some problems are simple exercises, but the others are really hard results borrowed from the current research literature). The bibliography does not exhaust all the publications relating to the theory of Gaussian measures. However, together with the bibliographical comments, it enables the reader to get a sufficiently complete vision of the history of the subject as well as entertain a more thorough bibliographical research. One can use this book as a source for several one- or two-semester courses (of
different levels) for graduate students. For example, Chapter 1 can form a core for an introductory course on finite dimensional Gaussian distributions. A course on infinite dimensional Gaussian distributions can be based on Chapters 2 and 3. Chapters 5 and 6 may be helpful for lecturing on nonlinear stochastic analysis. This book is an expanded and improved version of the Russian original based on the author's lectures at the Department of Mechanics and Mathematics of Moscow State University. Parts of the book have been written during my visits to other universities and mathematical institutes, in particular, in Rome, Paris. Bonn, Bielefeld, Pisa. Warwick, Stockholm, Edmonton, Minneapolis, and Haifa. I am very grate-
ful to L. Accardi, S. Albeverio, J. Baxter, G. Da Prato, G. Dell'Antonio, D. Elworthy, F. Gotze, N. Jain, N. Krylov, P. Malliavin, E. Mayer-Wolf, B. Oksendal. M. Riickner, B. Scbmuland, M. Zakai, and other colleagues from these institutions for the excellent working conditions. I have had many profitable discussions regarding various subjects treated here with V. Bentkus, S.G. Bobkov. N.V. Krylov,
M. Ledoux, M.A. Lifshits. Yu.V. Prohorov, A.N. Shiryaev, A.V. Skorohod, O.G. Smolyanov, A.M. Stepin, V.N. Sudakov, V.V. Ulyanov, H. von Weizsacker. A.Yu. Zaitsev, and O. Zeitouni. My special thanks are due to D.E. Aleksandrova. V. Bentkus, A.V. Kolesnikov, E.P. Krugova, N.N. Nedikov, O.V. Pugachev, T.S. Rybnikova, B. Schmuland, and N.A. Tolmachev, who read the drafts of the book and made many critical remarks.
CHAPTER 1
Finite Dimensional Gaussian Distributions The connection of literature purposes with the purely scientific ones. the desire to occupy imagination and at the same time to enrich life with ideas and knowledge create considerable difficult ies in composing the different parts of the book and hamper the unity of exposition. Alexander von Humboldt. Views of Nature
1.1.
Gaussian measures on the real line
We start our discussion of Gaussian measures by recalling the identity +x. 1
cr
2ir J
-x
exp
(t - a)2
\
2u-
) dt = 1.
(1.1.1)
known for all real a and o > 0. A standard way of verifying this equality is to evaluate the double integral ff e xp(-x2 - y2) dxdy in polar coordinates and apply Fltbini's theorem.
1.1.1. Definition. A Borel probability measure ry on R' is called Gaussian if it is either the Dirac measure b at a point a or has density ( (f - a)2 P(',a.a2): t i-' 1
a
27reXp
\
2o-2
with respect to Lebesgue measure. In. the latter case the measure y is called nondegenerate. The parameters a and a2 are called the mean and the variance of y, respectively.
The quantity a is called the mean-square deviation. For any Dirac measure (i.e., a probability measure concentrated at a point) we put a = 0. The mean and the variance of a Gaussian measure y admit the following representation: +x
x
a = J t y(dt). a2 = I (t - a)2 y(dt).
-x
-x
The measure with density p( , 0, 1) is called standard A mean zero Gaussian measure is called centered or symmetric.
1.1.2. Definition. A Gaussian random variable is a random variable with Gaussian distribution.
A Gaussian random variable with a centered distribution is called centered or symmetric. Clearly, an arbitrary Gaussian random variable with mean a and I
Chapter 1.
2
Finite Dimensional Distributions
variance o2 can be represented as o£ + a, where C is a random variable with the standard Gaussian distribution. Gaussian distributions are often called normal. Using equality (1.1.1) it is easy to find the Fourier transform (the characteristic functional) of the Gaussian measure -1 with parameters (a,a2). We have
7(y) := I exP(iyx)7(dx) = exP(iay - a2y2) IRI
The normal (standard Gaussian) distribution function lation
is defined by the re-
ft
p(s, 0, 1) ds.
4'(t)
The inverse function 4-1 is defined on (0.1). It is convenient to employ the follow-
ing convention: 4-1(0) = -oo and 4-1(1) = +oo. The rate of decreasing of 1 - 4) at infinity is estimated as follows.
1.1.3. Lemma. For any It > 0. one has
t
2n \ t
)e'0/2 < 1
- $(t) <
1
! e-
(1.1.2)
v n
PROOF. By virtue of the integration by parts formula, we get se-°2/21
1t
$
ds = e_t2/2 t
- I 1 e-a2/2 ds < es=
P/2 t
t
0
The lower bound is proved in a similar manner.
The next classical result has a lot of applications in the theory of Gaussian measures.
1.1.4. Theorem. Let s;,, be a sequence of independent centered Gaussian random variables with variances an. Then the following conditions are equivalent:
(i) The series
converges almost everywhere: n=1
(ii) The series in (i) converges in probability; (iii) The series in (i) converges in L2; X (iv) E a' < 00. n=1
PROOF. According to the dominated convergence theorem and the independence condition, either of conditions (i) - (iii) implies convergence of the product
x
x
J exP(iW dP = 11 exp(-On/2). n=1
n=1
which yields condition (iv). Conversely, condition (iv) implies condition (iii), since by virtue of independence and symmetry of the random variables in question, we have
rm
r J (G +...+Ckt»e)2dP=J -9 fk+;dP. Hence (ii) is satisfied as well. Thus, the only non trivial implication in our claim is
In order to prove it, note that the conditional expectation of the
1.2.
3
Multivariate distributions
square-integrable random variable t; = F,', £ with respect to the er-field A,", generated by t;l, ... , f,,,, coincides with F,', f, since the random variables fn are independent and have zero means. Therefore, the sequence of the partial sums of the series defining l; is a martingale with respect to {An}. Hence Theorem A.3.5 D in Appendix applies and yields the almost sure convergence. One more classical result related to one dimensional Gaussian distributions is the central limit theorem. We only give its special case that will be used below. 1.1.5. Theorem. Let {t;n} be a sequence of independent random variables with
one and the same distribution such that 1F{, = 0 and o2 = IEt;i < W. Put S _ t:l + ... + t;". Then, for any x, we get
P(
o <x} -+4(x)
In addition, the distributions of measure.
asn
Sn
of converge weakly to the standard Gaussian
PROOF. Since E i < oo, the function ,p(t) = IEettF' is twice differentiable. We have
p(t) = 1 -
I a2t2 + o(t2),
The characteristic functionals Vn of fixed t, we get
an
t -. 0.
are given by tp(
n
o tn
Hence, for any
\1
a2t2 +O( I )J =e -t'12 . llm pa(t)= n-x llmlLI1- 2o2n n-.x n I
Both claims follow from this equality (see [697, Chapter III]). 1.2.
Multivariate Gaussian distributions
1.2.1. Definition. A Borel probability measure 7 on IR" is called Gaussian if for every linear functional f on IR.", the induced measure y o f - l is Gaussian. We shall use the standard identification of the space of linear functions on IR" with IR". The inner product in IR" is denoted by (- , ) or by ( , ). Recall that the Fourier transform µ of a finite Borel measure p on IR" is defined by the formula
is R" - C1, µ(y) = 1 exp[i(y,x)] p(dx). R^
Recall that measures on lRn are uniquely determined by their Fourier transforms.
1.2.2. Proposition. A measure -y on IR" is Gaussian if and only if its Fourier transform has the form
5(y) = exPl i(y,a) - I
(1.2.1)
Chapter 1. Finite Dimensional Distributions
4
where a is a vector in lR" and K is a nonnegattve matrix. The measure y has a density if and only if the matrix K is nondegenerate. In this case, the density of the measure y is given by
xPROOF. Let f be a linear function on IR". Using the change of variables formula (see formula (A.3.1) in Appendix), one evaluates the Fourier transform of the measure v = y o f -' as follows:
1(t) = f exp(its) v(ds) = use
J R^
exp(it f (x)) y(dx).
Let us denote the vector representing the functional f by the same symbol. From (1.2.1) we get
v(t) = exp(it(a, f) - I t2(K f, f )) . which means that the measure v is Gaussian. Conversely, suppose that all such measures are Gaussian_ Denote their means and variances by a(f) and a(f )2, respectively. Then the following equalities hold true:
a(f) = f t yof-'(dt) = f f(x)y(dx). R"
IR'
a(f)2 = f (t - a(f))2-yof(dt) = f ((f .T) -a(f))27(dx). W
IItt
Hence the function f a(f) is linear, and the function f '- a(f )2 is a nonnegative quadratic form. Therefore, there exist a vector a and a nonnegative symmetric
operator K such that a(f) = (f, a) and a(f )2 = (K f, f ). This yields (1.2.1). The assertion about densities reduces to the one dimensional case, since we can use the coordinates corresponding to the eigenvectors of the matrix K. 0
1.2.3. Corollary. Let -y be a Gaussian measure on lR" with the Fourier transform (1.2.1). Then
a = f x y(dx),
(1.2.2)
Ut
(Ku, v) =
J(u.x - a) (v, x - a) y(dx),
Vu, v E 1R" .
(1.2.3)
It^
The vector a given by equality (1.2.2) is called the mean of the Gaussian mea-
sure -y, and the operator K defined by means of (1.2.3) is called its covariance operator.
Clearly, Gaussian measures on lR" can be described as the images of the standard Gaussian measure on IR" (i.e., the product of n copies of the standard Gaussian measure on IR') under affine mappings x ' -vl-Kx + a. On the linear subspace VW(lR") we define the inner product (u. V)., :=
(/ 'u, v'
' ,)
I
1.2.
'Multivariate distributions
5
The unit ball with respect to this inner product (i.e., the ellipsoid VK-U, where
U is the closed unit ball in IR") is called the ellipsoid of concentration of the Gaussian measure y. The ellipsoid of concentration can be written as
A(U)={hElR"J(Lx : Ihk, <1}. .
jhi, = suP{ (l, h): I E IR", l
- a)z y(dx) ^
111
Note the following useful identity.
1.2.4. Lemma. Let K = (K1,),_1 be a positive definite matriz and let o(K, ) be the density of the centered Gaussian measure with covariance K. Then z
where K;.1 = K11,.
8Kj,e(K.x) _ (1 - 26'') axax ,g(K.x).
(1.2.4)
PROOF. By the inversion formula for the Fourier transform we have @(K, x) = (2I
f exp(-i(x.y) - 1(Ky,y)) dy. ut ^
Differentiating in
K.., with i (2,7r)"
f
j, we get
ysy,eXp(-i(x,y) - 2(Ky,y)
\
I dy.
This coincides with the right side in (1.2.4) by a property of the Fourier transform. The case i = j differs only by factor 1/2. The following observation will be used below.
1.2.5. Lemma. Let y be a centered Gaussian measure on IR". For every real 0, the image of the measure y a y on the space IR" x IR" under the mapping (x, y) '-. xsin0 + ycos0 coincides with y. In the probabilistic language: if two independent random vectors { and p have equal centered Gaussian distributions, then the vector t; sin ti + p cos 0 has the same distribution.
PROOF. It suffices to evaluate the Fourier transform of the measure µ that is the image of y(&y under the mapping indicated above. The Fourier transform of y is given by I i-. exp [- z K(l )] , where K is a quadratic form. Hence,
f exp[il(x)] p(dx)
= Jf exp[il(usinB+vcos9)] y(du)y(dv) f
= J exp[il(u)sino] y(du)
J exp[il(r)coso] y(dv)
= exp[-1K(1)sinz(1] exp[-IK(l)cosztl] =exp[-2K(l)] whence our claim.
As we shall see below, the converse to this statement is also true. A useful property of the Gaussian random variables is that random variables S1, ... , t;' are independent, provided they are orthogonal in L2 and the random vector , t") has a centered Gaussian distribution. Moreover, the following more general fact is true.
Chapter 1. Finite Dimensional Distributions
6
1.2.6. Lemma. If a random vector
= (t;t,... ,t;n) has a centered Gaussian
distribution and a random variable t;t is orthogonal in L2 to the random variables
2.... ,1;,, , then i is independent of the a -field generated by 2..... .,,. PROOF. To simplify notation we consider the case n = 2. Let K be the covari-
ance matrix of . By condition, K is a diagonal matrix, since its entries are the numbers Et;;CJ, i, j < 2. This means that the measure on IR2 generated by the vector r; splits into the product of the measures generated by {1 and {2, which is equivalent to the independence of Ct and t; 2. Then {1 is independent of every Borel function of i;2, hence it is independent of the a-field generated by & 0
=
Note that the random vector
whose components r and 2 are
Gaussian random variables may fail to be Gaussian.
1.2.7. Example. (i) Let us define a function p on the plane as follows: in the first and the third quadrants p equals 2p, where e is the standard Gaussian density on IR2, and at all other points p is zero. Clearly, p is a non Gaussian (xt,x2) on probability density. Hence the two dimensional random vector (1R2, pdx) is not Gaussian. However, its both components are Gaussian random variables, since the half-planes with the boundary lines parallel to the coordinate axis have the same measure with respect to pdx as with respect to the standard Gaussian measure, which is equivalent to saying that xt and x2 possess standard Gaussian distribution functions. (ii) Let X and Y be two independent standard Gaussian random variables on a probability space (S2, P). Put Zt = (X, tX + 1 --t2 Y), t E (-1.1). Denote by -Yt the measure on JR2 induced by the random vector Zt. Let us define a probability measure P0 on the plane by the equality 1
P0(B) = 2
Jt(B)dt.
BE B(1R2).
Then the integrals with respect to this measure are evaluated by means of the formula
f
f (x) Po(dr) = 2
JR2
f Ef (Zt) dt = 2
fJf(xh(dx)df, -11W
where lE stands for the expectation with respect to P. Denote by F and p the coordinates on the plane regarded as random variables on (1R2, P0). It is readily seen that these are standard Gaussian variables. For example, denoting by 1E,, the expectation with respect to P0, for rt one has
Ep exp(isrt) = 2
2
f Eexp[is(tX(.) +
1 - t21'( ))J dt
)exp(_s2) dt = expr-2s2
1.
1.3.
7
Hermite polynomials
In addition, Ep0(Er7)=
J1Ex(tx+
1-t2Y)dt=
Jtdt=o.
_r
_1
However, the measure At is not Gaussian. This can be seen from a similar evaluation of the Fourier transform of the random variable l; + q, which gives Epo eXP[is(4 + rl)] =
I - exp(-2s2) 2s2
We end this section by proving the following result called the normal correlation theorem.
1.2.8. Theorem. Let 1; and q be two Gaussian random vectors in R" such that the vector p) is also Gaussian. Then the conditional expectation E(l;jrl) is a Gaussian random vector which can be written as A177 - Et)) + Ee, where A is a nonrandom linear operator. PROOF. It suffices to consider the case IEC = Err = 0. Moreover, it is sufficient to prove that for every component {, of C one has IE(l;;I77) = (Ai, q), where Ai is a nonrandom vector in R". In addition, we may prove this claim in the basis of R" formed by the eigenvectors of the covariance of n (which makes the nonzero components of q independent). It remains to note that one can find scalars ail n
such that {, - E A 71J is orthogonal to each >7i, j = 1,... , n, hence is independent i=l of i?. Indeed, we can put ) = E(l;,gq) and A, _ (ail,... ,A,, ). Then we get
0
(A ,r]).
1.3. Hermite polynomials
1.3.1. Definition. For any k = 0,1.... , the Hermite (or Chebyshev-Hermite) polynomials Hk on the real line are defined by the formula kkexpl
21expt-2 1
Hk(x)=(
F o r any multiindex a = (kl, ... , kn) with nonnegative integer entries, the Hermite polynomial H. on R" is defined by
H. (xl.... , xn) = Hk, (xl) ... Hk (xn), where Hk, are the one dimensional Hermite polynomials.
Hermite polynomials on the real line are obtained by orthogonalizing the sequence of the powers of x in L2 with respect to the standard Gaussian measure. These polynomials can be introduced in several different ways, e.g., by means of the expansion /
exp( Ax - 1),2 )
\
2
x
x
-k
O
Hk(x).
(1.3.1)
k!
To see this, note that exp( Ax - !A2) can be written as
rx
exp(2x2
k dk
- 2(x - A)21l = exp(2x2) ` k! dak expl k=0
\\\
2(x - Jl)2 / a=0
Chapter 1. Finite Dimensional Distributions
8
1.3.2. Lemma. Hermite polynomials on 1R1 have the following properties: (i) {Hk} is an orthonormal basis in the space L2(y1), where y1 is the standard Gaussian measure on 1R1;
(ii) Hk(x) ='Hk_1(x) =xHk(x)- k+1Hk+1(x). k> 1: (iii) for any numbers Al,... , An, one has n aki+...+k
t.l
ki!Hk,(A1) =
n
1
n
k expt,a; - 2 Et? C7tk, ...atn 1
PROOF. Claims (ii) and (iii) are easily deduced from the definition and (1.3.1).
Integrating the function exp(tx - 2t2) exp (sx - 2s2) in x with respect to the standard Gaussian density we get exp(ts). On the other hand, using (1.3.1), we get
Ek.n>O(n!k!)-1J2tnsk(Hn,Hk)L2(.t,). Comparing the coefficients at t and s,
we see thit Hk are mutually orthogonal and have unit norms. Note that Hk is a polynomial of degree k. Hence the linear span of Ho, ... , Hk coincides with the space of polynomials of degree at most k. Suppose that f E L2(yi) is orthogonal to all Hk. Then f is orthogonal to all polynomials. Hence all derivatives at zero of the function
F(z) =
J IR'
exp(izx)f(x)y1(dx)
vanish together with F(O), which implies that F is identically zero, since F is
0
holomorphic. Therefore, f = 0 a.e.
1.3.3. Corollary. The system of the Hermite polynomials Hk,.....k.,,
ki=0,1,...,
on 1Rn is an orthonormal basis in L2(-y,,), where yn is the standard Gaussian measure on IR".
PROOF. This follows from the fact that, given measures p, i = 1,... , n, on spaces X, and orthonormal bases {Sp' } in L2(µ;), the system of functions
,xn) forms an orthonormal basis in L2(gn 1µi).
O
Using Hermite polynomials, one constructs a decomposition of L2(yn) into the direct sum of mutually orthogonal subspaces Xk of polynomials. For any k = 0,1,..., denote by Xk the closed linear subspace in L2(-y,,), generated by the Hermite polynomials H. with Ia{ = k, +... + kn = k (note that these polynomials form an orthonormal basis in Xk). Denote by Ik the orthogonal projection in L2(yn) onto Xk.
1.3.4. Proposition. The subspaces Xk are mutually orthogonal and L2(yn) is their direct sum.
PROOF. According to Corollary 1.3.3, the linear span of the subspaces Xk is dense in L2(-y,,). Since the polynomials of the form Hk, (xi) ... Hk, (xn) with n
ki = k form a basis in Xk, it suffices to note that any two such polynomials of t=1
different degrees are orthogonal.
0
1.4.
The Ornstein-Uhlenbeck semigroup
9
Let E be a finite dimensional Hilbert space. Then the space Xk(E) of E-valued
Hermite polynomials is defined as the linear span of the mappings f v. where f E Xk, v E E. in the space L2 (_Y,,, E) of E-valued mappings. It is straightforward to see that Xk(E) is a closed subspace in the Hilbert space L2(-t,,, E). The orthogonal
projection of L2(y", E) onto Xk(E) is denoted by Ik as in the scalar case.
1.3.5. Remark. Suppose that f E C'°(1R1) is such that f(k) E L2(yl) for every k > 0. Then
Ik(f) = f fHk d'YI =
(
1
f
f(k) d,,.
Indeed, this follows directly from the definition of Hk and the k-fold integration by parts. Thus, we have
f= 1.4.
k
(
k k (f(k), l)L2(i,)Hk
The Ornstein-Uhlenbeck semigroup
Let - be a centered Gaussian measure on IR". The Ornstein-Uhlenbeck semigroup (Tt)t>o is defined on LP(y) by the following formula (known as the Mehler formula):
Te f (x) = f f (e-ex + v'-I-- e-21 y) y(dy) R"
Certainly, it has to be verified that these operators are well-defined. This is done as follows. We know that y is the image of the measure -to -y on IR" x 1R" under the mapping (x, y)'-' a-ex + 1 - e-2t y. Therefore, for every f E LP(y) we have
f I f (x) I D y(dx) = f J I f (e-x + V/e-2t y) l D y(dx) y(dy)-
(1.4.1)
By Fubini's theorem, it follows from (1.4.1) that x _.
f (e-tx +
ID
1 - e-21 y)l1 -y(dy)
R"
is -y-integrable. By Holder's inequality, we obtain that Tt f E LP(-y). Moreover, IITtf II LD(-,) s If II L`(,,)
Note that for every f E L1 (y) we have by the change of variables formula
f Ttf (x) -t(dx) = f f (x) y(dx), R^
(1.4.2)
R^
which means that y is an invariant measure for the semigroup (Tt)t>o. Note that since Tt f (x.) converges to fi dy as t - oo for every bounded continuous function f ,
it follows that -y is a unique invariant probability for M60By the same expression we define the action of Tt on the vector-valued functions
f E L'(y, E), where E is a separable Hilbert. space.
Chapter 1. Finite Dimensional Distributions
10
Simple additional considerations lead to the following result (the definitions of the notions connected with operator semigroups can be found in Appendix).
1.4.1. Theorem. For every p > 1, the family (Tt)t>o is a strongly continuous semigroup of operators on LP(y) with the operator norms IITtIIc(LP(,)) = I.
The operators Tt are symmetric nonnegative on L2(y). In addition, if E is a separable Hilbert space, then (Tt)f>o is a strongly continuous semigroup of operators with unit norms on the space LP(-y, E) of E-valued mappings.
PROOF. Since Ttl = 1 and IITtIIc(LP(,)) <- 1 for every t > 0, the operators Tt have norm 1 on LP(y). In order to prove the equality Tt+, f = TjT, f note that the measure y is the image of the measure -toy under the mapping
1-e-
(y,z)`..a-"
y+
1-e-2'
Z.
Therefore,
Tt(Taf)(x) = f T.f (,-Ix +
1 - e-2t y) y(dy)
=
ffi(ee_tx + e-'
=
fi(ex
+
1 - e-2ty + e-2t-2. w) y(dw)
1-
1 --e-2s z) y(dz) y(dy)
= Tt+af(x)
The symmetry of Tt on L2('y) is verified in a similar manner. Namely, the inner product
(Ttf, 9)L'(,) = f f f (e-tx +
1 - e-2ty) 9(x) y(dy) y(dx)
can be written, by the change of variables
u= etx +
1- a-2ty, v=- 1- e-2tx + e'ty
and the Gaussian property, as
JJf(u)g(etu -
V1--e--21v),-y(du) y(dv),
which is (f, Ttg)L2(,) by the symmetry of y. Now the equality Tt = T112Tt12 implies that Tt is nonnegative. For every bounded continuous function f, the mapping t '-. Tt f with values in LP(y) is continuous, which is readily seen from the Lebesgue theorem. If f E LP(-Y),
then there exists a sequence of bounded continuous functions fk convergent to f in LP(-y). By virtue of what has already been proved, sup IITtf - TefkllLp(,) <- If - fkllLP(,) , 0, r>o
k - oo,
whence the strong continuity of our semigroup. The same reasoning applies to vector-valued mappings.
1.4.2. Corollary. For all f E LP(-t), one has
thmllTtf
- f fdy I
=0. LP(,)
1.4.
The Ornstein-Uhlenbeck semigroup
11
PROOF. For bounded continuous functions f this claim follows from the Lebesgue theorem. In the general case, it suffices to find a sequence of bounded continuous
functions fi convergent to f in LP(-y) and note that
suoIITtf-Ttf,IILP(,)
Jfdi -, Jfd -t
asi-00. Let us mention some additional useful properties of the Ornstein-Uhlenbeck semigroup.
1.4.3. Lemma. For all f, g E Ll(ry), one has
Ttf
[Tt(fg)]2
y-a.e.
PROOF. The first inequality is obvious; the second one follows from the Cauchy
inequality applied to the integral representation of Tt(fg). The next theorem links the Ornstein-Uhlenbeck semigroup and Hermite polynomials. As above, let - be the standard Gaussian measure on )Itn.
1.4.4. Theorem. For all t > 0 and f E
one has
W
Ttf = re-khIk(f) k=o
PROOF. Let us denote by St f the right hand side of the equality we want to prove. Clearly, St is a continuous linear operator on L2(-y ). Therefore, in order to get the equality Tt = St it suffices to show this equality for the Hermite polynomials Hk. By Fubini's theorem, the proof reduces to the one dimensional
case, where it is easily obtained by induction in k. Indeed, it is true for k = 0. Suppose it is true for k - 1. Since T, Hk is a polynontial of degree k, then the equality TtHk = cHk will follow from the orthogonality of TtHk to all polynomials
Ht, I < k, in the space L2(ryt). In order to verify this orthogonality, we use the symmetry of Tt, the inductive assumption, and the relationship Hk J. H1, which give
(TtHk,Ht)L2(,) = (Hk,TtH1)L2(,t) = 0. Comparing the coefficients at x , we conclude that c = e-kt.
The Ornstein-Uhlenbeck semigroup gives an integral representation of the solution of the Cauchy problem for the parabolic equation
att = Du - (x, Vu), 8t
u(0, x) = f(=).
We shall see below that the operator A - xV is the generator of the OrnsteinUhlenbeck semigroup.
Let L be the generator of (T1)t>o on L2(-y ). This means (see Appendix) that
Lf = Iimt-'(Ttf - f) for all f such that the limit exists in L2(-y ). By the
same symbol we denote the generator of (T1)t>o on the Hilbert space L2 (-y", E) of E-valued mappings, where E is any separable Hilbert space.
Chapter 1. Finite Dimensional Distributions
12
1.4.5. Proposition. The domain of definition of L is D(L) =
{f:
k2IIlk(f)Il(,
i
k=1
On this domain. one has
Lf =
(1.4.3)
k=1
The analogous statement holds true for the operator L on the space L2(1". E). where E is any separable Hilbert space.
PROOF. Suppose that f is in the domain of L. i.e.. the mapping t - Tt f to L2(ry") is differentiable at zero. Then
ItLf = dt e k`Ik(f)I
= -kIa(f) t=o
(,
IILf III 2(. ) < oo. In addition, we get (1.4.3). Conversely, let f be a function such that this series converges. By virtue of the estimate
whence Ek k2IIIk (f) II i
It-1(e-kt
- 1)I < k.
we get ast-.0
Ttft f +E 'c k=1
3C
klk(f)I2
1
= L2(.,,)
J
k=01
-0.
Hence the mapping t - Tt f is differentiable at zero. Operator L is called the Ornstein-Uhlenbeck operator.
1.5. Sobolev classes
Let S1 C IR" be an open set, p > 1, r E IN. Everywhere below the symbol C,(')- (S2) stands for the set of all infinitely differentiable functions with compact supports in Q. By CC (IR") we denote the space of all infinitely differentiable functions on IR" with bounded derivatives of all orders, and by S(IR") its subspace that con-
sists of all functions whose products with all polynomials are in C, (lR"). Recall is defined as the that the Sobolev class WP'(SI) (an alternative notation is space of all functions f E Lp(f2) whose generalized partial derivatives (the derivaBy definition, the tives in the sense of distributions) up to order r are in generalized partial derivative (or Sobolev derivative) with respect to the variable x, is a function Or, f which is integrable on S1 and for every p E CC (1) satisfies the following integration by parts formula:
f a=,v(x)f(x)dx = - J(x)Of(x)dx. 11
U
Denote by W(o' (IR") the set of all functions f on IR" such that (f E II'"*(IR")
for each ( E Co (IR"). It is known that every locally Lipschitzian function f belongs to It' °' (IR") for all p > 1. In addition, if io is Lipschitzian on IR1 and f E II ,P''1 (IR" ), then ;p o f E II' o" (R") and 0, (, o f) = iPt (f )0, f
1.5.
13
Sobolev classes
Every function f E It"" (1R") has a version fo with the following property: for every vector h, the function t r-+ fo(x + th) is locally absolutely continuous for almost every x E R" (see [297, Corollary of Theorem 5.5, Ch. 2, §5)). According to the Sobolev embedding theorem (see, e.g., [3, Theorem 5.4]), if j and m are nonnegative integers and mp > n. then every function in tt' +`(R") has a modification with j continuous derivatives, i.e., there is a natural embedding It10C+m(lR") C CJ(R"). Also. one has ltloc1"(R'n) C C3(1R"). For a detailed exposition of the theory of Sobolev spaces we refer the reader to the books [3], [536]. For the theory of Gaussian measures, it is natural to introduce analogous classes with a Gaussian weight. Let y" be the standard Gaussian measure
on R". Denote by fr the space of all 1-linear functions on IR" with the Hilbert-Schmidt norm n
2
I12
JILI171,
at.....r,=I
where {e,} is the standard basis in 111".
1.5.1. Definition. Let p > 1 and r E W. The Sobolev space
is the
completion of the space CC (IR") with respect to the Sobolev norm 1/p / Iifllp,r = { / If(x)I" In(dx) +
fI1D'f(x)II, 7n(dx)}
.
J
Here the derivatives Of are considered as mappings with values in the spaces %I.
Let f E By definition, there exists a sequence f, E Co (R") convergent to f in LP(yn) such that the sequences {D' f, }, I = 1, ... , r are fundamental in Put
X
Dl f := lim Of.. For the first derivative Df we also use the symbol Of. It is readily verified that this definition does not depend on our choice of an approximating sequence. Indeed, for the sake of simplicity, let us consider the case r = 1. Suppose that a sequence {w,} of smooth compactly supported functions G in LP(-y", 11"). Let us show that converges to zero in Lp(yn) such that G = 0 almost everywhere. To this end, it suffices to prove that (G = 0 almost everywhere for every ( E C' (R"). Thus, we may assume that the functions <', vanish outside some common hall. Letting p" be the density of y,,, for all e. k E R", we get by virtue of the integration by parts formula
fexp{i(k. x)] (G(x), e) pn(x) dx _ slim JexP[i(k.x)J (Vj(x), e)pn(x) dx
_ -slim i(k,e) f exp[i(k.x)] y,(x)pn(x)dx + lim x whence (G, e) = 0 almost everywhere.
f
exp [i(k, x)]
(x)(x, e) p"(x) dx = 0,
Chapter 1.
14
Finite Dimensional Distributions
1.5.2. Proposition. Let p > 1 and r E 1P1. The Sobolev class coincides with the class of all functions f E Woe (]R") such that f E Lp(ry") and I = 1.... r (the corresponding derivatives coincide as well). IID'fllx, E PROOF. Let f E WP,T (y"), C E Co (an), and let {f, } be an approximating sequence of smooth functions. Then the sequence {(f; } converges to (f in
W , (R"). It is easily seen that the derivatives in the sense of WT(y) can be taken for the derivatives in the sense of W.:(IR"). To prove the converse, let us choose a sequence (, E Co (IR") with the following properties: 0 < (j < 1, (j(x) = 1 if Ixi < j, sups sup= IID'Sj(x)IIij, < oo for all I < r.
Put fj = (j f. Clearly, fl E W p,r(y") n Wp,r(IR"). Then f, - fin LP(y") and the sequences {D'fj) converge in the spaces LP(%,?{r) to the derivatives of f in the sense of the class W (IR" ).
1.5.3. Corollary. Let f E and let gyp: IRI -+ IR1 be Lipschitzian o f) = ,p'(f. with constant C. Then V(f) E WP-' (-y.) and )D f . In particular, one has I f I E WP'' (y") and
II IfI
IIp.1 =
F1fllp.'.
PROOF. It suffices to refer to the corresponding results for the class Wa (IR") and apply the previous proposition.
1.5.4. Proposition. The Sobolev class tions f E L2(y") such that
r E IN, consists of all func-
1 k'IIIk(f)IIL.(,n)
In addition, the linear space generated by all Hermite polynomials is dense in the Hilbert space W2,' (-'
).
PROOF. The convergence of the series in (1.5.1) implies the mean-square con-
vergence of the series E Ik(f) together with the derivatives up to order r. This is k=O
readily verified with the help of the identity H,(t) = /H,,t_,(t). It is not hard to prove that Hermite polynomials can be approximated by smooth compactly supported functions in the norm II II2,r Therefore, (1.5.1) implies the membership in the class W'2.r(y").
Note that any smooth compactly supported function f is approximated by linear combinations of Hermite polynomials in the norm of the space W'2.,(-r").
x
This follows from the mean-square convergence of the series E Ik (f) together k=o
with the derivatives up to order r, which can be seen from the explicit estimate for Ik(f) which is obtained by applying repeatedly the following integration by parts
1.5.
Sobolev classes
15
formula:
f k,+If Hk,(x,)All, ...,xn) r,(dx) j=1
=
f
a:,Hk,+1(x1) HHk,(x,)f(x1,-.. 3#_
_ - f Hk.+1(x,) fHk,(xi)0z,f(x1,... ,xn)yn(dx) #i
+f xiHk,+1(xi) IT Hk,(x))f(x1,... ,xn)y.(dx). j#i can be approximated by linear combinaTherefore, every function from tions of Hermite polynomials in the norm of this space. Since Hermite polynomials the convergence of the series in (1.5.1) for are mutually orthogonal in f E Wl-'(N) follows by evaluating the corresponding norms of Hermite polynomials.
1.5.5. Proposition.
(i) For all g E W2'1(%), f E W2.2(yn) one has
J (Vg(x), Vf(x))
dx) = - f g(x)Lf(x)yn(dx).
(1.5.2)
(ii) The equality D(L) = W2.2(yn) holds true. In addition, for all f E W2,z(yn) one has (1.5.3)
Lf(x) _ Af(x) - (x,Vf(x)).
PROOF. The equality D(L) = W2'2(-y) follows from Proposition 1.4.5 and Proposition 1.5.4. Note that by virtue of the integration by parts formula, equality (1.5.2) is equivalent to (1.5.3). Either of these equalities is readily verified for any Hermite polynomials and extends to the functions from the corresponding Sobolev classes by a standard limiting procedure. Indeed, if f E W2.2(yn), then the polynomials f,,,: _ E Ik(f) converge to f in W2.2(yn) according to Proposition 1.5.4. k=o
It follows by Proposition 1.4.5 that L f, -+ L f in L2(yn). Since V fn, V f in L2(yn, IR") (by the convergence f,,, - f in the Sobolev norm), we get (1.5.2). Certainly, (1.5.3) can be also verified directly for smooth f by differentiating Tt f, then integrating by parts in the term
f (Vf(e-tx+
71-e-2ty),e-2t(1_e-2t)
I/2y)yn(dy)
and letting t -+ 0 after this integration. Recall that the Ornstein-Uhlenbeck semigroup acts naturally on vector-valued functions. This circumstance is used in the following useful commutation identity, which is straightforward.
1.5.6. Proposition. For any function f C- W2.1 hn), one has
VTtf = e-`Tt(V f )
Chapter 1. Finite Dimensional Distributions
16
E) of mapFinally, note that in a similar manner the Sobolev classes ping with values in a d-dimensional Euclidean space E are defined. Identifying E with Rd, the mappings of such classes can be written as F = (F1..... Fd), F, E 1V'''(-y.). In Chapter 5 we consider Sobolev classes over Gaussian measures on infinite dimensional spaces. The reader will find there a characterization of the spaces WP''(-y,,) by means of certain integral representations. 1.6. Hypercontractivity
In this and the next section we consider several useful differential inequalities connected with Gaussian measures. A common feature of all these estimates is that the Ornstein-Uhlenbeck semigroup is involved either in the formulations of the results or in their proofs (or both). The following result known as the logarithmic Sobolev inequality has numerous applications in analysis and stochastics. In its formulation we employ the convention f (x)2 log f (x) = 0 if f (x) = 0.
1.6.1. Theorem. For every function f E W2't(ryn), one has
f f2 log IfId7n <- f (Vf.Vf)d7n+2 f f2d7n log\J f2d7.).
(1.6.1)
R"
R"
R"
R"
PROOF. Suppose first that f E Cb (IRn) and f > c > 0. Put :p = f2. Then V f = 1and the inequality to be proven is equivalent to the following one: pd-yn log
f ,alogIVId?'n R"
R"
J
Vdyn <
Note that Tt,p > c2. Since we have
TtvlogTtv -
J
J !Iv4o2dyn.
2
(1.6.2)
R"
r d?,, log / ypd7n,
t - xG.
R"
R"
the left side of (1.6.2) can be represented as
- f (ddt f Tt,GlogTtv d7.) dt, o
R"
which, by virtue of the semigroup property, can be written as
-
x
V (LT, v)
x( dt - f r \Rl
dtT,Vd-t.
dt.
Since the second term vanishes due to (1.4.2), we get
(LTtV) logTtPd.m I dt. Applying (1.5.2), we rewrite the same expression as
f f (VTp,O(logTt))d)dt=7,ipIVTtpl2d7n)dt. J(J
1.6.
Hypercontractivity
17
Put dy".
F(t) = J Using the identity VT, p =
and Lemma 1.4.3, which gives the estimate V
T,
we get e-21
f1 ,SLR" Tt
(Tt8=, X0)2 d - 7 5-
21
/ Tt 1 1(O1.'P)2) d7n '_11R"
- e -2t j I
JW
which yields (1.6.2) after integration in t over (0, oo).
Suppose now that f E W2.1 (-y,) and f > 0. Then the logarithmic Sobolev inequality follows by Fatou's theorem, since there is a sequence {yp,} C C b (IR") such that gyp, > 1/i and ip; f in and almost everywhere. In order to apply Fatou's theorem, note that t2 log t > -1 if t > O and that cp,(x)2 log w,(x) - 0 if +p,(x) --a 0. Finally, for an arbitrary function f E W2.1(y"), the logarithmic Sobolev inequality follows from the fact that If l E 1V2-1 (-f.) and I V If I I = I Vf I almost everywhere.
Note that the logarithmic Sobolev inequality can be written in terms of the Ornstein-Uhlenbeck operator L as
f f2loglfldy"<<-
f
In this form it is known to be equivalent (for any Markov semigroup) to the following hypercontractivity property.
1.6.2. Theorem. Let y be a centered Gaussian measure on R". Then the corresponding Ornstein-Uhlenbeck semigroup (Ti)t>o is hypercontractive, i.e., for
any p> 1,q> 1, one has IlTtfllq 5 Ilfllp
for aU t > 0 such that e2t > (q - 1)/(p - 1). PROOF. We may assume that 7 is the standard Gaussian measure. It suffices to prove this estimate for nonnegative functions f . Further, it will be enough to do this for smooth nonnegative functions f. Clearly, we may assume that f > c > 0, approximating nonnegative functions by functions satisfying this condition. Under the assumptions made, the function G: t IITt f IIQ(t), where q(t) = 1 + (p - 1)e2t,
turns out to be differentiable in t on [0, oo) (recall that Ttf > c). We shall show that G'(t) 5 0. Hence G(t) 5 G(0), whence the desired estimate. Note that the logarithmic Sobolev inequality applied to the function f'12 in place of f, and the integration by parts formula for the Ornstein-Uhlenbeck operator (see (1.5.2)) after
Chapter 1.
18
Finite Dimensional Distributions
dividing by r/2 give the estimate
fir log f dy., --1
J
f' d -v. (log f f r d7. )
R'
R'
R"
<
f fr-2IVfl2dyn = 2(rr
2
1)
f fr-1 Lf dyn.
(1.6.3)
R'
R^
Put F(t) = f(Tf)(h) dry,,. Then G(t) = F(t)l1eit) and we have R'
G'(t) = G(t) [-q'(t) log F(t) + 9(t)
F(t) 9(t)F(t),
Since q'(t) = 2q(t) - 2 > 0, it suffices to show that
q'(t) <0.
(1.6.4)
Noting that
F'(t) = f (Ttf)J(t) [q'(t)lortf +9(t) Tef
dyn,
we see that inequality (1.6.4) is equivalent to the inequality
-
F(t) log F(t) + f (Tef )4() log Tef ddy" +
R'
9 fit)
f (T'f
)°I:)-1
LTef dy,, < 0,
R'
which is exactly (1.6.3) for r = q(t) and Tt f replacing f, since q(t) = 2q(t) - 2.
1.6.3. Corollary. Let p > 2. Then the operator I,,: f F-+ I"(f) from L2(y) to L"(y) is continuous and III"(1)IIP <- (p-1)"I211f112.
(1.6.5)
In addition, for every p E (1, co), the operators I. are continuous on LP(y) and IIInllt(LPc7)? < (M - 1)"/'2,
(1.6.6)
where M = max(p, p(p - 1)-1).
PROOF. Let p = e21 + 1. Then, by virtue of the hypercontractivity, one has IITtfllp < IIf112,whence IITeI.(f)IIp < IIIn(f)l12 < 111112. Since
we get the estimate
III"(f)IIp < e"tllfI12 < e"tllfII, which is the first claim. The second claim for p > 2 follows from (1.6.5). If p < 2, then q = p(p - 1)-1 > 2. Hence, for every bounded measurable function f and any g E LQ(y), one has
f I"(f)gdy = f fln(g)dy <_ IIfllPIII"(g)IIQ whence lfln(f)IIP <- (q - l)"12IIfIIP.
<_ (9-1)"12llfllpllgll4, 13
Another important inequality - the Poincare inequality - can be derived from the logarithmic Sobolev inequality (or proved by a similar and even a shorter argument), but a direct proof is also available.
Several useful estimates
1.7.
19
1.6.4. Theorem. For every function f E
the following inequality
holds true:
f (f - f fdy")2dy" <_ f (Vf,Vf)dy" R"
R-
R^
PROOF. According to Proposition 1.5.4, it suffices to show this inequality for all
Hermite polynomials. This is straightforward taking into account the relationship
Hk'(t) = v Hk_l(t).
11
1.7. Several useful estimates
The Ornstein-Uhlenbeck semigroup can be also used to derive some exponential integrability results and estimate the tail distributions of Lipschitzian functions. However, we shall first show that interesting estimates of this sort can be obtained without any special analytic tools directly from the elementary properties of Gaussian distributions.
Let f : R' -. Rk be a locally Lipschitzian function. Then its derivative f' (in the usual sense) exists almost everywhere and f'(x) is a linear operator between
R" and Rk. 1.7.1. Theorem. Let t; and q be two independent Gaussian vectors in R" with one and the same centered distribution y such that the mapping f is y-integrable. Then for every convex function'P : Rk R' such that (Z j'{t;}(q)) is integrable, we have
E`l'(f(E) -
EW(2f'(e)(n)),
(1.7.1)
which is equivalent to the following inequality:
f
f (f (x) - R" f d-)
R"
y(dx)
<
f f e (2 f'(x)(y)) 7(dx)-t(dy)
(1.7.2)
R^R^
PROOF. Put l;(o) = l; sin 9 + q cos 0 for 0 E [0, 27r]. Then
t;'(B) := 8 (0) = l: cos0 - gsin0. By our assumption, f is Lipschitzian in 0, and we have with probability 1 that w/2 /2
fW - f(q) = f aof (E(o)) do = 0
J(f'((o))'(o))&. 0
By virtue of convexity of 4+, we get '/2
E4i(f(E) - f(n)) 5
n
f
>EW(2 (f`(E.(9)).C'(o)))
do.
0
Since C and q are independent, we have
Elk
f(n)) = f f 4'(f(x)
- f(y)) y(dx)y(dy)
Applying the convexity inequality to the integral in y, one gets
EW(f(C)
- f(n)) >- E41 (f (C) - Ef(q)).
(1.7.3)
Chapter 1. Finite Dimensional Distributions
20
By the Gaussian property, the random vector (t;(o), l;'(O)) has the same distribution as In particular, for all B we have
El((f'((o)).'(o))) =Rq'
2f'(C)(n)).
Hence we arrive at estimate (1.7.1). Inequality (1.7.2) follows from (1.7.1) by the change of variables formula. 1.7.2. Corollary. Suppose that the measure -y in Theorem 1.7.1 has a density. Let %P: Rk - IR' be a convex function and let f E 1j,'1. (IRn, Rk) be a mapping such that I'(2 f'(l;)(rl)) is integrable, where we denote by f' its derivative in the sense of Sobolev classes. Then estimates (1.7.1) and (1.7.2) hold true.
PROOF. The same proof goes through with the only difference that relation (1.7.3) requires a different justification. We may assume that -y is the standard Gaussian measure. Then it suffices to show that for a.e. (x, y) E R2n we have w/2
f(x) - f(y) = J (f'(xsino + ycosO),xcos9 - ysin9)dO. 0
To this end, it is enough to verify that the integrals of both sides multiplied by any smooth compactly supported function g on R2" coincide. Changing the order of integration and using the variables u = x sin B + y cos B, t, = x cos B - y sin B, we get on the right the expression 'r/2
f f (f'(u),v)g(usin g+vcos0,ucos8-vsin 0)du
dvd0.
a R7"
Clearly, this equals to the integral of (f (x) -f (y)]g(x, y) over IR2", since the equality is true for smooth f (due to (1.7.3)), hence also for f E I4'6(R". Rk), which follows directly from the definition.
1.7.3. Corollary. Suppose that in the situation of Theorem 1.7.1 k = 1. Then, for r > 1, one has )'E[a"
IE
f (S) - IEf
lE
2f,W(*1)Ir=A1r1
(f,
r]
+x where M, = we get
JItIrp(o, 1, t) dt. In particular, for the standard Gaussian measure 7n x r
[1(x) - J f dln 1'n(dx) < Mrt 2)r `
J IVf(T)Ir7'n(dx).
(1.7.4)
R^
Moreover, the integrability of f follows from the existence of the integrals on the right-hand sides of these inequalities.
PROOF. Suppose first that the function f is integrable and apply estimate (1.7.1) to W(x) = i'Ir. Then it remains to note that, for every linear functional 1, the random variable l(>)) is Gaussian with variance a,,(l)2.
1.7.
Several useful estimates
21
we may In order to show the integrability of the function f with I V f I E replace it by if I and assume that f is nonnegative. Let us take a sequence of functions wZ defined on IR' as follows: p (t) = t if Its < j. v,(t) = j sgn (t) if Itl > j. Put fj = pj(f ). Then for the functions f, estimate (1.7.4) is valid with the right-hand sides uniformly bounded in j, since IVf,I S IVII. By Fatou's theorem, we obtain
lim inf [f, - / fj dyn] < oc. Since f, -. f pointwise, one has lim inf J f, dyn < oc, whence, applying again Fatou's theorem, we get the integrability off.
1.7.4. Corollary. Let y,, be the standard Gaussian measure on R', k = I and
If (x) - f(y)I <- CIx - yl Then the function f is integrable and for all a <
the following inequality
holds true:
Jexp(oi - J If
z
f d-n
(i - ZA2C2 )
1/2 .
(1.7.5)
R' PROOF. Suppose first that fi dyn = 0. Let us apply inequality (1.7.2) to the function W(t) = exp(at2). Then the left side of (1.7.5) is majorized by the
/
f'(x)(y))z.
integral of expl a(2
For every fixed x, the function f'(x)(y) is a
Gaussian random variable with variance not larger than C2, since I f'(x)(y)I < CI yI. Therefore, the integral in y is majorized by the right side of (1.7.5). In the general
case, it remains to note that f is integrable according to the previous corollary, hence we can apply the case considered above to f - fi &t,,. El 1.7.5. Corollary. Let -y" be the standard Gaussian measure on 1Rn and let f be a locally Lipschitzian function. Then
I
expIf
R'
- f f d^fnj dyn
J exp( R'
R^
IVfI2) djt
.
(1.7.6)
PROOF. Let us apply (1.7.2) to 41(t) = et. Evaluating the integral on the righthand side of (1.7.2) in y (the corresponding integrand is the exponent of a centered Gaussian random variable with variance a21V f (x)I2/4) we get exp(rr21Vf(x)12/8), whence (1.7.6). The estimates above can be improved for numeric Lipschitzian functions. This improvement will involve an argument based on the Ornstein-Uhlenbeck semigroup
and similar in its nature to the one used in the proof of the logarithmic Sobolev estimate. In fact, it may be shown that analogues of the next exponential integrability estimate hold true for any hypercontractive (or, equivalently, satisfying the logarithmic Sobolev inequality) Markov semigroup.
1.7.6. Theorem. Let yn be the standard Gaussian measure on 1R" and let f be a Lipschitzian function with constant C. Then for all r > 0. one has
/
-t. (x:
II
If(x)-1 fdynl
r
>r) <2exp(-2C'=).
(1.7.7)
Chapter 1.
22
Finite Dimensional Distributions
If ffd i = 0. then for all A E 1R1 one has / A2Cz
f exp(Af)dy <exp[
2
(1.7.8)
PROOF. Clearly, it suffices to prove both estimates for smooth bounded functions. Let us establish first (1.7.8). Since A is arbitrary, the general case reduces to the case C = 1. The argument is similar to the one used in the proof of the logarithmic Sobolev inequality (see Theorem 1.6.1). Let (T1)t>0 be the Ornstein-Uhlenbeck semigroup with generator L. Using the semigroup property and formula (1.5.2) of
integration by parts for L, we get due to the relation lim T, f =
t--x
G(t) :=
r
J
exp(AT, f)
J f dy = 0 that
d f= 1 - JG'(s)ds
= 1+ A2
J I VT, f J 2 exp(ATsf) d Yt) ds.
t J\W
In addition, VT, f = e-'T, (V f ), whence I VT f I < e-', since I V f I < 1 by condition. Therefore, we arrive at the inequality
G(t) < 1+A2Je-2sG(s)ds. t
Denoting the logarithm of the right-hand side of this inequality by H(t) and differentiating in the lower limit, we get the estimate -Ate-2tG'(t) -Ate-2teH(t)
H'(t) =
cH(t)
1
/>
eH(t)
- -A 2e` 21
for all t > 0. Hence,
log G(0) < H(0) _ - J H'(s) ds <
A'
0
which is our claim. Estimate (1.7.8) applied to the function g = f - f f dy,,, and the Chebyshev inequality give the estimate
y ({x: exP(Ag(x)) > R})
exp(2A2).
Inserting A = r and R = e" into this inequality and taking into account an analogous inequality for -g, we arrive at (1.7.7). 0 Note that for Lipschitzian real f Theorem 1.7.6 improves Corollary 1.7.5. Now, following [571, we shall prove a generalization of estimate (1.7.8), which improves also Corollary 1.7.5. We shall start with the following well-known lemma.
1.7.
Several useful estimates
23
1.7.7. Lemma. Let f and g be two positive integrable functions on a measurable space (X,µ). Then r rlog r log g dlc - I f dµ { > J g d;L) .
JX
X
f
XX
\
X
(1.7.9)
provided that f log f and f log g are integrable. In addition. the equality is only possible if f = cg a.e. for some number c. PROOF. Suppose first that f and g have equal integrals. Note that the inequality log x < x-1 on (0, oc) implies the estimate f logg- f log f = f log(g/ f) g - f (take x = g/ f ). Integrating this estimate, we get the inequality
flog fdp> ffloggd,1.
Jx
x
Clearly, the equality is possible only if f log(g/f) = g - f a.e., which is equivalent to f = g a.e. In the general case, writing the latter inequality for the functions with equal integrals, we arrive at (1.7.9). f IIf II2 and 9II9Il j )
Note that the quantity f f log f dp is called the entropy of f.
1.7.8. Theorem. Let (X,µ) be a probability space and let F be a mapping defined on a linear subspace D(F) C L2(µ) and taking values in L2(µ) such that I
.1cp) = AI'(W) for all .\ > 0 and ;p E D(F). Suppose that
J X
eye' dµ -
r 1
dµ (log f eX
X
dµl < J1(w)2e '° dµ
/
(1.7.10)
X
for all ip E D(r) such that all the integrals above exist. Then, for every p E D(I')
with f p dµ = 0 and all a > 1. one has
(fexp(ar()2)diz)
(1.7.11)
PROOF. We may assume that the right-hand side in (1.7.11) is finite. Let us apply (1.7.9) to the functions f = e,' and g = exp(aF(cp)2 - 3). where Q = log Jexp(or()2). Then (1.7.9) yields
f ape' dµ -
fe'' dµ (log f e, dµ) > fe'[or( y;)2 - 3, dµ. (1.7.12) x X X Denoting the left-hand side of (1.7.12) by H(e"), we obtain from (1.7.12) and
X
(1.7.10) that
H(e'') > aH(e'') -,3f e`'dµ, X
Chapter 1.
24
Finite Dimensional Distributions
whence
H(e') <
3
1
re= dµ.
(1.7.13)
x t E (0, 1], and letting
Replacing ;,% by
3(t) = log f exp(atF(y)2) dp. x
we obtain from (1.7.13)
a(tz)
H{e`'} <
f e`+du.
(1.7.14)
x
Let us put
u(t) = 1 log
dµ.
x It is readily seen that H(e`=) = t2u'(t)e`u"}. Using this identity, we rewrite (1.7.14) as 1
U (t)
- 0-1
6(t2) t2
Note that the function l;r
3(t)
do)
\fx ((
t
= log
is nondecreasing on (0, 1], which follows easily by Holder's inequality. Hence u'(t) <
°31 for all t E (0,1]. The condition
J x
cpdu = 0 implies that lim u(t) = 0. Theree-0+
r
fore, u(1)
°
1
whence
Jx
0
e' dµ < e3l(0-l), which is (1.7.11).
1.7.9. Corollary. Let ry" be the standard Gaussian measure on IR" and let f be a function from 1i'2.1(y"). Then, for every a > 1/2, one has
fexp[i_J fdY"] df. < (fexp(aIVf2)dn)'"2' JV
IV
.
(1.7.15)
Rn
PROOF. It suffices to consider the case where f f dry" = 0. Let us take for D(F') the Sobolev class W2,1 (-r") and put
Applying the logarithmic Sobolev inequality to the function f = e"'I2, we get (1.7.10). Hence Theorem 1.7.8 applies.
Note that letting a = 1 we get
fexP[f -
f
1
f dry"] dry" <_ J exp(IVfI2) d- y,.
0
1.8.
Convexity inequalities
25
< C, then the right-hand side in (1.7.15) is majorized by the number exp(aC2/(2a - 1)). Letting a --+ oc we obtain (1.7.8). If IV f I
I.S. Convexity inequalities For any sets A, B C R" and any real numbers a, 3 put
aA+3B:={aa+3bIaEA,bEB}. If A = -A, then the set A is called symmetric. A set A C 1R" is said to be convex if Aa + (1 - A)b E A for all a. b E A and all A E [0, 1]. A convex symmetric set is called absolutely convex.
For every set A C 1R", there is the minimal convex set cony A containing A. This set is called the convex hull of A. The absolutely convex hull absconv A of the
set A is defined as the minimal absolutely convex set containing A. One readily verifies the following equalities:
convA={alai absconvA= {alai
la, EA,
1, kEIN}.
A convex set may not be Borel (e.g., one can add to an open disc a non Borel set on its boundary), however, it is always Lebesgue measurable.
1.8.1. Lemma. Let A be a convex subset of IR". Then either A is contained in a proper affine subspace or it has the nonempty interior. In the latter case, the closure of A coincides with the closure of its interior. In addition, the boundary 8A of the set A has Lebesgue measure zero and A is measurable. PROOF. We may assume that A contains the origin. If A contains n linearly independent vectors, then A has inner points, since it contains the simplex generated by these independent vectors together with the origin. Otherwise, A belongs to a proper linear subspace. If A contains an open ball U, then every point a E A belongs to the interior of the set cony (U U {a}), which is contained in A by virtue of convexity. Hence the closure of A coincides with the closure of the interior of A. Note that the boundary of any set is closed, hence is measurable. The last claim of the lemma is proven by induction in n. For n = 1 it is obvious. Let n > 1. If A has no interior, then the measure of its closure is zero according to what has been proven above. Suppose that A has inner points. Let H be the (n - 1)-dimensional hyperplane orthogonal to the vector el = (1.0.... ). By FLbini's theorem, it suffices to show that for almost every point t on the first coordinate axis, the set (t+fI)n8A has zero (n - 1)-dimensional measure. If t + R contains an inner point of the set A, then this follows from the inductive assumption, since in this case A n (t + II) is a convex set in t + 11 possessing the nonempty interior from the point of view
of t + H. It remains to note that there exist at most two points t for which the intersection defined above is nonempty, but has no interior. Indeed, according to the first claim of this lemma, such points are boundary points for the projection of the interior of A onto the coordinate line. However, the interior of A is a convex open set, hence its projection is a convex open set on the straight line, which may have at most two boundary points. 0
Chapter 1. Finite Dimensional Distributions
26
Let f and g be two Borel functions on R". Let us fix A E (0,1) and put y)ag(1
h(f,g)(x) = sup f (x
y)
i
yER' Note that the function h is measurable, since for any Borel function ,p on the plane, the set {x: sup W(x, y) > c} is the projection of the Borel set I (X, Y): w(x, y) > c}, Y
i.e., is a Souslin set (see Appendix). Denote by IIfIII the norm off in Lt(R").
1.8.2. Theorem. Let f and g be nonnegative Borel functions on R' such that 1If1I1 > 0, IIgI1I > 0. Then the following inequality holds true: (1.8.1)
Ilh(f,g)II, > Ilf11 I1gIIi-A.
PROOF. It suffices to prove (1.8.1) for bounded functions. Let us use induction
in n. Let n = 1. Since f = F sup If (, g = G sup Igl, where sup IFI = sup IGI = 1, estimate (1.8.1) reduces to the inequality IIh(F,G)IIh > IIFIIi IIGII;-A. By the concavity of logarithm, it suffices to show that IIh(F,G)111 > AIIFII1 + (1 - A)IIGII1.
(1.8.2)
Put A(t) = {F > t}, B(t) = {G > t}, C(t) = {h(F,G) > t}. Note that AA(t) + (1 - A)B(t) C C(t)
for t E (0, 1), and, in addition, A(t) and B(t) are nonempty (for t > 1 these sets are empty). Indeed, if F(a) > t, G(b) > t, then h(F, G)(Aa + (1 - A)b) > t, which is seen from the formula for h(F,G) (it suffices to substitute x = )ta + (1 - A)b, y = (1 - A)b into that expression). Therefore, Lebesgue measure m of the set C(t) is not less than Am(A(t)) + (1 - A)m(B(t)). This estimate is the simplest (one dimensional) case of the well-known Brunn-Minkowski inequality
m(M + (1 - A)B) > Am(A) + (1 - A)m(B), which is readily verified for finite unions of intervals, and then extends to arbitrary nonempty Borel sets (see 1120, Chapter 2)). Using Problem 1.10.11, we get (1.8.2).
Suppose that (1.8.1) is proven for n - 1. Put
x= (y, z), y E R', z E Rn-'. x x fo(z) = f f(y,z)dy,
go(z) =
f
g(y,z)dy,
It is not hard to prove that fo and go are Borel functions. Let us fix w E Rn-1 Then
f((y-s, -w)\a gf(s,w)\ i-a
h(f,g)(y,z)> Eup
Applying (1.8.1) to the functions t -+ f (t, z - w), t «-. g(t, w) of the real variable, we get
f h(f,g)(y,z)dy>_ -x
f f(y,z-w)dY) -x
(f g(y,w)dy) 'x
1.8.
Convexity inequalities
27
whence f h(f, g)(y, z) dy > h(j0, go) (z), since in the previous estimate one can take sup over w. Integrating in z, using the inductive assumption and Fubini's theorem, we arrive at (1.8.1).
Recall that the Brunn-Minkowski inequality states that m" (AA + (1
- A)B)1'" > Am,, (A)'/" + (1 - a)m"(B)1/",
for any Borel sets A, B in IR" with Lebesgue measure mn (see a proof in [120, Chapter 2], [237]). Theorem 1.8.2 implies this inequality in the case m"(A) = mn(B). Indeed, let us take f = IA and g = IB. Note that h(IA, IB) = IAA+(1-A)B,
since h(IA,IB) = 1 if and only if there is y E IR" such that y E (1 - ))B and x-y E AA. Hence we get from (1.8.1) that m"(AA+(1-A)B) > m"(A)"m"(B)'-'. We say that a nonnegative function o on 111" is log-concave (logarithmically concave) if
p(.1x+ (I
- A)y) ? p(x)Ag(y)1-'',
dz, y E IR", VA E [0,1].
Such a function can be written as p = exp(-V), where V: JR" - (-oo.+oo] is a convex function. The function V is finite and convex on the convex set dom(V) _ {V < oo} and is +oo outside dom(V).
1.8.3. Corollary. Suppose that p is a log-concave function on R". Let
po(u) = J
p(u, v) dv,
u E IRk.
IRn-k
Then go is log-concave on IRk.
PROOF. For every u1, u2 E IRk, A E (0,1), and v, w E IR"-k, we have
p(Aul +0 - A)u2, w) > p(nt.
u
.
v)'\p(u2,
1
v A
Hence, letting pl(w) = p(ul,w), p2 (w) = p(u2,w), we get p(Aul + (1 - a)u2, w) > h(pl, p2)(w).
0
It remains to apply Theorem 1.8.2. The next theorem gives the logarithmic concavity for Gaussian measures.
1.8.4. Theorem. Let µ be a probability measure on IR" with a log-concave density p. Then, for any Borel sets A and B and every A E [0, 11, one has
µ(,\A + (1 - A)B) ?
(1.8.3)
In particular, this holds true for the standard Gaussian measure.
PROOF. Put f = QIA, 9 = QIB, and C = AA + (1 - A) B. By condition, we have p((x - y)/A)'p(y/(I -'\))"
<_ p(x).
Chapter 1. Finite Dimensional Distributions
28
Since at y = (1 - A)x both sides of this inequality coincide. one has h(g, p) = P. Note that if A-1(x - y) E A and (1 - A)-1y E B, then x E C. Therefore, h(f, g) < Ich(t?, e) = Ice. It remains to apply Theorem 1.8.2. Probability measures on lR' which satisfy (1.8.3) are called log-concave: in fact. any such measure is given by a log-concave density on suitable affine subspace. The following important result is called the Anderson inequality for Gaussian measures.
1.8.5. Theorem. Let y be a centered Gaussian measure on 1R". Then. for any symmetric convex set A and any vector a, the following inequality holds true: y(A + a) < -y (A).
(1.8.4)
More generally, for every t E [0, 1], one has y(A + a) < -y (A + ta).
(1.8.5)
In particular, the function t - y(A+ ta) is nonincreasing on [0, +0C). PROOF. According to Problem 1.10.16 (or Lemma 1.8.1), every convex set in IR" is measurable with respect to every Gaussian measure. Therefore, it suffices to consider Borel sets. Moreover, it suffices to prove the claim for the standard Gaussian measure 'y". Let us put A = (t + 1)/2. Then A E [0,1]. Applying (1.8.3) to the sets A +a and A - a having equal y"-measures, we get (1.8.5), since A + to = A(A + a) + (1 - A)(A - a) by the equality 2A - 1 = t. Letting t = 0. we arrive at (1.8.4). 0
1.8.6. Corollary. Let y be a centered Gaussian measure on 111" and let f be a function on IR" such that the sets (f < c}. c E R. are convex and symmetric (e.g.. let f be convex and f (x) = f (-x)). Then, for any a E 1R". one has
f f (x)'y(dx) <
f(x + a) y(dx),
IR'
provided both integrals exist. More generally, the function
t-J
f (x + la) y(dx)
'R"
is nondecreasing on [0, 1], provided f ( + ta) E L1(y) for all t > 0.
PROOF. Note that -y(( f < t}) > y({ f < t} - a) for all t. since { f < t} is a convex symmetric set. Hence,
y(x: f(x) > t) < y(x: f(x+a) > t which yields the desired estimate if f is nonnegative (see Problem 1.10.11). If f > c, where c is a constant, then the claim follows by passing to the nonnegative function
f + Ic]. Finally, in the general case, it suffices to note that the functions ft = max(f, -k) satisfy the conditions of this corollary, are bounded from below, and their integrals with respect to the measures -y and y,, converge to the corresponding integrals of f. The second (more general) claim is derived in a similar manner from inequality (1.8.5).
I.S.
29
Convexity inequalities
1.8.7. Example. Let y be a centered Gaussian measure on R". Then
j Ixl a ry(dx) < r Ix + alP ry(dx), IV
Va E lR", p > I.
JR'
1.8.8. Lemma. Let C,.... , {" be random variables defined on a probability space (0, P) such that the distribution of the vector (l;l,... , t;") is a centered Gaussian measure with the covariance matrix A. Suppose that
Xo = 0. Then the
E X,_ 1, where X; is the linear span of following estimate holds true:
/
Pw:
1
i
):5
( fP(x,0,d2)dx -1
(1.8.6) JJJ
PROOF. Denote by vk the distribution of the vector (6, ... , tk) in IRk. The probability on the left in (1.8.6) equals vn(Vn), where V is the cube [-1, 11n. in Let us prove the claim by induction in n. For n = 1 this claim is obvious. Let n > I and assume that (1.8.6) is proved for all smaller n. Put
R.
n-I t=1
where en 1 X,,_I. Then Con is a centered Gaussian random variable with variance > d2. In addition, CO and are independent. By Fubini's theorem, n
Pmi
1)=P(lcmax11v
j
<1)I/n_1(dx)
The integral on the right is majorized by 1)v"_I(Vn_1), since, by virtue of Anderson's inequality (the one dimensional case), we have P(1C°
- rl < 1) < P(IC01 < 1)
for every r. It remains to note that 1
P(Ianl 5 1) < rp(t,0,d2)dt
0
and apply the inductive assumption.
1.8.9. Theorem. Let µ and v be two centered Gaussian measures on R". Then the following conditions are equivalent: (i)
f(yx)2j.s(dz) ? j(y,x)2v(dx),
b'yE
IRn.
(ii) there exists a centered Gaussian measure o such that µ = v * o; (iii) for every convex symmetric set A one has µ(A) < v(A).
Chapter 1.
30
Finite Dimensional Distributions
PROOF. If (i) is fulfilled, then the quadratic form
Q(y) = f (y, x)2 p(dx) - f (y, x)2 v(dx)
is nonnegative. Let a be the Gaussian measure with the Fourier transform exp(-Q/2). Then j = L ;-u, hence p = v * a. Suppose that p = v * or. Then, for every convex symmetric set A, one has
p(A) =
J IR'
v(A - x) a(dx) < v(A),
since v(A - x) < v(A) for each x, by virtue of the Anderson inequality. Finally, if (iii) is satisfied, then for every fixed y one has
p(x: (y,x)2
0. Therefore, (i) is satisfied (see Problem 1.10.11).
1.8.10. Corollary. Let C be an absolutely convex set in IR", -y,, the standard Gaussian measure on Ifi." an let T be a linear operator on IR" with the operator norm JITILc(R_) < 1. Then
yn(T(C)) < y"(C). PROOF. It suffices to consider the case where the operator T is invertible. In
this case let us put p(A) = v (T(A)), v = y,,. Since T = UITI, where U is an orthogonal operator and IT[ is symmetric, we may assume further that T is symmetric. Then we get
(y,y) = (7T-'y, TT-'y) 5 (T-'y,T-ly) = f (y,x)2p(dx) IR'
Hence we can apply Theorem 1.8.9.
1.8.11. Corollary. Let C and y" be the same as in Corollary 1.8.10. Then, for any linear subspace L C IR", one has
y"(C) S yn((CnL)+L-).
(1.8.7)
PROOF. Denote by v the image of y" under the orthogonal projection onto L. Then the right-hand side of (1.8.7) coincides with v(C), since v(C) = v(CnL) and (C n L) + Ll is the preimage of C n L under the orthogonal projection onto L. Therefore, the previous corollary applies.
1.9. Characterizations of Gaussian measures
There is an extensive literature on the characterizations of the normal distribution (see (1141, [389], [502]). We only mention a few results of this sort (see also Section 1.10). The first of them was obtained by B. V. Gnedenko [292] as a generalization of a theorem of S. Bernstein (62] proved under the additional assumption about the existence of the finite second moment of . For its proof and further generalizations, see [114], [389], [502], [532], [591].
1.9.1. Theorem. A random vector ty in iR" is Gaussian if and only if for every random vector?) that is independent of and has the same distribution, the random vectors C - rl and C + q are independent.
1.9.
Characterizations of Gaussian measures
31
The next result gives the converse to Lemma 1.2.5.
1.9.2. Theorem. A random vector in IR" is centered Gaussian if and only if for every pair ({1, W of independent copies C and every real number "p, the random vectors bl sin' + C2 cos V, 6 cos - C2 sin V are independent copies of C. This useful result proved by M. Kac [385] is a special case of Theorem 1.9.6 below. In fact, it suffices that the condition above be satisfied for some cp 96 rrk/2,
k E IN (see Corollary 1.9.8). Theorem 1.9.2 can be also deduced (see Problem 1.10.22) from the following characterization obtained by S. Kwapien, M. Pycia,
and W. Schachermayer [459], who proved a conjecture stated by S. Bobkov and C. Houdre. The proof given below is borrowed from [580], where a more general result has been proved.
1.9.3. Theorem. Let rl and t; be two independent random variables with a
I
common symmetric distribution such that
t)
<- P(I C I> t),
Vt>0.
(1.9.1)
Then these random variables are Gaussian.
PROOF. The major step of the proof is to show that
oo. Suppose this is done. Then IE ( 2) = by virtue of the symmetry and independence of and r). According to Problem 1.10.12, one has in fact the equality in (1.9.1) which means (by the symmetry of the distributions), that (t; + 77)/V"2 and t; have equal distributions. Let {t;,} be a sequence of independent random variables with the same distributions as C. By induction, one proves that the random variables 2-"/2 6n) have the distribution as well. According to the central limit theorem, t; is a Gaussian random variable. Let us show that Ef2 < oo. Put h(x) = cosx if Ixi < 7r, h(x) = -1 if ]xl > ir. For any random variable (, define the function f( by f((s) = Eh(s[;). By Lebesgue's theorem, this function is continuous. It is readily verified that 2
h(x + y) + h(x - y)
< h(x)h(y).
2
If ( and 9 are independent random variables with symmetric distributions, then integrating the inequality h(s(+ se) + h(s(- s9) < h(so)h(s9), 2
we get ff+e(s) < f((s)fe(s). Note that for some 6 > 0, the function ffl f is strictly positive on [0, 6], since lEh(0) = 1. The function h does not increase on [0, oc), hence (1.9.1) implies the estimate f(4+,,)1' 12(s) > ft(s). Therefore, A(s) <- fE
Let g(s) = ff(s)h/82, s E (0,6]. Rewriting the previous inequality as g(s)32 <
g( f) a7
we get g(s) <
g(;). Hence g(2> g(6)
Therefore, letting rk = 2-k/2b, we get IF,
Eh(rkf) > 9(6)"> k
1 - crk,
= e`, where c > 0.
Chapter 1.
32
Finite Dimensional Distributions
whence E(t;/7r)Ijr i< < rk 2Esin2(rkC/2) < c/2. By Fatou's theorem. we conclude that
0
oc.
Let us mention a classical result due to H. Cramer (1731, its proof can be found in several books, e.g.. in (114, Ch. 2].
1.9.4. Theorem. Let t; and q be two independent random variables such that C + q is normal. Then C and q are normal. In other words, if the convolution of two probability measures is Gaussian, then each of them is Gaussian as well. A useful characterization of the Gaussian property was found by G. Polya 1618]. We shall derive Polya's theorem from the preceding results. which were obtained, however, much later.
1.9.5. Theorem. Let £ and q be two independent random variables such that q, and (t;+q)/ / have equal distributions. Then t; and q are centered Gaussian.
PROOF. Let p be the distribution of t; and let v be symmetric with p about the origin. Observe that two independent random variables bl and ql with distribution p * v satisfy the initial condition and have a symmetric distribution. By Theorem 1.9.3, C1 is Gaussian. Applying then Theorem 1.9.4 we conclude that { is
Gaussian as well. An alternative proof is to show that £ has second moment and O apply the central limit theorem. The following characterization of Gaussian functions (not necessarily probability densities) is obtained by E. Carlen [1481.
1.9.6. Theorem. Let f E LP(IR2' ). where p E (I. x]. Suppose that f is a product function in the coordinates (x, y) as well as in the coordinates
x+y x-y V2_
V2_
i.e.. there exist functions W,. V2 and 1, 02 in LP(1R") such that
f(x,y) _ Wt(x)'P2(y) ='01
x + yIP2 x - y
a.c.
are Gaussian functions. i.e.. zp1(x) = exp[-K(x) + ile(x) +12(x) + c],
Then f ,'Pi . c02. 11'1. and
where c is a complex constant, K is a nonnegative quadratic form. and ii. 12 are real linear functions on )R' . and the functions 2. Vii. v''2 are of this form. PROOF. First suppose that each of the functions ;o, and w, is nonnegative. Note
that even in this case the result is not covered by the previous theorems if p > 1. Let (Pi)t>>o be the heat semigroup on LP(R2') defined by PP,p(x) = f f (y)pe (x - y) dy,
;,; E LP(lR2' ).
R="
where pt is the density of the centered Gaussian measure with covariance tI on IR2"
and t > 0. Denote by (Qt)t>o the heat semigroup on LP(IR"). It is easily verified and Qti1', are smooth and strictly positive and that, that the functions Ptf,
letting u = (x+y)/f, v = (x - y)/f, we have Ptf(x,y) = Q11PI(x)Q04'2(y) = Qt `i(u)Qt't2(v)
1. 10.
Complements and Problems
33
Therefore, ar, 8b4 log Pt f = 0. Using that 2d=, 05k = a4, auk - 0,, 0t.k + at4
au, a,.,
and that log Pt f (x, y) = log Q, s:'1(u) + log Q,tl,2(v), we see that log Q, tbt and log Qtik2 are second order polynomials. By the integrability condition, Qtrp, = exp[-K, + A; + c,],
where K; is a positive definite quadratic form, A, is linear and c, E 1R1. In particular, Pt f is integrable, which implies the integrability of f by virtue of Fubini's theorem. It remains to note that P, f -. f in L' (1R2") as t 0. To remove the assumption that f is nonnegative, let us apply the previous case to I P1 f I = 1QtV1 I IQ6p2I = IQt?GI [ M021. This gives Pi f = exp[i(t + K,], where St is
a smooth real function and Kt is a negative definite quadratic form. Now the same reasoning with the logarithmic differentiation as above applies. Letting t tend to zero, we get the claim. 0
1.9.7. Remark. It is clear from the proof of Theorem 1.9.6 that its conclusion remains valid if we consider the coordinates
(xsin8+ ycos8,xcos8 - ysin0) with 0 56 irk/2, k E N, in place of 0 = it/4.
1.9.8. Corollary. If the condition mentioned in Theorem 1.9.2 is satisfied for some p 54 Irk/2, k E N, then is Gaussian.
1.10. Complements and problems
Other characterizations of Gaussian measures Characterizations of the type given by Theorem 1.9.1 were investigated by G. Darmois [184], V. P. Skitovich (7051, and later by many other authors (see references in [114] and [5021). The next result is called the Darmois-Skitovich theorem. See 15021 for its proof. Observe that Theorem 1.9.1 and Theorem 1.9.2 follow from this result.
1.10.1. Theorem. Let £! .... ,
be independent random variables and let o;
and ,3=, i = 1,... , n. be nonzero real numbers such that the random variables E o, , 1=1
and E 8 , are independent. Then the F, 's are Gaussian. t=1
A wide class of Gaussian characterizations steams from the fact that certain classical inequalities (e.g., Young's inequality) have Gaussian functions as extremals. For example, as shown in [4851, if G(x, y) is a Gaussian kernel on lR"xlR", i.e., G(x, y) = exp[Q(x. y) + I (x, y)], where Q is a complex-valued quadratic form on n 2n and 1 is a complex-valued linear function on ]R2", then the norm of the integral operator between LD(lR") and LQ(IR"), generated by kernel G, is attained only on Gaussian functions, i.e., functions of the form cexp[B(x) + f (x)], where B is a quadratic form and f is linear (both complex-valued). For related results, see [46] and [148].
We shall give an interesting characterization of Gaussian measures via the Poincari inequality (found in [105] in the one dimensional case and generalized
Chapter 1. Finite Dimensional Distributions
34
considerably in [156] whose proof we reproduce). Let p be a probability measure on R" such that the norm is in L2(µ) and
I xixj p(dx) = b+j.
Put
U(p)=sup
ff2dp
1fIVfl2dp
,
fECl(R")nL2(p)
1.10.2. Proposition. One has U(p) = 1 if and only if p is the standard Gaussian measure.
PROOF. Note that U(p) > 1, since we can take f(x) = x1. Clearly, we have U(7") = 1 by the Poincare inequality. To prove the converse, put AX) = x,,+Ag(x),
where g E C"(R"j) and A E R'. Since U(p) = 1, one has J [xj + Ag(x))2 p(dx) <
J1e j + AV9(x) 12 p(dx),
which yields A2
Jg24+2A Jr xjgdp
A
r !!
Since this inequality holds for all A, it follows that
JxjgdP = J(evV9)d. Now let us take g(x) = cosy, x) and g(x) = sin(y, x), y E R". Put p(y) = A(y). Then the equality above shows that 8y,w(y) = -yjv(y). Since this is true for every
j = 1, ... , n, we get A(y) = exp(-1y12/2), whence the desired conclusion. It was shown by G. Hardy [327] that if a function f E V(1111) and its Fourier then each of them is a finite linear comtransform are both bination of the Hermite functions Hk(x) exp(-x2/2) and if m = 0, then both are proportional to exp(-x2/2).
Integro-differential inequalities related to Gaussian measures There is a lot of integro-differential inequalities similar in their nature and methods of proving to the logarithmic Sobolev inequality and Poincare inequality. We shall mention here some of them. The next interesting result was proved in [468], [470), and [350] by different methods. We shall give the short proof borrowed from [470].
1.10.3. Proposition. Let 'Yd be the standard Gaussian measure on Rd and Then, denoting by E the integrals with respect to -td, we have let f E (
Ef)2
"-t ki. )kE(IIDk k=O
f In 2
); J2e_tE(IIT*DfII) dt.
-1
.
0
1.10.
Complements and Problems
35
PROOF. Clearly, it suffices to prove this equality for f E Cb (lftd). Taking into account that Tt f (x) -. E f as t -. oo, we have E(Tt f )2 -. (JEf )2. Hence,
x + f dE(Ttf)2 = 1E (f2) +2 f E(Ttf LTtf) x
(]Ef )2 =
]E(f2)
0
0
x
_ E(f2) - 2f 1E(IDTtfI2) = E(f2) 1
x
-2 f,-2'E(jTDfj2). 0
0
Integrating by parts in t and using the identity dtE(ITtDfl2)
= 21E(TtDf,LTtDf) _ -2e-2tE(IITjD2fIIii2).
we arrive at the following representation:
(Ef )2 = IE(f2) -72e4tE(IID2uIIi2) IEIDfl2 +dt. 0
Iterating this integration by parts in t, we arrive at the desired representation. O The next result is due to [348], 13491.
1.10.4. Corollary. Suppose that f E W2.2n(yd). Then
(-lk'k+1
E(IIDkfil;t
2E'
< E(f2) - (Ef)2 <_
k=1
k=1
1.10.5. Corollary. Suppose that f E Then
f f d7d )
x
2
(
_
Rd
and IIIDkfIli2(,d,xk) -y 0.
k, ) k
IID"flli 2(,d.'1t.)
k=0
1.10.6. Proposition. Let yd be the standard Gaussian measure on Rd and let f E W1'2(yd) be positive a.e. Then, for every s E (0, 11, one has
f Ivfl2drdf (Lf)2dyd+s Rd
Rd
f
Ivfl2dids(1-s)f #dVd,
Rd
Rd
provided all the integrals exist. If f E W2.2 (7d) is positive a.e., then
f IVfl2d7d <- f (Lf)2dtid+s f f Ivfl2d.yd, Rd
Rd
Rd
provided the last integral exists.
PROOF. It suffices to prove both estimates assuming that f E C630 (IRd) and f > c > 0 for some constant c. Note that d
2L(IVfI2)
- (vf,v(Lf)) = Ivfl2+
02f
\8x,a ,.5=1
2.
(1.10.1)
Chapter 1. Finite Dimensional Distributions
36
The proof of this identity is left. as Problem 1.10.24. Hence,
IL(IvfI2) - (vf,v(Lf)) >- Ivfl2. If s E [0. 1), then we put yo(u) = (1 - s)-Iut-'. Ifs = 1. then we put ;p(u) = log u. Clearly, ;p'(u) = u-'. Applying the previous estimate to ,' o f replacing f and noting that V(,p o f) = f-'V f, we arrive at the estimate 1L(Ivf12) 2
- (vf,v(Lf)) ? IVfI2+ I (vf,v(Ivfl2)) - l2Ivf l;. 7
Integrating this estimate and using the integration by parts formula for L, we arrive at the desired estimates.
1.10.7. Remark. W. Beckner [47] proved the following generalized Poincare inequalities: let f E }i72.1 (td), where 'yd is the standard Gaussian measure on IRd,
and let 1 < p < 2, a-1 = vim. Then
f Ifl2dyd- /f Ivfl2d,d.
(1.10.2)
2/p
(1.10.3)
f If12dyd- I f If IDd7d)` <(2-p) f IVfI2d^,d.
The proof follows the same lines as the proof of the logarithmic Sobolev inequality and is left as Problem 1.10.25.
The following interesting inequality has been proved in [66] (as noted in [66], this inequality is equivalent to the isoperimetric inequality for Gaussian measures discussed in Chapter 4, i.e., either of the two inequalities implies the other one).
1.10.8. Proposition. Let yd be the standard Gaussian measure on IRd, let g be a smooth function on 1Rd with values in [0, 1], and let p be the standard Gaussian density on IR1. Then the following inequality holds true:
p(V1(lEg))-IEp(41-1(g)) <1EIVgl,
where lEi := f V, dyd.
In [148], among other related interesting results, it is proved that the equality in the logarithmic Sobolev inequality is possible only if f is proportional to for some a E 1R". Another result from [148) is related to Fisher's information. Let us recall that Fisher's information of a probability density p E U-71
(IR") is defined
as
1(A)
= f IVOWI2 dx. fft^
A(x)
where we put DA/A = 0 if g = 0. Fisher's information 1(e) of a random vector in IR" with the distribution density A is defined as I(A). The Blachman-Stam inequality states that for any pair of independent random vectors 1; and , in IR" and any A E (0.1), one has
Kurt; + 1 --A,)) < Al({) + (1 - A)I(r)),
1.10.
Complements and Problems
and equality holds just when
37
and rl are Gaussian with equal covariances. A
short proof of this result was given by E. Carlen 11481 (it follows easily from Theorem 1.9.6). For applications of differential inequalities to the central limit theorem, see [508]. The inequalities proved in this chapter will be extended to the infinite dimensional case in Chapter 4 and Chapter 5.
Problems 1.10.9. Let t: be a random variable such that IEe" = En" for every n E IN, where rr is a Gaussian random variable. Show that f is Gaussian as well. Hint: use Fatou's theorem to verify that Ee' C < x for every A. then notice that Ee'af = Ee`a". since both functions are holomorphic in A.
1.10.10. Let (X.µ) be a measurable space with p(X) < c. Show that p-measurable
function f is integrable with respect to p if and only if En µ(x: If (x)I > n) < ,c. In particular. f E nq>1 LP(p) if and only if µ(x: If(x)I > n) = o(n-k) for all k E N. 1.10.11. Let (X, p) be a space with a finite measure and let f be a p-integrable function. Show that
mflfI
f If(x)I i(dx) =1
> t)dt =
x J
p(Ifl > t)dt.
Hint: verify this equality for f with finitely many values and approximate general f by such functions (see (536. Theorem 1.2.3]): another proof is to write the left side as the integral of t against the distribution of If I and apply the integration by parts formula for functions of bounded variation (see 1214, Theorem 111.6.221).
1.10.12. Assume that 4 and n are two random variables such that Er;2 = Eq2 < x t) for all t > 0. Show that P(IFI > t) = P(IgI > t). and P(IC( > t) < 1.10.13. Show that if a sequence of centered Gaussian random variables (C,,) converges in probability to a random variable t;. then is a centered Gaussian random variable and lim E.
1.10.14. Prove that the linear space generated by the exponents of the linear functions is dense in L2(y). where i is the standard Gaussian measure on Ht". 1.10.15. Show that Hermite polynomials on the straight line can be defined by Hk(x)
)k f (x+iy)k'1(dy)
where y, is the standard Gaussian measure on IR'.
1.10.16. Show that every convex set in HI" is measurable with respect to every Gaussian measure.
1.10.17. Let y be a centered Gaussian measure on III" with the covariance operator K and let A be a linear operator on HI". Prove the following relationships: f (Ax. x) y(dx) = trace AK. a..
I (Ax.x)2
ItraceAK]2 + 2 trace J(AK)2J.
ta"
Assuming that A > 0 use the Chebyshev inequality to prove the estimates
y{ x: (Ax.x) > I} < traceAK.
Chapter 1. Finite Dimensional Distributions
38
?{s: I(Ax.x) - trace AKI >
1} :5 2 trace (AK )2.
1.10.18. In the situation of Problem 1.10.17, assuming that 2v'-K-.AV < I prove the equality
f ei(dx) = ldet(I - 2fiA%rK-)]
[det(I
- 2AK)]
1J2.
MI,
1.10.19. In the situation of Problem 1.10.17 prove that J (x, yi) ... (x, yak)'Y(dx) = G12k 1(0) (bi ,
.
.
, ysk ).
R'
where G(x) = exp[- (Kx.x)] . y, E R".
1.10.20. Let . be a standard Gaussian random variable. Evaluate IE(iAC2), A E R'. Hint: find E(3C) for 3 E (-x,1/2) and consider the analytic continuation. 1.10.21. Let U. be the unit ball of IR" and let ?" be the standard Gaussian measure
on R". Show that ),,(U") -» 0 as n -. x. 1.10.22. Deduce Theorem 1.9.2 from Theorem 1.9.3 and Theorem 1.9.4. using symmetrizations as in the proof of Theorem 1.9.5.
1.10.23. Show that the maximum of the integral H(f) _ -
r f (x) dx over J f (x) log r
all positive probability densities f on R' such that. J x f (x) dx = 0 and I x2 f (.r) dx = 1 is attained on the standard normal density. Hint: use Lemma 1.7.7 and consider the function g(x) = exp(a + bx + Cr2). 1.10.24. Verify identity (1.10.1).
1.10.25. Prove estimates (1.10.2) and (1.10.2). Hint: see [467J.
CHAPTER 2
Infinite Dimensional Gaussian Distributions My personal opinion is that the number of dimensions of space is a very, very subtle matter. Probably, space is actually dimensionless.
N. Lusin. From a letter to V. I. Vernadskii
2.1. Cylindrical sets The definitions of the elementary notions used below in connection with locally convex spaces can be found in Appendix. In fact, we only assume the acquaintance with the following notions: a locally convex space, a continuous linear mapping, a seminorm, a Banach space, a Hilbert space, a product of a family of real lines. Linear combinations of sets in linear spaces are defined in the same manner as in case of JR". The same meaning is given to the terms convex set, symmetric set, absolutely convex set, convex hull, and absolutely convex hull. In particular, for any set A in a linear space, its convex and absolutely convex hulls are defined by the equalities cony A = {alas +
+ akak,
ai E A, ai > 0, a1 +
+ak = 1, k E NJ,
(2.1.1)
absconvA = {atal
a, E A, a, E 1R1. all +
+ Iakl < 1, k E IN}. (2.1.2)
Recall that we call cylindrical sets (or cylinders) the sets in a locally convex space X with the dual X', which have the form
C = {x E X : (11(x).....111(5)) E Cp}.
1, E X',
(2.1.3)
where Co E B(IR") is called a base of C. Certainly, this representation is not unique (e.g., the space IR3 can be represented as the preimage under a projection of 1R.1 as well as of 1R2). The linear space
L = fl" 1Ker li has finite codimension k < n. If we choose in X a k-dimensional linear subspace Lo, algebraically complementary to L, then the set C is written as
C=C1+L. where C1 is some Borel subset of Lo.
Notation. (i) Denote by C(X) the a-field, generated by all cylindrical subsets
of X. In other words, E(X) is the minimal o-field, with respect to which all continuous linear functionals on X are measurable. 39
Chapter 2. Infinite Dimensional Distributions
40
(ii) Let F be some family of functions on a set X. Let us denote by E(X,F) the minimal a-field of subsets of 9, with respect to which all functionals f E F are measurable. In other words, E(X,. F) is generated by the sets of the form if < c},
f E F. c E R'. In particular, e(X) = E(X, X' ). (iii) The image of the measure µ on a measurable space (X.M_x) under a measurable mapping f : X Y to a measurable space (Y,.My) is denoted by p of
and defined by the equality
ltof-'(B) := µ(f-'(B)). BE My. If X is a locally convex space, Y = X and MN = E(X) or Mv = 8(X). then for any h E X the mapping x h-. x + h is measurable. The image of the measure It under this mapping is denoted by µi, and is called the shift of the measure to the vector h. By definition, l1h(A) = p(A - h).
It is clear that E(X) is contained in the Borel a-field of B(X), but may not coincide with it (e.g., this is the case for the product R` of the continuum of the real lines). However, for separable Frechet spaces (in particular, for separable Banach
spaces) the equality E(X) = B(X) holds true (see Appendix). For example. this equality is true for the countable product of the real lines ft'. For any measure (nonnegative countably additive function) p on E(X ). we denote by E(X), the Lebesgue completion of E(X) with respect to p. In other words, A E E(X),, precisely when there exist sets B1, B2 E E(X) such that B1 C A C B2, µ(B2\Bj) = 0.
2.1.1. Lemma. Sets of the form
{xER": (x1....,x")EB}.
BEB(IR"). HE IN.
generate B(R") = E(R" ). PROOF. This follows from Theorem A.3.7 in Appendix, since the coordinate functions are continuous and separate the points of the space R".
2.1.2. Lemma. A set E belongs to E(X) precisely when it has the form
E = {x E X : (13(x).....1"(x)....) E B). where 1, E X', B E B(I
(2.1.4)
).
PROOF. It is readily verified that the sets of the form (2.1.4) form a a-field. Since this a-field contains all cylindrical sets, then it contains E(X ). On the other hand, for any fixed countable collection {l, } C X', the family of all sets B E B(IR ) for which the set E given by formula (2.1.4) belongs to E(X), is a a-field as well.
Since this a-field contains all sets of the form {x E R": (x3.....x") E Co}, Co E B(R"), then, by virtue of Lemma 2.1.1, it coincides with B(1R ). 2.1.3. Lemma. Let C be a cylinder in a locally convex space X with a compact base. Then, for any continuous linear operator T: X Rd. the set T(C) is closed.
PROOF. Let C = P-' (K), where K is a compact set in R" and P: X - IR" is a continuous linear mapping. Without any loss of generality, we may assume that
R" = P(X). In this case there exists an n-dimensional linear subspace Y C X,
2.1.
Cylindrical sets
41
which is an algebraic complement to Ker P. The mapping P is an algebraic, hence also topological, isomorphism of the finite dimensional spaces Y and R". Therefore, the set Q = Y n P-' (K) is compact in X. In addition, C = Q + Ker P. It remains to note that T (Q) is compact, hence T (C) = T(Q) + T(Ker P) is a closed set. 0
Note that a finite dimensional projection of a cylindrical set may fail to be a Borel set (even for sets in the plane; see the section about Souslin sets in Appendix).
2.1.4. Lemma. Let p be a measure on E(X), where X is a locally convex space. Then, for any set A E E(X). its linear span. its convex hull, and its absolutely convex hull are measurable with respect to p (i.e., belong to E(X)},).
PROOF. According to formulas (2.1.1) and (2.1.2) and an analogous formula for the linear span. it suffices to show that for any collection of sets C, E E(X) and any Borel set A C IR' . the set A = ` '\1c1 + ... + AkCk. (A1.... , Ak) E A. ci c- Ci }
is measurable with respect r to p. By Lemma 2.1.2, there exist sets B, E 8(R') and
a countable family (f, } C X' such that C, = f -'(B,). where f = (fl): X 1R" . Therefore. it suffices to prove our claim for the measure v := po f-' on lfft." and the sets
B={albs+...+Akbk, (Al.....Ak)EA. biEB,}CR-. Indeed, if we find sets E1, E2 C B(IR C) such that EI C B C E2 and v(E2\EI) = 0, then
f-'(EI) C f-'(B) = A C f-'(E2) and p (f -' (E2 )\ f -' (EI)) = 0. For the measure v, the claim follows from Theorem A.3.15 in Appendix. since B,x xBkxA is a Borel set in the corresponding product space, and the mapping (x1, ... , xk, A) - A1x1 + + Akxk is continuous. O
2.1.5. Lemma. Let p be a measure on E(X ), where X is a locally convex space. Then. for any set A E E(X ),, and any e > 0, there exists a set E of the form (2.1.4) such that B is compact in IR.", E C A and p(A\E) < E. In addition. there exists a cylinder C, with a compact base such that p(A A C,) < e. PROOF. By definition. there exists a set S E E(X) with S C A and µ(A\S) = 0. According to (2.1.4), the set S has the form
S = f-'(So). So E B(IR") f = (f,): X - R", ff E X. By Theorem A.3.11 in Appendix applied to the measure pof - ' on lR", there exists
a compact set B C So such that po f -' (So\B) < e. Now one can put E = f -'(B).
In a similar manner, there exists a cylinder D such that p(A i D) < e/2. Let D = P-' (Bo), where P: X 1R" is a continuous linear operator and Bo E B(IR"). Since v := poP-' is a Borel measure on IR", then there exists a compact set K C Bo,
for which v(Bo\K) < r/2. Letting Cf = P-'(K), we get p(A v C.) < e.
0
2.1.6. Lemma. Let p be a measure on E(X), where X is a locally convex Then, for any convex p-measurable set A and any e > 0. there exists a convex cylindrical set C. with a compact base such that p(A L CC) < E. If A is space.
absolutely convex, then C, can be found with the same property.
Chapter 2. Infinite Dimensional Distributions
42
Pttoot:. By definition, there exists a set B E E(X) such that B C A and µ(B) = li(A). By Lemma 2.1.5, we may assume that B has the form B = U', B,,, where B. = f -' (K ), the are compact in 1R" and f : X R°` has the form f = (f,), f, E X. In addition, there exists a cylindrical set AL containing A such that µ(A,) - µ(A) < e. Let AE have the form AE = P-'(C0),
Co E B(IRd),
where P: X
Rd is a continuous linear mapping. By the convexity of A, we have cony B C A. According to Lemma 2.1.3, the sets are closed in W. Hence the convex hull of their union is a Borel set. This means that
E := P(conv B) E B(IRd).
Note that E is convex and that B C P-1(E) C A,,, since E C P(A) C P(A,,).
P_' (E) up to 6 with respect to the measure p. It remains to note that there exists a compact set K in E with It o P-1(E\K) < e. Since cony K is convex and compact in E (for E is a convex set in Rd), the cylinder C, := P-' (conv K) approximates A up to 2e. The
Thus, we have approximated A by the convex cylindrical set
reasoning is similar for absolutely convex sets.
2.2. Basic definitions
2.2.1. Definition. (i) Let E be a linear space and let F be some linear space of linear functions on E, separating the points in E. A probability measure. y on E(E, F) is called Gaussian if, for any f E F, the measure y o f -' is Gaussian. (ii) Let X be a locally convex space. A probability measure y defined on the a-field E(X), generated by X', is called Gaussian if, for any f E X', the induced measure yof _' on 1R' is Gaussian. The measure y is called centered (or symmetric) if all the measures -yo f ', f E X', are centered. (iii) A random vector is called Gaussian if it induces a Gaussian measure. Note that although in case (i) the space E is not supposed to be endowed with a topology, this case in fact reduces to (ii), since one can introduce on E the locally convex topology a(E, F), generated by the family of seminorms p f(x) = I f(x) 1, f E F, as explained in Appendix. In this case F turns out to be the dual to E with this topology, and we arrive at case (ii). Some of the results discussed below do not use any topology on the space with measure, while a number of fundamental
results involve explicitly the topological structure. For this reason, for the sake of uniformity of exposition, everywhere below we consider Gaussian measures on locally convex spaces. However, nothing from the theory of locally convex spaces is used in this chapter, except for the notion of a continuous linear functional. Note that one need not require in the definition that, the measure y be probability: this is obvious for nonnegative measures and is true even for signed measures (see Problem 2.12.15).
It follows directly from the definition that the image of a Gaussian measure under a continuous affine mapping is a Gaussian measure (recall that an affine mapping is a linear mapping plus a constant vector).
2.2.2. Lemma. Let y be a Gaussian measure on locally convex space X and let T: X -. Y be a linear mapping to a locally convex space such that loT E X'
2.2.
Basic definitions
43
for any I E Y'. Then the measure yoT-' is Gaussian on Y. The same is true for the of fine mapping x'-+ Tx + a, where a E Y.
Recall that the Fourier transform (the characteristic functional) of a measure µ defined on the o,-field £(X) in a locally convex space X is given by the formula ii:
X. - C', W) = f exp(if(x)) li(dx) X
2.2.3. Example. For every measure p on £(X) and every h E X, we have Wh-(I) = eulh}u(l),
V1 E X'.
PROOF. It suffices to note that, by the change of variables formula (see Appendix), we have 11(x) Ph(dx) =
f f(x+h)µ(dx)
for every £(X)-measurable bounded function f. Some additional information about the Fourier transforms is presented in Ap-
pendix. Here we only need the fact that any two measures on £(X) with equal Fourier transforms coincide.
2.2.4. Theorem. A measure y on a locally convex space X is Gaussian if and only if its Fourier transform has the form
7(f) =exP(iL(f)
- IB(f,f))1
(2.2.1)
where L is a linear function on X* and B is a symmetric bilinear function on X' such that the quadratic form f +-- B(f, f) is nonnegative.
PROOF. Let -' be a Gaussian measure. It follows from the definition that f E L2(y) for any f E X'. Hence one can put
L(f) = Jf(x).v(dx).
B(f,g) = J[f(x) - L(f)] [g(x)
f E X*,
- L(9)] y(dx),
f, 9 E X.
It is clear that L is a linear function and B is a bilinear symmetric function. In addition, the quadratic form f '- B(f, f) is nonnegative. Now equality (2.2.1) follows by the elementary properties of the one dimensional Gaussian distributions.
Conversely, let y be a measure on £(X) with the Fourier transform given by (2.2.1). By the change of variables formula, we get
fexp(ivt)'of'(dt) = f exP(iyf (x)) y(dx) = exp [iyL(f) - 2y2B(f, f )] , whence it follows that -y is Gaussian.
2.2.5. Corollary. A Gaussian measure y on a locally convex space X is centered precisely when y(A) = y(-A) for any A E £(X). This is equivalent to the relation L = 0 in (2.2.1).
Chapter 2. Infinite Dimensional Distributions
44
PROOF. It suffices to note that the Fourier transform of the measure A .-. y(-A) is the function which is complex conjugate toy and that measures on E(X) with equal Fourier transforms coincide.
2.2.6. Corollary. The product 'y 0y2 of two Gaussian measures yl and y2 on locally convex spaces X1 and X2 is a Gaussian measure on X1 xX2. In the case where X 1 = X2 = X., the convolution yl * 72 of the measures - and y2; defined as X , (x. Y) i - x + y the image of the measure y1 ®y2 under the mapping X x X (see Appendix) is Gaussian as well.
2.2.7. Definition. Let X be a locally convex space and let u be a measure on
E(X) such that X' c L2(µ). The element a. in the algebraic dual (X')' to X'. defined by the formula
aµ(f) = f f (x) µ(dx),
(2.2.2)
x is called the mean of p.
The operator R,: X' -+ (X')', defined by the formula R,.(f)(9)
f [f (X)
- av(f)] [g(x) - aµ(9)],i(dx),
(2.2.3)
x
is called the covariance operator of p. the corresponding quadratic form on X' is called the covariance of t. Notation. Let y be a Gaussian measure on a locally convex space X. We denote by X,; the closure of the set
If - a. (f ), f E X' }, embedded into L2(y), with respect to the norm of L2(-y). The space obtained in this way and equipped with the inner product from L2(y) is called the reproducing kernel Hilbert space of the measure y. Put IhIH{,) = sup{l(h): I E X', R.,(1)(1) 5 1},
H(y) _ {h E X: IhIH(,) < oc}, U, = {h: lhlnc < 1}. The space H(y) is called the Cameron-Martin space. In the literature, it is also called the reproducing kernel Hilbert space and denoted by RKHS, but we use neither this term nor notation. The mapping R, is also defined on X;:
R,: X; -. (X' )', R,(f)(g): = f f(x)[9(x) - a-,(g)] 'y(dx),
f E X;. g E X'.
x Clearly, for any centered measure, this is merely the extension of R, from X' to X;. However, independently of whether the measure y is centered or not, for
each f E X', the functional R., (f) coincides with the functional R, (f - a, (f) ) generated by the element f - a,.(f) of the space X; (although the functional f itself may not belong to this space if a, 0 0).
2.2.
Basic definitions
45
In the case where the functional R,(f), where f E X.;, is generated by an element of the initial space X, this element is denoted by the same symbol as the functional. In this case
(R,(f),l) = (1,R,(f)), VIE X'. It is clear that in relation (2.2.1) we have B(f.f) = R, (f)(f) for all f E X. Put
a(f)
R (f)(f) = 1L(')2
fEX
If f E X, we put
a(f)
f (x)2 (dx).
Finally, the element R, (f) is also denoted by R, f .
2.2.8. Lemma. Every element g E X; is a centered Gaussian random variable with variance a(g)2 = II9II lV(,)
PROOF. The claim follows from Problem 1.10.13. but for the reader's convenience we include the proof. Let {fn } be a sequence of elements of X' for which the sequence {fn - a, (fn)) converges tog in L2(y). Then it converges to gin measure, whence
fex(itfn(z)
Jexp(itg(x)) ry(dx) = lim
-
7(dx)
x
X
= lim exP[-2t2a(fn)2J Therefore, there exists the limit d2 = lim This means that the measure yog is Gaussian with variance d2 = a(g)2 and zero mean.
nx
2.2.9. Remark. The examples of Gaussian measures discussed below show that the elements a, and R. (f) for f E X' are not always represented by vectors from the space X. However, it will be shown in the next chapter that this cannot happen for Radon Gaussian measures. In particular, such examples are not possible for Gaussian measures on separable Banach spaces.
2.2.10. Proposition. A probability measure 'y on the a-field E(X) in a locally convex space X is centered Gaussian precisely when for every real cp the image of
the measure y® on X x X under the mapping
XxX
(x,y)-xsinV+ycosW,
coincides with y. PROOF. The necessity of this condition is verified by the aid of formula (2.2.1). The proof of the sufficiency reduces to the one dimensional case. Indeed, the Fourier
transform of the image of the measure vow, where v = yo f-1, under the mapping
Chapter 2. Infinite Dimensional Distributions
46
of the form indicated above from 1R2 to 1R1 has the form /'
t
J
J exp [it(u sin p + v cos gyp)] v(du) v(dv)
]-t(dx)1'(dy) = f f exp[it(f(x)sinV +f(y)cos v)JexP[itf(z)]
=
-y(dz) = 'v(t).
0
Hence one can use Theorem 1.9.2.
2.2.11. Corollary. Assume that X is a locally convex space equipped with the o-field E(X). A measurable mapping t; with values in X is a centered Gaussian random vector in X if and only if for every pair (1i1,t;2) of independent copies and every real number gyp, the mappings f l sin V + 1;2 cos cp,
t; l coW- bsin ,p
are independent copies of l;.
There is a useful characterization of the Gaussian property by means of the triple convolutions. Let us introduce the following notation. Let X be a linear space. For any p E 1R1 put 1
2
a=3+3cos5o, Q
=
3
- 3 cos O - 7 sin p,
1
1
7 = 3 - 3 cos p + Note that a + Q + y = 1 and that the matrix
To=
f 1
sin cp.
fa Q 7 7ap Q 7 a
is orthogonal: it defines the rotation by angle cp around the axis with the direction
vector (1/f,1/f, 1103). This matrix generates the operator
T,.X: X3-.X3 (x,y,z) -T;,X(x,y,z) = (ax+$y+ryz,ryx+ay+Qz,OX +ryy+az). 2.2.12. Theorem. Let X be a locally convex space and let u be a Gaussian measure on X. Then the measure µ3 = µ®µ®µ is invariant with respect to T,,.x for any gyp. Conversely, if u is a probability measure
on E(X) such that the measure p3 is invariant with respect to T,,X for some W 2kir/3, then µ is a Gaussian measure.
2.2.
47
Basic definitions
PRooF. The first claim follows from the equality a + 0 + y = 1, which yields that the Fourier transform of the measure u'i o T-,1 coincides with the Fourier transform of the measure µ3. For the proof of the second claim let us fix f E X' and put v = ,uo f-t. Then
43 = v3oTR where the operator T ..,R;, on IR3 is given by the matrix T,. To simplify the subsequent calculations we put F = 1 and T :n;(x,y,z) =
Then the equality of the Fourier transforms of the measures v3 and v3oTR, yields the identity
F(x)F(y)F(z) = F(x,)F(yv)F(z,)-
(2.2.4)
Note that F does not vanish. Indeed, otherwise by the continuity of F there exists
a point a such that F(a) = 0 and F has not zeros on [-a, a]. However, equality (2.2.4) applied to the vector (a, 0, 0) yields
0 = F(a) = F(aa)F(8a)F(ya), which leads to a contradiction with our choice of a, since by condition q = sup(lal,181, Iyl) < 1. Let us pick 6 > 0 such that for all x from the closed interval I = [-6, 6] the values F(x) belong to a simply connected region U in the complex plane whose closure is compact and does not contain the origin. In this region U we can find a branch of logarithm and put G(x) = Log F(x), x E I. Note that F(x;o) E U for all x E I, since x,. E I. By virtue of (2.2.4) we have
C(x) + G(y) +G(z) = G(x,;) +G(y,) +G(z,,,),
ex, y, z E I.
(2.2.5)
This identity implies that G is infinitely differentiable. Indeed, integrating (2.2.5) in y over I we get 26G(x) + 26G(z) + f66 G(y) dy = f 6[G(ax+By+yz)+G(yx+ay+/3z)+G(,Qx+yy+az)]dy. It follows from the condition that the numbers a, Q, y are nonzero. Hence the first
integral on the right can be rewritten as 1
QX+36
ux-,36
G(s+yz)ds,
which is a function differentiable in x. Two other integrals admit analogous representations. Thus, the function G is differentiable. By induction one readily verifies its infinite differentiability. Differentiating three times the equality
G(x) = G(ax) + G(Qx) + C(yx), which follows from (2.2.5), we get G,,, (x) = a3Gm(ax)
+ ^`( '3X) +,Y3G..(yx)
Chapter 2. Infinite Dimensional Distributions
48
By virtue of the equality a2 + 32 + y2 = 1 one has the estimate
[al3+1313+[y13 <_q<1. Therefore, IG,,,(x)I <
sup
q -q I=
(G,,,(u)I.
u5gIxI
whence the equality G"' = 0 on 1. Therefore, on the closed interval 1. we have F(x) = exp(a + bx + cx2), where a, b. and c are some complex numbers. Since F(0) = 1, {F(x)I < 1 and F(x) = F(-x), we get F(x) = exp(imx - a2x2),
(2.2.6)
where m and a are real. Then this equality extends to an arbitrary interval [-b, b]. Indeed, having proved it for some closed interval Io = [-bo. be], we can extend it to a bigger interval [-bo, b1 ], since the image of the interval [0. bo] under the mapping defined in (2.2.6) has no self-intersections and belongs to a simply connected open set not containing the origin, which enables one to use the reasoning above. Clearly, this yields (2.2.6) for all x E [-bo, oo). In a similar manner we get (2.2.6) for all -6o]. Equality (2.2.6) shows that the measure v is Gaussian. xE 2.3.
Examples
2.3.1. Theorem. Let y be a Gaussian measure on a separable Hilbert space X and let X' be identified with X by means of the Riesz representation. Then there
exist a vector a E X and a symmetric nonnegative nuclear operator K such that the Fourier transform of the measure,) equals
x - exp(i(a,x) - 2(Kx,x)).
(2.3.1)
Conversely, for every pair (a.K) of the aforementioned type. the function (2.3.1) is the Fourier transform of a Gaussian measure on the space X. In addition, a is the mean of the measure y and K is its covariance operator. PROOF. Suppose that y is a Gaussian measure. Then its Fourier transform is given by equality (2.2.1), where x - L(x) is a linear function and x e -. B(x.x) is a quadratic form. By Lebesgue's theorem, the function 7 is continuous. This implies the continuity of both functions L and B. Therefore, there exist a vector a and a bounded symmetric nonnegative operator K such that
L(x) = (a.x).
B(x.x) = (Kx,x).
It is clear that the operator K is compact. Indeed, if x. - 0 in the weak topology, then `y(x,,) -- I by Lebesgue's theorem. Hence Ilv'-K-x,11 - 0. Now we have two possibilities to see that K is a nuclear operator. The first one is to show directly that
J(x.x).r(dx) < oc.
(2.3.2)
x This follows from Fernique's theorem about the exponential integrability (see Theorem 2.8.5 below). Then, assuming that a = 0 (which simplifies the formulas), for
2.3.
Examples
49
any orthonormal basis {en} in X, we get
E 1 (, , n=1
X)
n=1X
oo.
X
The second possibility is to use Theorem 1.1.4. Indeed, we may assume again that a = 0 (which is achieved by shifting the measure). Since the operator K is compact and symmetric, there is an orthonormal basis {en} consisting of the eigenvectors of K. Then the orthogonal centered Gaussian random variables (x, en) on (X, y) are independent and F,(x,en)2 = (x,x) < Oc n=1
for any x. By Theorem 1.1.4, this implies (2.3.2). It is readily seen that a is the mean of 'y and K is its covariance operator.
Conversely, let a be a vector in X and let K be a symmetric nonnegative nuclear operator on X. Denote by en the eigenvectors of K corresponding to the eigenvalues kn. Let {{n} be a sequence of independent standard Gaussian random variables. Note that the series
a'
f
knyn(w)eR n=1
converges in X for almost all w. Indeed, by the monotone convergence theorem, x x E knt;n(w)2 < oc a.s., since E kn < oo. Denote by y the distribution of the n=1
n=1
random vector t; in X. It is easy to see that y is a Gaussian measure and its Fourier transform is given by (2.3.1). Finally, note that the desired result (in both directions) could be deduced from the well-known Sazonov theorem [668] describing the Fourier transforms of measures on Hilbert spaces (see, e.g., [800]).
2.3.2. Corollary. If dim X = oc, then the function yo(x) = exp(cannot be the Fourier transform of any countably additive measure on X.
IIxII2) z
Note that if the measure -y is centered, then the operator K in representation (2.3.1) is determined by the identity
(Ku.v), = J(ux)x(vx)., y(dx).
(2.3.3)
X
Therefore, the closure of X = X' in L2(y) coincides with the completion of X with respect to the norm I IvrK-xllx Hence if {en } is the orthonormal basis in X formed by the eigenvectors of K, corresponding to the eigenvalues kn, then this completion can be identified with the weighted Hilbert space of sequences r 5 (xn): `
x
l
knx2 < or1. n=1
The operator K has the natural extension to this completion X., and one has H(y) = K(X.). However, if K is considered only on the initial space X, then H(y) = v'K-(X).
Chapter 2. Infinite Dimensional Distributions
50
For arbitrary a, in the natural coordinates associated with {e,,}, formula (2.3.1) reads as follows:
Ix
y(y) = expl i
k"yn2
a"1!n - 2 n=1
n=1
Let a = 0. Then for a cylindrical set C = P,,-'(B), where P. is the orthogonal projection onto the linear span H. of the first n vectors e, (identified with IR") and B E B(H, ), one has
?(C) =
J
exp(- Jx2) dx1 ...dyn. 1
9 J=1
2.3.3. Remark. Let y be a centered Gaussian measure on a separable Hilbert space X and let K be defined by (2.3.3). When we identify X' with X, this operator
coincides with R but in order to avoid confusion we do not denote K by R, (in addition, R, is defined on the larger space X,;). If X is infinite dimensional, then the domain of the operator K-112 has measure zero. However, for every v E X, one can define K-1"2v as the element I(v) E X; corresponding to the vector VKv E H(-Y).
This notation is consistent, i.e., if v = Ku, where u E X, then I(v) coincides as an element of X. with the continuous linear functional on X generated by u. Indeed, for any y E X, one has
I(v)(x)(y,x), y(dx) = (1kv,y)x =(Ku,y)x =
(u, x). (y, x),, -y (dx).
J x x Note also that (I(v),I(w))L3(.,) _ (vnv,v' OH(.) _ (v,w)x.
2.3.4. Example. Let 1 be a ball in 1R'. It is known (see [3, Theorem 5.4]) that the Sobolev class W2-"(S2) is embedded into Cs(S2) provided 2r > n. Hence the L2(Q) is a Hilbert-Schnlidt operator (see Lemma natural embedding A.2.13(ii) in Appendix or 13, Theorem 6.53]). Therefore, there is a Gaussian mea-
sure y on L2(S2) with the Cameron--Martin space W2-''(ft). If r > m + n/2, then C f'2'"(fl) is a ry is concentrated on W2-"(f ), since the embedding Hilbert-Schmidt operator (see [3, Theorem 6.53]).
2.3.5. Example. Let yn be the standard Gaussian measures on IR1. Then the measure
x
7=®7n n=1
is centered Gaussian on IR". In addition, X,; ?' !2, H(y) = 12. PRooF. Note that the measure y is defined on the a-field generated by the x,,, which coincides with the Borel a-field of the coordinate functions x = (xn) space IR" according to Lemma 2.1.1. It is known that every continuous linear functional f on 1R" is a finite linear combination of the coordinate functions (see Appendix). Therefore, for some n, the induced measure Yo f -1 coincides with the measure
(
-)) oFn 1, where FF(xl,....x,,) = f(x1.... ,x,,,0,0....). For any
=1 elements f = (fn) and g = (gn) from (1R")* one has R-r(f)(g) _
f J \n=1 ix) t
x
x 9nxn) 'Y(dx)
_ , fngn
2.3.
Examples
51
This means that the closure of (R')* in L2(y) can be identified with 12, since
it is the set of all functions of the form E cxn, where (ca) E 12 and the series n=1
converges in L2(y). It is clear that H(y) = 12.
As the results of the next chapter show, Example 2.3.5, quite special at the first glance, represents in fact, up to a linear measurable isomorphism, all centered Radon Gaussian measures.
2.3.6. Example. Let y be the countable product of centered Gaussian measures yn on Rl with variances on > 0. Then -y is a centered Gaussian measure on R". In addition, y is a centered Gaussian measure on every weighted Hilbert space
Ea = Ix E Rx :
l Ilxlla = F Qnxn < 00J,
anon < oc.
where n=1
n=1
PROOF. It suffices to note that y(E.) = 1, which follows from convergence of the series
faxr(dx) n=1
In addition, the coordinate functions
and the monotone convergence theorem. generate E(E0) = 8(Ea).
2.3.7. Example. The measure mentioned in Example 2.3.6 is concentrated on the space co of the sequences convergent to zero precisely when
Eexpl
< oc,
202
Ve > 0.
(2.3.4)
n=1
The convergence of the series in (2.3.4) for some e > 0 is equivalent to the equality y(l") = 1.
PROOF. Let series (2.3.4) converge for some e > 0. Then on -. 0. Therefore,
for any 6 > 0, there exists r > 0 such that n' I yn ((-r, r]) > 1 - 6. Indeed, according to (1.1.2), one has 2
2022
a'I)eXp(-2o2n ) 51-yn([-r,r]) 5 2 rnexp(
2rr(an 2n By the convergence of the series in (2.3.4), we readily get that x r2 an exP
lim r-.z r n=1
ton
n
). (2.3.5)
=0.
This means that y(l") = 1. Conversely, if y(l") = 1, then for some r > 0, one has nn 1 yn ((-r, r]) > 0. By estimate (2.3.5), this implies the convergence of the
series E (an/r) exp(-r2/(2on)), hence the convergence of the series in (2.3.4) for n=1
sufficiently large E. If the series in (2.3.4) converges for all e > 0, then one can construct a sequence of positive numbers en 0 such that
x exP n=1
En
ton
< 00.
Chapter 2. Infinite Dimensional Distributions
52
As it was proved above, the product v of the one dimensional Gaussian measures with variances on/£n is concentrated on 1'C. Since the measure y is the image of v under the linear mapping (xn) F-+ ( £nxn), which sends l" to co, we get y(ca) = I. If the series in (2.3.4) diverges for some e > 0, then a ball U of a sufficiently small radius in the space co has y-measure zero. Then y(U + h) = 0 for any vector h with finitely many nonzero coordinates, since the shift to such a vector transforms y into an equivalent measure. Since co can be covered by a countable union of such shifted balls, we get y(c0) = 0. Finally, note that both claims could be deduced from the Borel-Cantelli lemma (see 1697, p. 253, Ch. II, §10]), using the independence of the coordinate functions as random variables on (1R', y) (see Problem 2.12.27).
2.3.8. Example. Let pa, a E A, be a family of Gaussian measures on locally convex spaces X,,. Then the product-measure p = (8) p,, is a Gaussian measure aEA
on X = fi X.. The Cameron-Martin space of p is the Hilbert direct sum of the U
spaces H(pn), i.e..
H(p) = lh = (ha) E X : ha E H(pa). IhI
IhalH(,,.) < 00}.
f(v) _ a
the space X;, is the collection of all functions of the form
x
x fu.,
X
a(faj2 < 00,
an E A. fa E Xµ.a,.
n=1
n=1
and a, (f)=Ea ,(fa)forany f =(fe)EX'. It
PROOF. The first claim follows from Corollary 2.2.6 and the fact that the dual
to X is the direct sum of the spaces (Xe)', i.e. the general form of a continuous linear functional on X is f(x) _ E fa(xes), where x = (xa) and only finitely many fa E (Xa)' are nonzero. By the definition of a product-measure (see Appendix), this yields the announced description of X. Further, for any f of the foregoing form. one has R,.(f) (Ri,.(fa)), whence the claim about H(p). The expression for aµ is obvious.
The concept of a Gaussian measure is closely related to that of a Gaussian random process. The latter is defined as a family t = (t;t )tET of random variables such that all their finite linear combinations are Gaussian. Indeed, the measure induced by such a process on the path space 1RT with the topology of pointwise convergence is Gaussian.
2.3.9. Proposition. (i) Let be a Gaussian random process on a set T. This means that for all tl,... tn E T, the random vector has a Gaussian distribution. Then the measure pE induced by £ on the space of all functions 1RT with the topology of pointwise convergence is Gaussian.
(ii) Let T be a nonempty set and let X = IR". Then every function F of the form (2.2.1), i.e.,
f -.exp(iL(f) -
2R(f,f)).
2.3.
Examples
53
where L is a linear function on (IRT)' and B is a symmetric bilinear function
on (IRT), such that f r~ B(f, f) is nonnegative. is the Fourier transform of some Gaussian measure on X. (iii) Let y be a Gaussian measure on the space X = lRT with the topology of pointwise convergence. Then a., E X and R,(X,) C X. PROOF. The first claim follows directly from the fact that every continuous with an aplinear functional on IRT can be written as x -+ cj x(t 1) + ... + propriate choice of c, and t,. In order to get (ii). it suffices to apply Kolmogorov's theorem on the existence of a measure with the consistent finite dimensional projections (see Theorem A.3.21 in Appendix). The consistency condition required
in this theorem is satisfied. Indeed, let t1.....t" E T. It is easily seen that the function
y=
exp
ir,)
-
l=1
J=1
J=1
is the Fourier transform of a unique Gaussian measure PP,,_,. ,t on IR". Let zr be the natural projection of IR"t' to IR'. Then the Fourier transform of the o xn 1 is the function y i-. Pt,,.......1(y o it"), which coincides measure PP,,._. with since the functional yo ir, on 1R" is given by the inner product with the vector (yl.... , y-0). Thus, Pf,.....t..,, o rrn' = Pt,,.,, ,t The last claim is obvious from the description of the dual to IRT as the linear 0 span of be, t E T, where be(x) = x(t). The function B appearing in representation (2.2.1) for the measure y on Rr corresponding to the Gaussian random process { is uniquely determined by the covariance function K of the process which is defined by the formula
K(s,t) = cov
fir) = IE
lE"r),.
The covariance function is connected with the form B by the equality
K(s,t) =
(2.3.6)
where b,(x) = x(s). This is clear from the fact that linear combinations of such
functionals exhaust the dual to R". In order that the function K( . ) on T x T generate a nonnegative quadratic form B on (IRT)* by means of (2.3.6) it is necessary and sufficient that for all finite collections s i , ... , s". t I.... , t, of points in T, the matrix (K(s t,))" be nonnegative definite. Such functions on T xT are called nonnegative definite. The simplest example is K(t, t) = 1, K(s, t) = 0 if s 34 t; this corresponds to the product of one dimensional Gaussian measures. Now we shall see that the previous example is universal in the sense that every Gaussian measure can be regarded as a measure on an appropriate product of the real lines.
2.3.10. Example. Let y be a Gaussian measure on a locally convex space
X and let T = X. Define an embedding X -. 1Rr by the formula x '. x( ), x(t) := t(x), t c T. Then the image of the measure
under this embedding is a
Chapter 2.
54
Infinite Dimensional Distributions
Gaussian measure 7' on R° and
y'l y E RT: (y(tt),... ,Y(tn)....) E B)
=yIxEX:
t,EX`, BEB(R").
PROOF. Every continuous linear functional on RT has the following form: x c, E Rl, t; E T. The composition of such a functional clx(t1) + ... + with the embedding X -. RT is continuous on X. Hence this composition has Gaussian distribution with respect to the measure y'. The last claim is true by construction.
2.3.11. Example. Suppose that in the situation of Proposition 2.3.9(ii) we have
T=[0,11, L=O, B(6a, 66) = mina, 6), where ba(x) = x(a), and let B be extended by linearity to (IRT). (i.e., the linear combinations of the functionals 6Q). It is readily verified that the associated quadratic form is nonnegative. The corresponding measure P"* on the path space is called the Wiener measure. For this measure PR the following equality holds true:
J
1x(t) - x(s)]'P"'(dx) = 3It - s]2.
(2.3.7)
X
In addition, (Pw)'(CIO,1]) = 1 and the measure defined by the formula
E .-. Ptt (E0), E = Eo n C[0,1j, Eo E E(RT ), is Gaussian on C[0,1] (it is also called the Wiener measure and denoted by Pt{'). PROOF. It follows from the definition that the functional x -. x(t) - x(s) on (RT Put") is a centered Gaussian random variable with variance It - s], since
B(bt -
6,) = t +s - 2mint,s) = it - s].
This gives (2.3.7). According to Theorem A.3.22 in Appendix, the space C[0,1) has Plt.-measure. Hence the formula full outer
En C(0,1] -P"_(E), EE E(RT), defines a probability measure on the a-field of the sets of such a form. However, by Theorem A.3.7, this a-field coincides with the Borel a-field of CIO, fl. It is clear that. Pty' is a Gaussian measure on C[0,1] as well, since each element from C[0,1)' is the limit of a sequence of linear combinations of Dirac's functionals b in the *-weak topology (see Problem A.3.36 in Appendix). The process considered in the previous example is called a Wiener process or a Brownian motion. In a similar manner one defines a Wiener process on any closed
interval. For a Wiener process one can take the function wt(w) = W(t),
t E [0, 11,
on the probability space (fI, 8, P"), where S2 = C[0,1), B = B(C[O. 1)). The following are the characteristic properties of the Wiener process: 1) the increments wt, - we...... wt are independent for all to < t1 < ... < t,,-.
2.3.
Examples
55
2) wt - w, for s < t has Gaussian distribution with mean 0 and variance t - s; 3) the trajectories t - wt(w) are continuous for a.e. w.
2.3.12. Corollary. The Fourier transform of the measure P' on C[0,1] has the following representation:
PK (a) = exp (
\
2
J
J min(s, t) A(ds) A(dt))
,
11
o0
where the dual to C[0,1] is identified with the space of all Borel signed measures A on [0, 11.
PROOF. If A is a finite linear combination of Dirac's measures bt,, then the claim follows from the construction of Pu'. It remains to be noted that for every Borel measure A on [0,1] there is a sequence of measures A, that are finite linear combinations of Dirac's measures and converge weakly to A in the sense that f iO d,\, -+ f od A for every continuous function V) on [0,1] (see Problem A.3.36 in Appendix).
Note also that for any cylindrical set
l C= { x E C[0,1]: (x(tl),... ,x(tn)) E B y, B E B(R"), where 0 < t1 < t2 < ... < t,,, one has the equality P1v(C) = cn f
exp(-1 Z(ul, ... ,
s
1
dul ... dun, II
where
-
U1
(u2 - ul)2
(u,, - un-1)2
tl
t2 - ti
to - tn_1
C = [(2Tr)"tI (t2 - t1) ... (tn - to-1),
1/2 .
2.3.13. Remark. Another frequently used construction of the Wiener measure on C[0,1] is this. Let {fn} be a sequence of independent standard Gaussian random variables on a probability space (1, P) and an orthonormal basis in tJan(s)ds. the space L2[0,1]. Put en(t) = Then for a.e. w, the series 0
x x(t,w) :=
Et:n(w)en(t) n=1
converges uniformly in t E [0,1]. The measure P0 induced by this random series in C[0,1], i.e., the image of P under the mapping w i x( , w), coincides with Pu. We shall derive this fact from more general results in Chapter 3; however, direct proofs are available, see, e.g., [430]. Given the convergence of this series, the equality
Po = P'v is immediate. Indeed, it suffices to show that every continuous linear functional I on C[0,1] has equal distributions with respect to the two measures.
Chapter 2.
56
Infinite Dimensional Distributions
k
Moreover, it suffices to consider 1(x) _ E c,x(ti). Then I is centered Gaussian with respect to both measures. The variance of l with respect to P0 is
x
k
i=1
k
2
/
/
E cjcj>en(ti)en(t.i). i.j=1
n=1
n=1
This is exactly the variance of I with respect to Pw, since x x
en(ti)en(tJ) _ (I(o.l,],con)L2(0.1i (Iio.t, i, n)L=[0.1] /
n=1
n=1
=
l] = min(ti, ti )
(I(o.c,l
Let us evaluate the eigenvectors and eigenvalues of the operator Rwwith kernel
min(t, s) on L'[0, 1], which is the covariance operator of the Wiener measure P' considered on L2 [0, 1]. Since this is a symmetric compact operator, its eigenvectors
form an orthonormal basis in Lz[0, fl. Let fo min(t,s)fa(s)ds = Afa(t) a.e. Then :
r1
sfa(s)ds+tJ fa(s)ds=Afa(t)
a.e.
o
It is easy to deduce from this relationship that fA admits a twice continuously differentiable modification satisfying the differential equation
AA IM = -h(t), fa(0) = fa(1) = 0. Indeed, if A 0 0, then fa has an absolutely continuous version, whose derivative satisfies a.e. the equation
fp(s) ds = Afi(t) If A = 0, then by a similar reasoning we conclude that fa = 0 a.e. Therefore, we get a countable collection //
z
t fn(t)=fsin =, A, =7r-zfn-2/, nEIN.
In particular, this evaluation shows that the embedding to L2[0,1J of the space H(PH ), which coincides as already explained with /(L2(0, 1]), is not a nuclear operator (but, clearly, it is a Hilbert-Schmidt operator). It will be shown in the next chapter that the space H(P44') does not depend on whether P1S' is considered on C[0,1] or on L2[0,1], so that we describe the Cameron-Martin space of the Wiener measure on C[0, 1J as well.
2.3.14. Lemma. The space H(PH') coincides with the Sobolev class K p'1[0,11 of the functions f on [0, 1] such that f is absolutely continuous, f' E L 2[0' 1] and f (O) = 0. In addition, If I H(PW) = Ilf'II 0(o.1]
PROOF. Since H(PH') is the same for P'' on C[0,1] and L2[0,1], we shall deal with L2[0,1) in this proof. The function f belongs to a,(L2[0,1]) precisely when
2.3.
57
Examples
its coefficients cn in the basis {f} satisfy the condition E c2. (n-1/2)2 < oo. The =1
latter is equivalent to the existence of a function g E L2[0,1) such that
r
(9,VIn)2 = 7rCn 1 n -2
,
where an(t) = f cos(t/ an) (note that lip,,) is an orthonormal system). If such a function exists, then its primitive is f. Conversely, if f is absolutely continuous, f (0) = 0 and g = f E L2[0,1], then we get I
I
Tr(n-2)cn=- f f(t)II'n(t)dt= f 9(t),n(t)dt=(9,'0n)20
0
Since the system {tb, } is orthonormal, one has
x
(g,1JJn)2 < no. It is readily
n=1
verified that If 1IL2[o,1) To this end, it suffices to note that {1P.} is an orthonormal basis in L2[0, 1]. For another proof see Example 3.3.9. 0
The Wiener process in IR" is defined as wt = (wi , ... , wi ), where wk are independent real Wiener processes. The measure Pu' induced by this process on C([0,T],lR") (or on C([a,b],1R"), L2([a,b],JR"), etc.) is called the Wiener measure as well. The corresponding Cameron-Martin space coincides with the space W,1 ([0, T], ]Rn) of all absolutely continuous R"-valued functions h on [0, T] such that h(0) = 0 and Ih'[ E L2[0,1]. The Cameron-Martin norm is given by II h' II L- ()o.T).Qt") These facts can be either deduced from the one dimensional case or proved by a similar reasoning.
2.3.15. Example. (i) The fractional Brownian movement is defined as a centered Gaussian random process t;° on [0. T] (where T E (0, oo]) with the covariance function
K(s, t) = s° + t°
2
Is - tI°
a E (0, 2].
(ii) The stationary Ornstein-Uhlenbeck process t = (t;t)tET on a set T C 1R1 is defined as a centered Gaussian process with the covariance function
K(s, t) = e-It-al. The stationary Ornstein-Uhlenbeck process can be expressed by means of the Wiener process by virtue of the formula tt = e-twea,. For this process, every random variable ft is standard Gaussian. (iii) Let Q > 0 and a > 0. The Ornstein-Uhlenbeck process with the parameters $ and a and the initial normal distribution having mean mo and variance ao is defined as a Gaussian process with mean m(t) = e-3tmo and the covariance function
[am+ Ge3(t+(e23mmn8) - 1).
cov(r,a) = e -(t+a)3 2 -
2
If or = co = 1, mo = 0 and 0 = 1/2, we arrive at the stationary Ornstein-Uhlenbeck process. Below we shall find a representation of the Ornstein-Uhlenbeck by means of stochastic integrals.
Chapter 2. Infinite Dimensional Distributions
58
(iv) The fractional Ornstein-Uhlenbeck process is a centered Gaussian process with the covariance function
K(t, s) = exp(-fit - s1°), 0< o < 2.
(v) The Brownian bridge is the process w° := wt - twl on T = to, 1]. Its covariance function has the form min(s, t) - at. (vi) The Wiener field (or Brownian sheet) is a centered Gaussian process on T = [0,1)d with the covariance function K(s, t)
= fld min(s t;). i=l
The Wiener field introduced in [157] is also called the Chentsov-Wiener field
In the case where the parametric set T of a random process { = (et)trT on a complete probability space (0,.F, P) is endowed with a a-field T, one can talk of the measurability of the process in both variables. The process f is called measurable if the function (t, w) lt(w) is measurable with respect to the a-Held T®.F. The next result shows that any measurable Gaussian process with the trajectories from LP induces a Gaussian measure on L.
2.3.16. Example. Let (T, T, a) be a measurable space and let _ (tt)tET be a measurable Gaussian process on T. Suppose that
/ 1l (t, w)]" a(dt) < oo for a.e. w, where p > 1.
(2.3.8)
y(B) = P(w:
(2.3.9)
T
Then the formula
E B),
B E E(L"(a)),
defines a Gaussian measure on L'(a). In addition,
J;(t. .)Ipa(dt) E L'(P)
for all r E [1,00).
(2.3.10)
T
Finally, (2.3.8) is equivalent to the condition
f
[K, (t, t)p/2 + Il(t)ID] a(dt) < 0,
(2.3.11)
T
where Kt is the covariance function of the process t: and m is its mean.
PROOF. Let 1/q + I/p = 1. Recall that L9(a) = LP(a)*. Let us show that for any tb E L9(a), the function n(w) =
Je(tw)1'(t)a(dt) T
is a Gaussian random variable. The measurability of this function is readily deduced from the measurability off in both variables and Fubini's theorem. Let ib,, = ibIA,,,
A. = {t: l'(t)2E4(t,w)2 < n}. It suffices to prove our claim for t, replacing V' for each n E TV, since the corresponding random variables converge a.s. to the initial n. Denote by G the one. In this case r) E L2(P), since closed linear subspace in L2(P) generated by the random variables t;(t, ), t E T.
2.4.
The Cameron-Martin space
59
Since G consists of Gaussian random variables, it suffices to show that g E G. Let theorem n = rtl +,, where, 1 G. By
Er]fi = f [Jet.
a)rTi(u) P(dw)J
o(dt) = 0, T Q whence , = 0. Therefore, the right-hand side of (2.3.9) m a Gaussian measure on LP(o). The claim about the integrability of follows from Fernique's theorem 2.8.5 and its Corollary 2.8.6, proved below. In order
to apply Fernique's theorem note that if a is separable (i.e., L' (a) is separable), then the norm of the space X = LP(a) is E(X)-measurable. In the general case, it follows by the measurability of the process that there exists a sequence of functions gk E LQ(o) with unit norms such that k
f
P-a.e.
Indeed, it suffices to show this for the processes t;n(t,w) = 1H
where
B = {t: EIt;(t,w)IP < n}. Hence we may assume that the process (t,:.r) is in LP(o ® P). Approximating this process in LP(o . P) by finite sums of functions p(t)B(w}, where p E LP(a) and 0 E LP(P), we get a sequence {yin} C LP(a) such that, for almost all w, the function t;( , w) belongs to the closed separable subspace E of 11(a) generated by It remains to note that the norm on E is given by sup I, for some 1,, E E' with unit norms. n
Let us prove (2.3.11). If { is centered, then (2.3.11) follows from (2.3.8), since by the Gaussian property, there exists c> > 0 such that
xF(t.t) p
= (EIt;(t.w),,)pl2
Now in the general case it suffices to note that m E LP(o) by the estimate I m(t) I P < EI&IP, which is integrable by (2.3.10). Conversely, (2.3.11) implies (2.3.8), since
C(t,w) - m(t) is a centered Gaussian random variable, hence there exists c2 > 0 such that EIW,w} - m(t)IP < c2 (EIC(t,w) -
m(t)1.,)p12
= c24Kt(t,
t)ip12.
11
2.4. The Cameron-Martin space 2.4.1. Lemma. A vector h from X belongs to the Cameron-Martin space H(ry) precisely when there exists 9 E X, with h = R, (g). In this case, IhIH() = 119110t,)-
PROOF. If IhIH(,) < oc, then by the Riesz theorem there exists an element g E X,* such that
f(h) = (f -a,(f),9)L,(-,) for all f E X', whence h = R,(g). Conversely, we have IhlH(,) = 1I911L2(,) < oo for every such h. 0
Chapter 2. Infinite Dimensional Distributions
60
If h = R.,g, then we say that the element g is associated with the vector h or is generated by h. In this case we use the notation
h:= 9. The relationship determining h is
f(h) = f [f(x)
- a,(f)]h(x) -r(dx).
d f E X'.
X
The Cameron-Martin space H(y) is equipped with the inner product (h, k)H(,)
(h. k)L2(,) = Ji(x)(xh(dx). x
The corresponding norm is
IhIH(.) = Aft=(,)Note that the inner product of any Euclidean space H is uniquely determined by its norm, since 4(x, y) = Ix + y12 - x - y12. It is clear from the previous lemma that if R_ (X,*) C X. then H( .) = R, (X,*).
In this case H(y)with the norm IR,(f)IH(,)= JR(f)(f) turns out to be a Hubert space, and the mapping R, is an isomorphism between X,' and H(-y).
If a,= 0. then (f)IH(-0 = Ilfll00) 2.4.2. Proposition. Let y be a Gaussian measure on a locally convex space X and let g E X.. Then the measure v on X given by the density
P(x) = exPl 9(x) - 20(9)2 with respect to the measure y is a Gaussian measure with the Fourier transform
v(f) =eXP(iRr(g)(f)+ia,(f)- 2a(f)2
(2.4.1)
PROOF. Since 9 is Gaussian, the function exp IgI is integrable with respect to the measure -y. Hence the function @ defines a finite measure v. Let f E X'. Put
k = exp[ia,(f) - 20(.9)2] Let us consider the following function of real argument z:
,p(z) = k f exP[i(f(x) - a.,(f) - zg(x))] y(dx). 11
By virtue of the inclusion f - a. (f) - zg E X. one has
-2 f (f - a,(f) - zg)2dy)
= kexp1-Z0,(f)2
-
Z20(g)2
2+zR,(9)(f)
It. remains to note that p admits a holomorphic extension to the complex plane.
By Lebesgue's theorem, W(z) -y 1(f) as z -* i. On the other hand. the limit on the right-hand side of the foregoing equality coincides with the right-hand side of equality (2.4. 1 ).
The Cameron-Martin space
2.4.
61
2.4.3. Corollary. Let y be a Gaussian measure on a locally convex space X.
Then, for any h E X such that h = R_(g) and 9 E Xi, the measures y and 7h = 'Y( - - h) are equivalent and the corresponding Radon-Nikodym density is given by the expression
oh(x) = exp(g(x) - 2IhIH(2.4.2) PROOF. Formula (2.4.1) and the equality
f(h) = R+(g)(f), V f E X', yield that the Fourier transform of the measure v with density ph coincides with the Fourier transform of the measure yh. Note that the formula for the density of a shift of a Gaussian measure is called the Cameron-Marlin formula.
2.4.4. Lemma. Let y be a Gaussian measure on IR". Then, for any vector h E R, (W), one has 2 - 2exP1 - 1 jhI l(.) ! S IIy - yhll < 2' J 1 - exp(-I Ih1H where
II
-
II
is the variation distance.
PROOF. It is clear that it suffices to prove the claim for nondegenerate centered
measures. In case of a degenerate measure we pass from the whole space to the support of the measure and note that for any vector h outside the support one has the equalities Ihi H(,) = oo and IIy - yhll = 2. Note that any centered nondegenerate Gaussian measure -y on R" is the image of the standard Gaussian measure p under some invertible linear mapping T and y,,, - y = (ph - p)oT-', IThIH(,) = IhIH(v) Hence we may assume that y is the standard Gaussian measure on IR". Moreover, we may assume that the vector h is proportional to et. Then everything reduces to the one dimensional case where the desired estimates are verified explicitly by the aid of Proposition 2.12.6 (see below). 2.4.5. Theorem. Let y be a Gaussian measure on a locally convex space X. (i) Let h E X be a vector such that
IhIH(,) = suP{f(h): I E X', R,(f)(f) < 1} = oo: Then the measures yh and I are mutually singular; (ii) If IhIH(,) < oc. then the measures y and yh are equivalent. In particular.
H(y)={hEX: yh-y}={hEX: IhiH(,)
(2.4.3)
PROOF. (i) It is easily verified that for any continuous finite dimensional linear operator P and any h the following relationship holds true: I1 7h - 111 ? II(7°P-')Ph -.'OP-'1I.
By condition, for any n there exists f E X' such that R,(f)(f) = 1 and f (h) > n. Then the measure y o f has variance 1. Applying Lemma 2.4.4, we get 11 (y o
f-')t(h)
-yof
' 11 ? 2 - 2exp(_
1
nZ
,
Chapter 2. Infinite Dimensional Distributions
62
whence 11_Y,, -'YII = 2, which is equivalent to the singularity (see Problem 2.12.21). (ii) As we already know, for any h with IhlH(.,) < oo, there exists g E X,' such
that R,(g) = h. Hence claim (ii) follows from Corollary 2.4.3. In addition, we get relationship (2.4.3).
2.4.6. Proposition. Assume that y is a Gaussian measure on a locally convex space X. Then the set
U = { h E X : IhlHttii < 11 is weakly bounded and weakly closed in X. In particular, if X is sequentially complete, then this set is weakly compact and the Cameron-Martin space H(y) with the norm I I H(,,) is a Hilbert space continuously embedded into the space X. -
PROOF. Let f E X. Then
f(h) = (f - a,(f), provided h = R,(g), g E X. Hence If(h)I < Ilf -a,(f)IIL=(,) if IhIH(.,) < 1, since in this case h = R,.(g), where 9 E X,; and II9hIL2(,) = IhIH(,) <- 1.
Hence f is bounded on U,,, which means that U. is weakly bounded. Now let h be a limit point of U,, in the weak topology and let {ha} C U be a net that converges weakly to h in X. Then f (h) = lim f (ha), whence If (h) I < 1, provided R,. (f) (f) < 1. Therefore, h E U . It follows from what has been proved that the embedding of H(y) into X is continuous. In the case when X is sequentially complete, H(y) with the norm I IH(.,) is complete as well. Indeed, if a sequence {hn} is fundamental in H(-y), then it converges in X to some vector h, and the reasoning above shows that h E H(y), whence one readily gets the convergence of {hn} to h in H(-y). Hence the weakly compact subset U,, of H(y) is weakly compact in X. For arbitrary Gaussian measures, the condition that X is sequentially complete is essential for the completeness of the Cameron-Martin space even if X is a separable normed space (see Remark 3.11.20 below). However, as we shall see in Chapter 3, this condition can be omitted for Radon Gaussian measures. 2.4.7. Theorem. Let y be a centered Gaussian measure on a locally convex space X. Then its Cameron-Martin space H(y) coincides with the intersection of all linear subspaces of full -f-measure (and also with the intersection of all linear subspaces from E(X) of full y-measure). In addition, y(H(y)) = 0 if Xy is infinite dimensional. PROOF. Let L be a linear subspace of full measure and h E H(-Y). The measures y and -yh are equivalent according to Corollary 2.4.3. Hence y(L-h) = 1. Therefore,
h E L. Conversely, suppose that h does not belong to H(-Y). Then IhiH(,) = 00, hence there exist fn E X' such that a(f,) = 1 and fn(h) > n. Since
x n=1
x
n-Z
x
n-2
Ifn(x)I y(am) :5 n=1
2.4.
The Cameron-Martin space
63
we get the linear space
L = {x E X : the series
n-2 fn(x) converges, n=1
having full measure and belonging to ,6(X). By construction, h does not belong to the set L. Finally, if XY is infinite dimensional, then there exists an infinite orthonormal
sequence {f} C X. It is easy to see that for every M > 0 the set S x E X : sup Ifn(x)I < M} ll n
has -y-measure zero. Therefore, the set
x
{xEX:
l fn(x)2 < 001
n=1
has -y-measure zero as well. In addition, it contains H(-y), since f,, (h) = (fn,9)L2(,)
ifh=R,,(g),gEX;. 2.4.8. Theorem. Let 7 be a Gaussian measure on a locally convex space X. For any p E (1, coo), r > p and any function f E L'(-y), the mapping (H(7), I H(,)) -. L°(y), h - f( + h), is continuous. If p = 1, then the same is true for all functions f E L'°(it). I
PROOF. This result is deduced from Corollary 2.4.3 as follows. For every cylin-
drical function f of the form f (x) = p (11(x), ... , ln(x)), where w E Cb(En) and li E X', the mapping above is continuous by Lebesgue's theorem. Let f E L'(y), where r > p. There exists a sequence {f3 } of cylindrical functions of the indicated type that converges to f in Lr(y). There exist numbers t and s such that t > 1, tp = r, t-1 + s-1 = 1. Recall that the element h = R71 h E X, corresponding to the vector h is a centered Gaussian random variable on (X, ry). Hence, 2
f exp(sh) dy = exP(2 Ih1y(7)). According to Corollary 2.4.3 and Holder's inequality, we get
JIf(x + h) -fj(x+h)Ip7(dx) = f If(z) - fj(z)I"exp(h(z)
r
- 2IhIH(,)) ti(dx) {JexP(s(z) - ZIh12 (7))
if If(z) - fi(z)Ir7(dz)J 1lt
5
{f If(z)-fj(z)I'7(dz)}
I(dr)}1/e
l1/t
_ exp(s 211hI2.(,)),
which tends to zero, as j - oo, uniformly in h E H(y) with IhIH(,) < R, for every fixed R > 0. Therefore, the mapping h s-+ f(- + h) is continuous. In the case when p = 1 and f E L" (ry), the proof is analogous with the only difference that the sequence {fj} should be taken to be convergent to f in measure with
Chapter 2. Infinite Dimensional Distributions
64
sup J f,1 < ] 1f1 1. , and instead of Holder's inequality one has to use the LebesgueVitali theorem on uniformly integrable sequences.
2.4.9. Corollary. Let y be a Gaussian measure on a locally convex space and let A be a set of positive -y-measure. Then them exists c > 0 such that cU C A - A. where U, is the closed unit ball in the space H(-I). PROOF. The function h y ((A + h) n A) is positive at zero and continuous by Theorem 2.4.8 applied to the indicator function of A, since
y((A+ h) n A) =
flA(x - h)I.4(x)7(dx). x
Therefore, there exists c > 0 such that - ((A+h)nA) > 0 for all h E cUH. Clearly,
hEA - A for such h. 2.4.10. Proposition. Let y be a Gaussian measure on a locally convex space X such that its Cameron-Martin space H(y) is separable with respect to the norm Suppose that Q is a set of full y-measure. Then, for y-a.e. x, the set
H(y)n(n-x) is dense in the space H(y) with respect to the norm I
in H('). Denote by PROOF. Let us take an orthonormal basis the countable set of all finite linear combinations of the vectors e, with rational coefficients. Since h E H(y), one has y(fl + h,,) = 1. Hence y(sl n (ft + h,)) = 1. Therefore, the set f2 :_ {x E 11: x + h E 0} has measure I as well. Put Y = nn lfl,,. Then y(Y) = 1. If x E Y. then n - x contains h for every n. whence the conclusion.
2.5. Zero-one laws
2.5.1. Lemma. Let y be a centered Gaussian measure on a locally convex space X. The collections of the functions of the form. Hk, (i1) ... Hk, U.), where the Hk, 's are Hermite polynomials and li E X', is dense in L2(y).
where V is a bounded Bore! PROOF. The functions of the form function on R", are dense in L2 (-y). Hence the claim follows from the corresponding finite dimensional result.
2.5.2. Theorem. Let y be a Gaussian measure on a locally convex space X such that R,(X `) C X. Suppose that a set A E .6(X), satisfies the condition
y(A+h)=y(A), b'hER,(X'). Then either -y (A) = 1 or y(A) = 0. In addition, if f is a y-measurable function such that for every h E R, (X*) one has
f(x+h) = f(x) then f coincides a.e. with a constant.
y-a.e..
2.5.
Zero-one laws
65
PROOF. To simplify notation let us suppose that a, = 0. Let hl = R,(11), hn = R,(ln), where li E X' and the vectors h, are orthonormal in H(y). By condition and the Cameron-Martin formula, the function
F(ti,...,tn)=y(A-t1h,
f exp(: A
tilt(s) -
211yt`h=ilH(,l)
-y(dx)
is constant. Therefore, for any collection ml,... , m of nonnegative integers not vanishing simultaneously, we have
am,+...+m F
8t""...8tn^(0,...,0)=0. Since the vectors h; are orthonormal, Lemma 1.3.2(iii) yields
f
Hm, (1, (x))
... Hm., (1.(x)) IA(x) y(dx) = 0,
x
where Hk is the k-th Hermite polynomial. Thus, the function IA is orthogonal to all the polynomials H,,,, ( 1 1 ). .Hm,. (ln), where m; are not zero simultaneously and
l; E X' are mutually orthogonal in X. This means that IA is a constant. This constant can be only 0 or 1. Considering the sets (f < c}, we get the claim for functions.
It is clear that it is sufficient to require the condition of Theorem 2.5.2 for all vectors from R,(Y), where a linear space Y is dense in X' with the L2-norm.
2.5.3. Corollary. Let the measure y be the same as in Theorem 2.5.2 and let A be a -y-measurable set such that
y(A\(A+ h)) = 0, `dh E R,.(X').
(2.5.1)
Then either y(A) = 1 or y(A) = 0. PROOF. Let h E R, (X' ). Then -h E R, (X' ). By (2.5.1), we have the equality
(A\(A - h)) = 0. Since y_h - y, we get y((A+ h)\A) = 0, which together with (2.5.1) yields y(A + h) = y(A). Note that in both claims the measurability of A + h follows automatically from that of A, since the measures y_h and y are equivalent. The following more general fact can be easily seen from the proof of the previous theorem.
2.5.4. Corollary. Suppose that in Theorem 2.5.2 the space H(y) is separable, {en} is its orthonormal basis and a set A measurable with respect to the measure -y has the property that
A + ren = A up to a set of -y-measure zero for all rational r and all n E IN. Then either y(A) = 0 or y(A) = 1. It follows from Theorem 2.5.2 and Corollary 2.4.9 that for any measurable linear
subspace L in a locally convex space with a Gaussian measure y (satisfying the condition in Theorem 2.5.2) the following alternative takes place: either y(L) = 0 or -y(L) = 1. Our next theorem shows that the same is true for any Gaussian measure. We use the argument suggested by Fernique [709] for the larger class of
Chapter 2. Infinite Dimensional Distributions
66
stable measures. Recall that an affine subspace in a linear space X is a set of the form v+ L, where v E X and L C X is a linear subspace.
2.5.5. Theorem. Let -y be a Gaussian measure on a locally convex space X and let L be a -t-measurable affine subspace in X. Then either y(L) = 1
or y(L) = 0.
PROOF. It suffices to prove the claim for linear subspaces L. Further, we may assume that L E E(X). Indeed, if L has positive measure, then, by Lemma 2.1.5, there exist a compact set K C R'° and a continuous linear mapping F: X -. R'° such that the set A = F-1(K) C L has positive measure. Clearly, the linear span S of K is Borel, hence Lo = F-'(S) is a positive measure linear subspace from E(X). Suppose first that y is centered. Let f and q be two independent random vectors in X with distribution y (e.g., one can take the measure P = y&y on X xX and put t (x, y) = x, q(x, y) = y for (x, y) E X X X) Put -
Stn={f ¢L}nit? +n. EL}, nEIN. By the linearity of L, t (w) and q(w) belong to L precisely when l; and q + 714 do. we Hence, taking into account that q + nt has the same distribution as get
P(q + n{ E L) - P({.e E L} n (q + n{ E L})
= P( 1 + n21; E L) - P({ f E L} n Jr) E L}) = P(C E L) - P(f E L)2. Using again the linearity of L, we see that the sets Stn, n E IN, are disjoint. Therefore, P(f E L) - P(f E L)2 = 0, whence the desired conclusion.
In the general case, let y1 be the image of -y under the mapping x ' -. -x. Then yo = -y * yi is a centered Gaussian measure. If y(L) > 0, then, clearly, -yo(L) > 0. Therefore, -yo(L) = 1. Since yi(L) = y(L) > 0, there is x E L such that 0 y(L) = y(L - x) = 1. 2.5.6. Corollary. Let y be a centered Gaussian measure on a locally convex space X and let L be a linear subspace in X. Then for any a ¢ L one has y.(L + a) = 0,
where y. is the inner measure. PROOF. Let B C L + a, B E E(X). By Lemma 2.1.4, the linear span Lo of the set B - a C L is measurable with respect to y. According to Theorem 2.5.5, the measure of the set Lo + a is either 0 or 1. The latter, however, is impossible, since y(Lo - a) = y(Lo + a) (due to the symmetry of y), but (Lo - a) n (Lo + a) = 0. It 0 remains to note that B C Lo + a.
2.5.7. Theorem. Let -y be a Gaussian measure on a locally convex space X such that Ii;, (X *) C X. If G is a y-measurable additive subgroup in X. then either
y(G) = 0 or y(G) = 1. PROOF. It suffices to be shown that if y(G) > 0, then C + H(y) = G. Indeed, for any h E H(-y), according to Corollary 2.4.9, there exists n E IN such that 0 n-'h E G- G. Hence h E G, i.e., H(y) C G.
2.6.
Separability and oscillations
67
2.5.8. Theorem. Let ^y be a Gaussian measure on a locally convex space X with mean a, and let C be an additive -y-measurable subgroup in X of full measure. Then 2a., E C and H(ry) C G. Then the measure -yo is symmetric, hence,
PROOF. Let '10 =
1 = -y(G) = ,yo(a. + G) = 7o (-(a, + G)) = 7o(-a, + G). Therefore, the set (-a, +G) (1(a, +G) is nonempty, which gives the first claim. For the proof of the second one let us note that, for any h E H(-f), the set h - a- + G 0 has full measure, hence the set (h - a., + C) f (-a, + G) is nonempty. We shall return to the zero-one law in the next section and in Chapter 5, where we discuss measurable polynomials.
2.6.
Separability and oscillations
2.6.1. Definition. Let (1, B, P) be a probability space and let T be a metric space. A random process £(t,w), t E T, w E 11, is called separable if there exist an at most countable set S C T (called a separant of the process) an a set Qo C f2 such
that P(5l0) = 1 and, for every open set U C T and every closed set Z C s.', the following equality holds true:
{wES2o: l;(t,w)EZ, dtEU}={wEflo: t;(t,w)EZ, dtEUf1S}. (2.6.1) Note that S is automatically dense in T, in particular, T is separable. Equality (2.6.1) is equivalent to the following condition: for every w E Ilo, t E T and e > 0, letting d be the distance in T, the point l: (t, w) belongs to the closure of the set {t;(s,w), s E S, d(s, t) < e}. Let f be a function on a space T equipped with a metric or semimetric d (recall that in the definition of a semimetric there is no condition that d(x, y) = 0 implies
x = y). The oscillation of the function f at a point r is defined as the quantity (possibly, infinite)
of(r)lim
sup G.sEK(r.b)
If(t)-f(s)I=6lim
I
sup
tEK(,.6)
inf.6)f(t)], J(t)-tE K(r
where K(r, 6) denotes the open ball in T of radius b centered at r. In a similar manner one defines the oscillation on a set M C T:
af(M) = lim sup{If(t) - f(s)I, t, s E M6, d(s,t) < b}, 6-0+
where M6 is the open 6-neighborhood of M, i.e., the collection of all points t E T such that there exists t' E Al with d(t, t') < 6. It is easily verified that a function on a metric space is continuous at a given point precisely when its oscillation at this point is zero. In the case where the function f is random, its oscillation at every fixed r or M turns out to be a function on a probability space. Hence the question about the measurability of this function arises. A simple sufficient condition of measurability is given in the next lemma.
Chapter 2.
68
Infinite Dimensional Distributions
2.6.2. Lemma. Let (T, d) be a separable metric space and let (1, B. P) be a t E T, w E i2) is a separable random probability space. Suppose that = process on T. Then the functions
-
w'
are measurable for every x and every set M C T.
Pttooe. Let S C T be a countable separant of the process t;. It is readily seen from the definition of separability that, for every open set U C T, for all w E S2o, one has
suptt(w) = sup t t(w), tEU')5
tEU
ttinf ut;t(w) =
inf
(2.6.2)
t(M),
and the functions defined by the right-hand sides of these equalities are measurable, since S is countable. Hence,
sup lCe(w) - ,(w)I = supet(w) - inf ,(w) tEt:
t.eEU
is a P-measurable function. Therefore, the function sup t,sE ti, d(t,s)<6
is P-measurable as well. Indeed, this function equals supW,.(w), where n
Vn(w) = sup I6(w) - Ww)I, t.sEU
is the sequence of all open balls with rational radii r < 6 and centers s E S such that U C U. This implies the measurability of cq.(,,,)(Af) for any and
subset M of T. For Gaussian processes, the oscillations turn out to be nonrandom. This surprising fact discovered by It6 and Nisio [370J (and in the form presented here proved in [375]), is directly connected with the zero-one law.
For any function f on a metric space (T, d), let us put lim sup f (s) = lim sup f (s) and lim inf f (s) = lim
s-t
s-t
6-,0+ sEK(t.6)
inf
6-0+ sEK(t.6)
f (s).
2.6.3. Theorem. Let (T, g) be a separable metric space and let t; = (l;t)tET be a centered separable Gaussian random process with the continuous covariance function (t, s) ,--r E&t .-
Then there exists a nonrandom function a : T -. [0, oc] such that, for almost all w, one has for all t
a(t) = lim sup{ISU(w) -u,v E K(t,e)}. In addition, for every fixed t E T, one has
lim sup t. (w) = to (w) + a(t) a.e..
(2.6.3)
liminf&(w)=tt(W)Ia(t) a.e. 3-t 2
(2.6.4)
s-t
2
2.6.
69
Separability and oscillations
PROOF. We may assume that the covariance function is uniformly continuous, passing to the metric pa(s, t) = p(s, t) + lEl{ - t;r12, which determines the same topology. Let S = {s.,} be a countable separant of the process. In addition, we may assume that the probability space is IRT, equipped with the measure y induced by the process (since the properties stated depend only on the finite dimensional distributions of the process by its separability). Then the process itself has the form t;t(w) = w(t). As noted above, the dual to 1RT coincides with the linear span of the functionals be. Hence R.,(f) E IRT for all f E (IRT )'. Moreover, the uniform continuity of the covariance function implies the uniform continuity of the function h = R,(f). Indeed, otherwise for some e > 0, there exists a sequence of pairs a,,, b E T, such that h(bn) e and IEI(b IZ < 2-1. Then the functionals
x - E 1 [x(b;) - x(a;)( are uniformly bounded in X;, but tend to infinity at the element h, which contradicts the inclusion h E H(-y). Let Al C T. We shall denote the oscillation of the function t --. £r (w) on the set Al by a(M, w). Recall that we are in the situation where tr(w) = w(t). For the uniformly continuous function h, one has a(Al, w + h) = a(Al, w). By Theorem 2.5.2, the function a(M,w) coincides a.e. with some constant a(Al) (maybe, infinite; note that Theorem 2.5.2 applies also to functions taking value +oo). Let g be the countable family of all open balls with the rational radii and centers at the points of S and let 520 be the full measure set corresponding to S by Definition 2.6.1. By the separability of the process, for every t E T, there is a sequence of balls Vj from g with radii r. such that Vj+1 C V3, rl -+ 0, t = nim1 vi and
a(t,w) = lim a(Vj,w),
`
u; E 520.
This gives the first claim. Therefore, for any fixed t E T, the random variable 71, := limsup(t;a - bt] a-1
is degenerate. By virtue of symmetry of the distribution of the process Ca := t:s -4t, the random variable limsup[-ta + t;t] has the same degenerate distribution as r7t,
a-t
hence it coincides a.e. with nt. It remains to note that the sum of these two random 13 variables is a(t,w). The Ito-Nisio theorem yields the existence of a modification that is continuous with respect to an appropriate metric on T (such a modification is called natural).
2.6.4. Proposition. Suppose that in the situation of Theorem 2.6.3 one has
a(t)
Then (T, pl) is separable and the process has a separable modification with the continuous sample paths on (T, g,). For such a modification one can take
i at(w) =
Llim
2
sup t;a(w) + lim t,(a.t)
inf
s(w)].
PROOF. It follows by (2.6.3) and (2.6.4) that for every fixed t one has t;n at = tt a.e., i.e., we get indeed a version of t;. It follows by the definition of a(t) that for
Chapter 2.
70
Infinite Dimensional Distributions
all t and w one has £ °t (w) = lim
1
sup Ww) - 1 a(t) = lim inf l;,(w) + a(t). 6-00(8.t)
6-0e(s.t)<6
(2.6.5)
Let us show the continuity of C °t(w) on (T,ei) for any w. Let t E T be fixed and let 6 > 0. According to (2.6.5) we have sup
&;°t(w) <
sup 9(s.t)<26
(3.r" <6
<
sup
i,(w) - I
inf
2 Q,($.t)
a(s)
.(w) - 2a(t) + 2
Therefore, Jim
sup
°f (w) < lim
sup
6-OQ(5.t)<26
6-'UPi(..t)<6
fs(w) - 1a(t) = V" (W). 2
inf t°t(w) 6-Opt(s.t)
In a similar manner one verifies that Jim
whence the
continuity at t. Let us show that (T, pl) is separable. Indeed, let us take a countable subset Q in the space (T, p) such that the set of the pairs (q, a(q)), q E Q, is dense in the graph of the function a on (T, p). This is possible, since the space (T, e) xR' is separable. It is clear that Q is dense in (T, pi). By the continuity of the sample paths of the constructed process, it is separable on (T, ei ). D
Finally, let us show that the separability of a process can be achieved by a of any process
suitable choice of a modification (called separable modification) with the separable reproducing kernel Hilbert space.
2.6.5. Proposition. Let ({t)tET be a centered Gaussian random process on a set T. Suppose that T with the semimetric d(t, s) = lEJl:t - I2 is separable. Then, on the same probability space, there exists a separable centered Gaussian random process (rtt)tET on T such that, for any fixed t, one has ft = qt a.e. PROOF. Let us choose a countable set S = {s2} which is dense in T with respect to the semimetric d. We may assume that d separates the elements in S. For every n E IN and every j E IN, we put
A(j.n)=K(s,
where K(s,r)={t: d(t,s)
The family {A(j.n)}, j E N. is a partition of T. For every partition of this kind,
let us consider the Gaussian random process t;"(t) = (sj), t E A(j,n). Then ]Eli -O < 2-". By the Chebyshev inequality x x
E P(w: n=1
n22-2".
(w) - t;t(w)I > 1/n) < n=1
Hence, for every fixed t, with probability 1 the sequence , converges to &. Now let us put 77, = line sup ti . It is clear that we get a modification of 4. Since d separates
the points sj. we have for s E S. The verification of the separability of the version constructed is left to the reader as Problem 2.12.22. 0
2.7.
Equivalence and singularity
71
It follows from the proof above that for a separable centered Gaussian process
(WrET with a separant To, any countable set T, dense in (Td), where d(t.s) = Elf: - C ,I2, serves as a separant. Indeed, it suffices to take a separable version t of C, for which T, is a separant and note that P(l t = t:t, V t E To U Ti) = 1. 2.7. Equivalence and singularity
2.7.1. Lemma. Let p be a Gaussian measure on a locally convex space X. A E E(X) and µ(A) > 0. If the measure v = I4µ/µ(A) is Gaussian. then µ(A) = 1. PROOF. According to Example 2.3.10, the claim reduces to the case X = IRT, in which, as we know. one has the equality R,,(X,) C X. Indeed, this reduction is a direct corollary of the equality in Example 2.3.10. Let h E H(p). For sufficiently
small t, according to Theorem 2.4.8, we have µ(A n (A + th)) > 0. Hence the measures v and veh are not mutually singular. Indeed, otherwise there is a subset
B C A such that v(B) = 1 and vth(B) = 0. By definition, this means that µ(B) = p(A) and µ(A n (B - th)) = 0. Then µ((A + th) n B)) = 0, since h E H(p). Hence µ(A n (A + th)) = 0. which is a contradiction. Therefore, h E H(v), whence with << v. This means that µ(A\(A + th)) = 0. Applying Corollary 2.5.3. we get our claim.
The next theorem of Hajek and Feldman is one of the central results in the theory of Gaussian measures.
2.7.2. Theorem. Any two Gaussian measures on one and the same locally convex space are either equivalent or mutually singular.
PROOF. Let µ and v be two Gaussian measures. Then µ = µt +102, where µ2 is mutually singular with v and µl = PI ,t << v for some A E E(X). Let us employ the mappings T,,.x and the notation µ3 introduced in Theorem 2.2.12 above. It is readily verified that µi << v3 and that the measure (p3 - µ) is mutually singular with v3. Note that the mapping T`,,X preserves these two properties and Tr'.X(Y3) = U3
T,,. .X(µ's) = p3,
Therefore, 3
1
3
t't10T,.x = lti If µ(A) > 0, then, by Theorem 2.2.12, the measure p1/µ(A) is Gaussian. By Lemma 2.7.1, this measure coincides with p, which implies the relationship µ
v.
If µ(A) = 0, then p and v are mutually singular. Finally. note that if µ and v are not mutually singular, then by the same reasoning v << p, hence these two measures are equivalent.
This theorem gives no constructive criterion to determine which one of the two cases takes place. Quantitative criteria for the equivalence of Gaussian measures will be obtained in Chapter 6 in the context of absolutely continuous linear transformations. Here we only give some general remarks in this direction. 2.7.3. Proposition. Let µ and v be two Gaussian measures on a locally convex space X.
(i) If µ - v, then H(,a) = H(v) as sets.
Chapter 2.
72
Infinite Dimensional Distributions
(ii) If it and v are centered measures and p 1 v, then is. 1 vb for all a, 6 E X.
(iii) If p and v are centered measures and p - v, then X = X. PROOF. (i) Let h E H(p). Then vh - Ph - p - v, whence h E H(v). In the same manner we conclude that H(v) C H(p). (ii) The general case obviously reduces to the case where b = 0. If a E H(p), then pQ - p, whence p" 1 v. If a ¢ H(p), then, by Theorem 2.4.7, there exists a linear space L E ,6(X) such that p(L) = 1 and a L. Then pa(L+a) = 1. On the other hand, v(L + a) = 0 according to Corollary 2.5.6. (iii) Clearly, equivalent measures have equal collections of measurable functions and sequences of measurable functions convergent in measure. If a sequence {fn} from X* converges in L2(p), then it converges in measure p, hence also in measure v, whence the convergence in L2(v) follows by the Gaussian property. Interchanging
p and v, we get the claim. In the finite dimensional case, any two Gaussian measures with full supports are equivalent. The situation is totally different in infinite dimensions.
2.7.4. Example. Let ry be a centered Gaussian measure on a locally convex space X. If dim (X.*,) = oo, then, for any nonzero real numbers r and q with Iri A jqj, the measures ryr(A) = ry(rA) and .y (A) = y(qA) are mutually singular. PROOF. Let {lc? %) be an orthonormal sequence in X;. Put Sn = n-1 C? By the Gaussian property, the random variables Sn are independent. According to
Lemma 2.7.5 below, Sn -+ r-2 a.e. with respect to yr and Sn -+ q-2 a.e. with respect to -y4. Therefore, these two measures are mutually singular.
0
The proof of the next classical result (the large numbers law) can be found in 1697, Ch. IV, §3, p. 363, Theorem 1).
2.7.5. Lemma. Let C. be a sequence of independent centered Gaussian random variables with variance d2. Then S,,
I
n
d2
a.s.
n ;_1
Note that this lemma means that the product -y of the standard one dimensional Gaussian measures is concentrated on the surface of an "ellipsoid".
2.7.6. Example. Let {on } and {Nn } be two sequences of positive numbers, p the countable product of the Gaussian measures on the real line with densities p( 0, a:R) and v the countable product of the Gaussian measures on the real line with densities p( , 0, 3 ,1,). The measures p and v are equivalent precisely when the following product converges:
x n=1
2anQ
n+3'
which is equivalent to the condition 2
(an n=1 0n
-1
(2.7.1)
2.7.
73
Equivalence and singularity
PROOF. This claim follows directly from Kakutani's theorem given below. In order to apply that theorem, let us note that
I(a,Q)
f p(t.0,a2)
p(t,0,32) dt =
2a,3
a2+Q2
Put On = an/Dn and notice that the convergence of the product fl ii equivalent to the convergence of the series
n=1 V
x 0(I
n=1
-
2B
1+6,
is
, which, in turn, is 1 ' ex
equivalent to the convergence of the series F (0n - 1)2. n=1
Holding the notation of Example 2.7.4, let us put 1
r n E,= {x: limn1-E1:;(x)=c}, c>0. nx JJJ
i=1
111
Since the set E1 has full measure, we have y(E,) = 0, provided c 1. Therefore, the mapping x +-+ cx takes some measurable sets to nonmeasurable ones. A similar effect occurs in the following surprising example discovered by Cameron and Martin in [1441.
2.7.7. Example. Let f : (0, oo) - [0,11 be an arbitrary function. Then there exists a 7-measurable set E C X such that cE is measurable for every c > 0 and y(cE) = f (c),
V c > 0.
PROOF. Let r: (0, +oc) -+ [-oo, +oo1 be an arbitrary function. Put
Ae = {x E X: l;l(x) > t},
t E [-00,+001.
It is clear that cAt = Act for all c > 0. Let
E = U (Ar(t) fl E) . t>o
Note that
cE = U(cAr(t) n cE) = U(Ac,.(t) fl E.2t). Hence cE is the union of the measurable set A,(,-2) n El and a subset of the set X\E1, which has measure zero. Therefore, the set cE is measurable with respect to y and
x ^Y(cE)
7(Ac,(c-2))
-
27,
1 z J exp(-2s ) ds.
cr(c-2)
It remains to choose the function r in order to get 1 - 4 (cr(c-2)) = f(c) for all c > 0. Obviously, this is possible.
74
Chapter 2. Infinite Dimensional Distributions
2.8. Measurable seminorms 2.8.1. Definition. Let -y be a Gaussian measure on a locally convex space X. A function q measurable with respect to £(X )., is called a £(X ).,-measurable seminorm if there exists a £(X).,-measurable linear subspace X0 C X of -f-measures such that q on X0 is a seminorm in the usual sense.
It is clear that any £(X).,-measurable seminorm can be redefined on a set of y-measure zero in such a way that it will be a seminorm in the usual sense on the whole space. Indeed, let Y be an algebraic complement to X0 in X. Put qo(x + y) := q(x) if x E Xo, y E Y. Then qp has the required properties. Note that sometimes modifications of measurable seminorms in our sense are also called measurable seminorms. However, we shall not employ this terminology.
2.8.2. Example. Let y be a Gaussian measure on a locally convex space X and let a sequence {f,,} C X' be such that q(x) = sup l f (x)l < oo almost everywhere with respect to y. Then q is a £(X),-measurable seminorm. PROOF. Let X0 = {x: q(x) < oo}. According to Problem A.3.34 in Appendix, the set Xo is in E(X). In addition, it is a linear space of full measure on which q is a seminorm. The measurability of q is obvious.
It will be shown in Chapter 4 that every y-measurable seminorm on a locally convex space X with a centered Gaussian measure y coincides almost everywhere with a function of the form
q(x) = sup[fn(x) +
f,, E X:,, a,, > 0.
2.8.3. Theorem. Let -y be a Gaussian measure on a locally convex space X and let q be a E(X).,-measurable seminorm on X. Then the restriction of q to the space H(y) with the norm I IH(,,) is continuous.
PROOF. The set V = {x: q(x) < n} has positive measure for some n E IN. According to Corollary 2.4.9, for a sufficiently small positive number r, the set V. - V, contains the ball of radius r from H(-y). Hence the seminorm q is bounded on the unit ball in H(y) which is equivalent to the continuity of q. 2.8.4. Proposition. Let -y be a Gaussian measure on a locally convex space X and let q, and 42 be two -y-measurable seminorms such that q, = Q2 a.e. Then q, = q2 on H(y). PROOF. It follows from the definition that there is a linear subspace Y of full measure on which both q, and q2 are seminorms in the usual sense. Let E :_
{x E Y: ql(x) = q2(x)} and let It E H(y). Then, for every n E IN, one has x y(E - nh) = 1, since y(E) = 1. Hence the intersection of E with n (E - nh) n-I has measure 1, in particular, there is some x in this intersection. This means that q, (x + nh) = q2(x + nh) for all n E IN. Clearly, h E Y. Since any seminorm on the two dimensional space spanned by x and It is continuous, we deduce from the
equality ql(n-tx+h) = q2(n-lx +h) that q,(h) = q2(h). The following result about the exponential integrability of seminorms is the celebrated Fernique theorem [2411.
2.8.
Measurable seminorms
75
2.8.5. Theorem. Let 1 be a centered Gaussian measure on a locally convex space X and let q be a E(X).,-measurable seminorm. Let us pick r such that
c=-y(g5r)>1/2. Put a= 24r2 log
c c if c < 1. Then
f exp(ag2) dy < 1(r, c) < oo, where I(r,c) depends only on r and c. If c = 1. then q E L'(-y). PRooF. Let r, t be two arbitrary numbers. According to Proposition 2.2.10 and the change of variables formula (see formula (A.3.1) in Appendix) we have
-y(q 5 r)-y(q > t) = f f
7(dx) 7(dy)
g(x)t
=
ffry(du),y(dv) ff
since q(u) >
t q
and q(v) >
t
q(u)>!. q(r)>!
ry(du) y(dv),
r,
, provided
is a seminorm, one has
q(u) >
q(u + v) - q(u - v)
Vu, v.
2
Therefore, we arrive at the following inequality which is crucial in the proof:
tT))2. 7(q 5 r)-t(q > t) < (-y(q>
(2.8.1)
Since q < oo almost everywhere, there exists a nonnegative number r such that c := '?'(q 5 r) > 1/2.
Then Po :=
< 1.
'Y(q << r)
We may assume that c < 1, since in the case where c = I the claim is trivial. Put
tn=r+tn_t\, to=r,
nEIN.
It is straightforward to verify the equality t" = r(1 + V25) ((-,/'2-)-+' - 1).
(2.8.2)
Let us define numbers p,,, n E IN, by virtue of the relations cpa. 7(q > Applying estimate (2.8.1) to the pair (r,tn), we get
(2.8.3)
(2.8.4)
Pn <- P2.-1
Applying consecutively relationships (2.8.3) and (2.8.4), we arrive at the inequality
?'(q>tn)SC(
1-c C
)
2"
Chapter 2.
76
Infinite Dimensional Distributions
Now let us choose a as follows:
a=
log
1
c (2.8.5)
1-c
247-2
Taking into account (2.8.2), we get
Jexp(oq2) dy <
r
exp(ag2) d -Y +
air
< cexp(ar2) +
exp(aln+l) Y(tn < q:5 In+j) n=n
exp(atn+1)y(q > ln) n=0
x
cexp(ar2) +
c n=0
(1-c}
2.,
exp[40r2(1 + V12)2 2-]
C
=cexp(ar2)+c
exp(2-[log'c+4ar2(1+v' )2]
n=0
I<00,
C
since 4(1 + f)2 < 24. 2.8.6. Corollary. Let y be a Gaussian measure on a locally convex space X and let q be a E(X)-measurable seminorm on X. Then there exists a > 0 such that exp(ag2) E L'(-t). PROOF. Let tj be the image of y under the mapping x -x and let ryo = 7*yr. Then -to is a centered Gaussian measure. Let us choose a > 0 for this measure as indicated in the previous theorem. Since
ffexP(aq(x - y)2) y(dx) y'(dy) < oo, Xx there exists y such that the function x -. exp(aq(x - y)2) is y-integrable. Then, for any oo < a, one has exp(aog2) E L'(y), since aoq(x)2 < aq(x - y)2 + aoa(a - ao)-'q(y)2. Indeed, one has q(x) c q(x - y) + q(y), 2q(x - y)q(y) 5 e2q(x - y)2 + E-2q(y)2,
where e2 = a/ao - 1. 2.8.7. Remark. It follows from the proof of Fernique's theorem that its conclusion remains valid in a slightly more general situation. Namely, let be a mapping from a probability space (f2, F, P) to a linear space E equipped with a seminorm q such that the function q(l;) is measurable and the mapping (x
Y )'-"
(q( (X)_(Y)) W(x)+ (Y) ))
is measurable and has the same distribution as (x.y)
on
( Il x fZ ,
P (9 P )
(q(. (x)),q(l;(y))).
2.8.8. Example. Let { Xn } be a sequence of random vectors in a locally convex space X having a Gaussian distribution y in the countable product X" of X. Suppose that q is a seminorm on X such that x = (xn) q(x,,) is y-measurable
2.8.
Measurable seminorms
77
oo almost sure. Then there exists a > 0 such that
for every n and n
Eexp(a(SUP q(Xn))2) < oc. Unlike the finite dimensional case, a measurable seminorm on a space with a centered Gaussian measure can coincide almost everywhere with a nonzero constant.
2.8.9. Example. Let -y be the countable product of the standard Gaussian measures on the real line regarded on lR". Put q. (x) = (1F_Then q = lim sup qn is a -y-measurable seminorm such that q = I almost everywhere. n-x PROOF. By virtue of the large numbers law, we have qn - 1 almost everywhere. The set L = {x: limsupgn(x) < oc}
n-X
is a linear subspace, since every function qn is a seminorm. In addition, q is a seminorm on this space. The measurability of L and q is obvious. The following theorem extends Anderson's inequality to the infinite dimensional case.
2.8.10. Theorem. Let y be a centered Gaussian measure on a locally convex space X and let A E E(X), be an absolutely convex set. Then, for any a E X such that A + a E £(X ), , the following inequality holds true: y(A + a) < -y(A).
More generally, if A + to E £(X), for all t E [0, 1], then
y(A + a) < y(A + ta),
d t E {0,1).
PROOF. Applying Lemma 2.1.6 to the measure y + y_Q and using the finite dimensional Anderson inequality, we get the claim. 2.8.11. Corollary. Let 7 be a centered Gaussian measure on a locally convex space X and let f be a function on X such that the sets if < c}, c E RI . are convex
symmetric (e.g., let f be convex symmetric). If f and f ( + a) are y-integrable. then
J f (x) 'y(dx) < X
J
f (x + a) y(dx),
da E X.
X
More generally. if f ( + ta) E L'(1) for all t > 0, then the function
J f (x + ta) y(dx) x is nondecreasing on [0, +x). In particular, this is true for every £(X )-measurable seminorm f. t
PROOF. The same reasoning as in the finite dimensional Corollary 1.8.6 applies.
Chapter 2. Infinite Dimensional Distributions
78
2.9. The Ornstein-Uhlenbeck semigroup Let -y be a centered Gaussian measure on a locally convex space X. In the same way as in the finite dimensional case the Ornstein-Uhlenbeck semigroup (T, )t>0 on LP(y) is defined by the formula
Ttf (x) =
J
1 - e-2t y) y(dy).
f (e-tx +
(2.9.1)
X
The same formula defines the Ornstein-Uhlenbeck semigroup (Tt)t>o on the spaces LP(y, E), p > 1, of mappings with values in a separable Banach space E. The next theorem is proved by the same reasoning as the one used in the finite dimensional case.
2.9.1. Theorem. For every p > 1, the family (Tt)t>0 is a strongly continuous semigroup on LP(y) with the operator norm IITtIIr(La(,)) = 1'
The operators Tt are nonnegative on L2(y). If E is a separable Banach space, then (Tt)t>o is a strongly continuous semigroup on LP(y, E), p > 1.
Denote by E the closed linear subspace in L2(y) generated by the Hermite polynomials of the form Hkt (l1) ' ... Hk,,, (lm ) , 1, E X', m = 0, 1, ... , n. Let Xo be the one dimensional space of constants + km having degree k1 +
and let Mk denote the orthogonal complement of Ek_I in Ek, k E IN. Then we get the following decomposition of L2(y) into a direct sum of mutually orthogonal subspaces: L2(y) _ ® Xk,
(2.9.2)
k=0
Denoting by Ik(F) the projections of called the Wiener chaos decomposition. F E L2(y) onto Xk, we get the decomposition
x
F = E I,. (F).
(2.9.3)
k=0
In each subspace Xk one can choose explicitly an orthonormal basis. To this end, let us take an arbitrary orthonormal basis (Wo En in X; and consider the Hermite polynomials Hat....,am:k,.....k :=Hkt(fa,)...Hk-(dam), a, EA, k, =0,1,... By virtue of Lemma 2.5.1 and the properties of the finite dimensional Hermite
polynomials, the collection of the functions we get is an orthonormal basis in L2(-y), + km = n and, for every n, its subset formed by the polynomials of degree k1 +
is a basis in X,,. Unlike the finite dimensional case, if k > 0, then the space Xk is infinite dimensional.
Clearly, if the Hilbert space Xy is separable (which is always the case for Radon Gaussian measures, as we shall see later), then the basis constructed above is countable. In addition, it is possible to choose an orthonormal basis in X,;
consisting of elements { E X. Note that in any case one can do this for the
2.10.
Measurable linear functionals
79
restriction of y to any countably generated a-field £1 C E(X) (recall that every set from £(X) is determined by some countable collection of continuous linear functionals). By analogy one defines the spaces Xk (E) of mappings with values in a separable Hilbert space E, i.e., Xk(E) is the closure of the linear span of the mappings f (x)v,
f E Xk, v E E. in L2(y, E). The following statement is easily obtained from the corresponding finite dimensional theorem.
2.9.2. Theorem. For all t > 0 and f E L2(y), one has
x Ttf = > e-kt jk(f). k=0
We shall see in Chapter 5 that the Ornstein-Uhlenbeck semigroup has a strong smoothing property (obvious in the finite dimensional case). Here we consider a simple example.
2.9.3. Example. Let f E LP (-y), p > 1. Then, for any t > 0 and for y-a.e. x, the function h - Tt f (x + h) on the space H(y) with its natural Cameron-Martin norm is continuous. The same is true if f E LP(-y, E), where E is a separable Hilbert space.
PROOF. The function y '-. f (e-tx+ 1 - e- t y) is in LP(y) for y-a.e. x. Hence it suffices to show that, for every function g E LP(-y), the function
V: h'-' f g(y+h)y(dy) x is continuous on H(-y). Writing p by the aid of the Cameron-Martin formula as D(h) =
Jgexp(i
- 2IhIH(,)) d-,,
we notice that, for any q > 1, the mapping h
exp(h
-
1h12H
with values in L9(y) is continuous on the balls in H(y). This yields the continuity of V. The vector case is analogous. 0
2.10. Measurable linear functionals Recall that a function f on a linear space is said to be affine if f = I + c, where 1 is linear and c E 1R1. By analogy one defines affine mappings with values in linear spaces.
2.10.1. Definition. Let y be a Gaussian measure on a locally convex space X. A function f on X is called a measurable linear functional (or, more precisely, -ymeasurable linear functional) on (X, y) if there exist a linear subspace L of full y-measure and a -y-measurable and linear in the usual sense function fo on L such
that f = fo y-a.e. The notion of a y-measurable acne function is defined by analogy.
Chapter 2.
80
Infinite Dimensional Distributions
Note that one can always redefine a y-measurable linear functional f : X -- IR' in such a way that fo will be linear on all of X (such a version will be called proper linear). Indeed, using any Hamel basis in X, we can extend fo to a linear function on X. Clearly, all such extensions are -y-equivalent functions. It is clear that the absolute value of a measurable proper linear functional is a measurable seminorm. Let as consider the following instructive example.
2.10.2. Example. Let -y be the countable product of the standard Gaussian measures on the real line and X = IRX. Then, for any E 12, the series E converges y-a.e., hence defines a measurable linear functional on X. However, only for finite sequences is this functional continuous.
2.10.3. Proposition. Let -y be a centered Gaussian measure on X. Any -jmeasurable linear functional f on X is a centered Gaussian random variable with variance IIfIIi=(,,)
PROOF. The random variables C(w) = f (x) and q(w) = f (y), where w = (x, y) E X x X, are independent on (X x X, yO y) and have the same distribution as f on (X. y). By linearity of f and the Gaussian property, one has
IEexp[J
] =Jfexp{iti(5-.=)] y(dx),(dy) = lEIite].
Therefore, (+ q) / f has the same distribution as 1;, which implies that t; is centered Gaussian (see Theorem 1.9.5). The following important result is a direct corollary of Theorem 2.8.3.
2.10.4. Theorem. Let y be a Gaussian measure on a locally convex space X and let 1 be a -y-measurable properly linear function on X. Then the restriction of I to the space H(y) with the norm I Ix(,) is continuous. -
The reader should be warned that in this theorem one has to deal with proper linear functions, since in the typical (infinite dimensional) situations H(y) has ymeasure zero, hence on this set measurable functions can be redefined in an arbitrary way. It is remarkable and nontrivial that. (as we shall see below) any proper linear y-measurable function is uniquely determined by its restriction to the set H(y). On the other hand, we shall see that in the infinite dimensional case there exist linear functions which are not measurable with respect to Gaussian measures.
2.10.5. Lemma. Let y be a centered Gaussian measure on a locally convex space X and let f E X; be a proper linear function such that R, f E X. Then
f(h) = (R,f,h)xt7! = Jf(x)J(x)7(dx). Vh E H(y).
(2.10.1)
PROOF. The second equality in (2.10.1) holds true by the definition of the norm in H(y). For the proof of the first one let us take a sequence { ff } C X' convergent
to f in L2(y). Let h = R,k E X, k E X;. By definition, we have k = h and
f.(h) = fn(R,k) = f f.,(x)k(x)y(dx). The right-hand side of this formula tends to the right-hand side of the equality we want to prove. Passing to a subsequence, we may assume that f, f almost
2.10.
Measurable linear functionals
81
Since the set Xo = {x: f(x) = lira is a linear subspace, nix by virtue of Theorem 2.4.7 it contains H(-y). Therefore, fn(h) - f (h) for all everywhere. h E H(-y).
2.10.6. Corollary. Let y be a centered Gaussian measure on a locally convex space X. If a sequence of -y-measurable proper linear functions f., converges to zero converge to zero in the in measure y, then the continuous linear functionals
norm of H(y)'. PROOF. By the Gaussian property, the convergence in measure implies the convergence in L2(-y). Hence the claim follows from the Cauchy-Minkowski inequality
applied to the right-hand side of (2.10.1) and the fact that IhIH(,) = 1140(,) for any h E H(y). 2.10.7. Theorem. Let y be a Gaussian measure on a locally convex space X, f and g two 1-measurable linear functionals. Then they are either distinct almost everywhere or equal almost everywhere. If y is a centered measure with R, (X') C X and f and g are proper linear, then the second of the two possibilities above takes place precisely when f = 9 on H(-y).
PROOF. We may assume that f and g are proper linear. Then L: = {f = g} is a measurable linear space. According to the zero-one law, either y(L) = 0 or y(L) = 1. In the latter case H(y) C L by Theorem 2.4.7. The last claim follows from Theorem 2.5.5.
2.10.8. Corollary. Let y be a centered Gaussian measure on a locally convex space X such that R,(X*) C X. Suppose that I is a linear y-measurable function (or a -t-measurable proper linear functional) such that it vanishes on some dense subset of the space H(y) (equipped with the norm I IHi,i). Then 1 = 0 -y-almost everywhere.
PROOF. This follows directly from Theorem 2.10.4 and Theorem 2.10.7.
2.10.9. Theorem. Let y be a centered Gaussian measure on a locally convex space X such that R, (Xy) C X X. Then the following conditions are equivalent: (i) f is a -t-measurable linear functional;
(ii) f E X4; (iii) there exists a sequence (fn) C X' convergent to f in measure. PROOF. It is clear that condition (ii) implies (iii). Let C X' converge to f in measure. There exists a subsequence { fR } (denoted by the same symbol), which converges to f y-a.e. Denote by L the set of all the points x E X such that Then L is a linear subspace of full -y-measure. Let us define there exists lim
foon Lby
nx
fo(x) = lim n
x
Thus, (iii) implies (i) (clearly, (iii) implies also (ii)).
Suppose now that f is a measurable linear functional on (X, y). We shall assume from the very beginning that f = fo, where fo is a linear functional from the definition 2.10.1. By Fernique's theorem, one has f E L2(7). According to Theorem 2.10.4, the functional f is continuous on the Hilbert space H(7). Let us choose an orthonormal basis {e0} in H(y) and denote by its part (which is
Chapter 2. Infinite Dimensional Distributions
82
at most countable), on which f does not vanish. Let en = Rkn, kn E X;. The vectors kn form an orthonormal sequence in X.Y, hence we can consider the element
x f(en)kn E X,.
g= n=1
Note that f = g on H(,), since for all e E {eQ}, not belonging to {en}, by virtue of Lemma 2.10.5 one has kn(e) = (en, e) H(,) = 0.
According to Theorem 2.10.7, f = g almost everywhere with respect to the measure -y. Thus, f E X. An analogous result holds true for measurable affine functions g. since g = f +c,
where f is a measurable linear function and c E R'. Note that without extra assumptions conditions (ii) and (iii) above are equivalent and imply (i).
2.10.10. Corollary. Let y be a centered Gaussian measure on a locally convex space X with the separable space X,*. Then there exists an orthonormal basis 1G) in X; consisting of elements of X. In addition, for any ^y-measurable linear E 12 such that functional 1, there exists a sequence
x
l(x) _ E enl;n(x) n=l
'y-a.e.
2.10.11. Theorem. Let 'y be a centered Gaussian measure on a locally convex space X such that R,(X.Y) C X. Then, for any continuous linear functional 1 on the Hilbert space H(^y), there exists a unique (up to equivalence) measurable linear functional 10 on X that coincides with l on H(-y). In addition, Itin1I Li(,) = 11111
PROOF. Let us take an arbitrary orthonormal basis {e0 } in H(y) and denote by {en} its part (which is at most countable), on which l does not vanish. Let en = R4n, where ,, E X. We can take for C,, the corresponding proper linear versions. The vectors l;n form an orthonormal sequence in the Hilbert space X;Put cn = 1(en). Then (cn) E 12. Therefore, we get the measurable linear functional 10 = E enlnn=1
Since tn(ek) = bnk, one has lo(en) = c = l(en). In addition, for any element e E {eQ}\{en} with e = Rk, where k E X,*, we have lo(eQ) = 0, since t;n(e) = (en,e)H(, = 0 by Lemma 2.10.5. Uniqueness follows from Theorem 2.10.7.
2.10.12. Corollary. Suppose that the conditions of Theorem 2.10.11 are satisfied and that {fn} is a sequence of 'y-measurable linear functionals. Then the following conditions are equivalent: (i) fn --* 0 in L2(?'); (ii) f,, 0 in measure; (iii) the restrictions of proper linear versions of the fn's to H(7) converge to zero in the norm of H(-y)'. In addition, for either of these conditions it suffices that fn - 0 on a set of positive measure.
2.11.
Stochastic integrals
83
PROOF. Clearly, (i) implies (ii). If (ii) is satisfied, then the functions fn E Xy are centered Gaussian random variables. Hence their convergence to zero in -rmeasure yields the convergence in L2(7). By Lemma 2.10.5, we get (iii). By the same lemma, IIfnIIH(y) = IIfnIIL2(y) for proper linear versions, which gives the implication (iii) = (i). Finally, the convergence of fn to zero on a set of positive measure implies, by the zero-one law, the convergence almost everywhere, since the 0 set of convergence is a measurable linear subspace.
The results of this section show that, for a centered Gaussian measure y satisfying the condition R,(X;) C X, the Cameron-Martin space H(-y) is naturally isomorphic to the dual of Xy in the sense that all continuous linear functionals on Xy have the form I i- lo(h), where h E H(y) and to is a proper linear modification of 1. With this identification, the norm of any measurable linear functional in L2(y)
is equal to the norm of the restriction of its proper linear version on the Hilbert space H(y). Measurable linear functionals on the classical Wiener space are discussed in the next section. 2.11. Stochastic integrals
Let us discuss measurable linear functionals on the Wiener space (C[0,1), PH'). Recall that for every function cp E L2(0,1), the Paley-Wiener-Zygmund stochastic integral Jcp =
JPtt)dwt 0
with respect to the Wiener process wt is defined as follows. If p is a step function of the form n
tl = 0 5 t2 5 ... 5 to+1 = 1,
cp(t) = F, Cil(t,.ii+,J, i=1
then we put 1
n
Jcp = f w(t) dwt
ci(wt,+1 - w1,). i=1
o
It is clear that Jcp is a Gaussian random variable with mean zero and variance n
E c?(t,+1 - ti) = i=1
Therefore, J has a unique extension to an isometry from L2[0,1J to the subspace in L2(Pu'), generated by all centered Gaussian random variables. Note that the Paley-Wiener-Zygmund stochastic integral cannot be defined as a Stieltjes integral, since the trajectories wt have unbounded variation almost sure. However, if we choose (C[0, 1J, Ply') as a probability space for wt, then Jcp becomes a measurable linear functional. Indeed, the functionals ci(wt,y, - wt.)
1n: w F-» i=1
Chapter 2. Infinite Dimensional Distributions
84
are continuous on C[O,1] and Jcp is obtained as their limit. Conversely, let I be a P'",-measurable linear functional. Put h = R4t'I. As shown above, h E IA o'', hence rp = h' belongs to L2 [0,11. In order to show that 1 = J
1
(Rw1, )H = (h, ip), = f h'(t)O'(t) dt = J ye(t) 0
(t) dt.
0
Note that the right-hand side of this equality coincides with
f 1 w(t)dt'(t) = Jv(' ). 0
In order to see this, notice that it is easy to deduce from the definition of J that there exists a refining sequence of partitions of the interval [0, 1] by certain points ti , ... t', such that the functions ti's-1
[tk,tk+l] k=1
converge to ;p in L2[0, 1] and
JP(w) = n-.oo lim E w(t%) [w(tk+1) - w(tn)]
Pu-a.e.
k=1
Since the domain of convergence is linear, it contains
W02''.
It is clear that on
W02.1
1
the limit equals f
(t)w'(t) dt according to our choice of tk
.
0
Let us give another proof of the fact I can be represented as a Paley-WienerZygtnund stochastic integral. We know that there is h E W02'1 [0.1] such that f (h) = f f (w)I(w) PW (d.j) for all continuous linear functionals f on CIO, 1] (or on L2[0,1] if Pµ is considered on L2[0,1)). Let us prove that
1Jw(t)dwt() Pn -a.e.
1(W) = 0 1
To this end, it suffices to show that f h'(t) dw= is the measurable linear functional 0
corresponding to h (since Rt4 (1) also equals h and Rw is injective). In turn, this will be implied by the equality t
f (h) = f [f (w) f h'(t) dwt(w)] P" ' (dw) 0
(2.11.1)
2.11.
85
Stochastic integrals
1
for all f of the form f (w) = w(r), r E [0, 1]. Noting that w, = f I[0,,, (t) dwt and 0
using that the Paley-Wiener-Zygmund integral is an isometry, we get 1 r
r [w(r)
J0
h(t) dult(w)] P") = (1[o,;, h')La(o.i; = f hi(t) dt = h(T). 0
which is the left-hand side of (2.11.1). If we deal with the space L2[0,1[, then a similar reasoning applies to the functionals f of the form w'-. (w, I[O.r))c2[0.11Note that the Paley-Wiener-Zygmund integral of f E L2[0' 11 coincides with the element I (f) in (L2(0, iJ) P,,. defined in Remark 2.3.3. Therefore, we get the following picture. Let P11 be the classical Wiener measure on C[0, 1J. Then its Cameron -Martin space coincides with the Sobolev class w2'1 = If: f is absolutely continuous, f (0) = 0 and f' E L2[0.1] }.
the measurable linear functionals are precisely the stochastic integrals
J(t)dwt.
V E L2[0.1],
0
t((
and for any h E 14'0'1, the following classical Cameron-Martin formula holds: I
1
=exp\f h'(t)dw- f 0
In the next chapter devoted to Radon measures we return to the discussion of measurable linear functionals. T
Jf(tw)dwt with re-
Let us recall the definition of Ito's stochastic integral 0
spect to the Wiener process wt for a random process f(t,w), defined on the same probability space (S2, P) as wt and satisfying the following two conditions: 1) the progressive measurability: for every t E [0, T], the function f (s, w) on 10, tJ x n is measurable with respect to B[0, t] x ft, where Ft := e({w s < t}). 2)
lEf
7,
If(t,w)12dt < oc.
0
For step functions of the form =1
where to = 0 and f (t ) is measurable with respect to .Ft,, we put
I(f)(w) = f f(t.w)dwt o
f(t,-i,w)(wt,(w)
- wt,_,(w))
'=1
7
The random variable 1(f) has mean zero and variance E f 1 f (t, w)12 dt. Hence 0 the linear mapping I extends uniquely to an isometry of the space of all processes
Chapter 2.
86
Infinite Dimensional Distributions
with the aforementioned two properties to L2(P). In a similar manner one defines stochastic integrals of the matrix-valued random processes with respect to the Wiener process {W} = {(w) , ... , wd) } in R". The stochastic integral of an Rd-valued process { qt } = { (71f 1,
...
,
ntd) } with respect to (14' } is defined as d
Tr
T
J i7t dWt = d f rte 0
dwA,
0
provided the components q,' satisfy conditions 1) and 2) above. Having defined stochastic integral, we can consider stochastic integral equations
t=40+J
J B(i.) ds,
(2.11.2)
0
0
where in the case of Rd the mapping A takes values in the space of matrices, B is Rd-valued and , o E Rd is a constant vector (or a random vector independent of 1171 1). This equation is the interpretation of the stochastic differential equation (2.11.3) d{t = A(t:t) dWt + B(t t) di. It is known that equation (2.11.2) is uniquely solvable in the class of progressively measurable processes if A and B are Lipschitzian. A detailed discussion of stochastic integrals and stochastic differential equations can be found, e.g., in the books [361], [430]. An important role in stochastic analysis is played by the following libs formula, which, in its simplest form, states that, given a diffusion process satisfying (2.11.3) satisfies and a twice continuously differentiable function y:, the process ii = the stochastic differential equation drlt =
dwt + ['PV )B(it) + 21 dt.
(2.11.4)
A similar formula is valid for multidimensional diffusions and mappings. For example, if A = I, B = 0 and f : [0, T] X Had RI is continuously differentiable in t and twice continuously differentiable in x, then a r
k=j
2
8x2dt.
(2.11.5)
See [822, Ch. 12] for proofs.
As an application of this formula, we shall obtain a stochastic representation of a function of the Wiener process, which turns out to be very efficient in the study of the distributions of nonlinear functionals on spaces with Gaussian measures. Let {14't} be a standard Wiener process in Rd. Denote by (PP)t>0 the corresponding transition semigroup given by
P, (x)=E (x+llt)= f W(x+fy)-ld(dy)=t-d12f
V:'(z)Pd(ztx)dz
Rd
Rd
for bounded measurable functions y;. Note that Pty; is infinitely differentiable if t > 0 and
VPtv(x) _
f f iIG(x+ fy)ld(dy) Rd
2.11.
87
Stochastic integrals
2.11.1. Proposition. Let T > 0 and f E W2.1(ya ), where -yd is the law I). Then, for every
of WT (the centered Gaussian measure with covariance T r E (0,T], one has
f(IV,-)=1Ef(Wr)+
J0
OPr-tf(IVt)dWt
a.e.
(2.11.6)
PROOF. Suppose first that f E Cb (IRd). Let us fix r > 0 and apply Ito's formula to the process il(t, Wt), where rt(t, x) = Pr_t f (x), 0 < t < r. We get dn(t, Wt) = OP1_tf (IVt) dWt,
2 AP,-t f (x). Therefore, by the equality rl(0, 0) = P,. f (0) _ (t, x) IEf (W,), we arrive at (2.11.6). In order to justify this equality for a wider class of functions, let us note that, for every g E W2.1 (-ydl), one has since
r
IEv_tfI2dt = 0JRdJ
Pr-tVf(x)12 yddx)dt I
r
rr
r fIVf(x +y) 12
J
t7d(dx)dt =
fIVf(z)I2i(dz)dt J
0 Rd
0 Rd Rd
r
<J
J 0 Rd
(dz)dt |
(T/r)d/2IQf(z)I2
which implies the existence of the stochastic integral in (2.11.6). Moreover, let us take a sequence of functions f j E Co (Rd) such that f; - f in W2.1 ('rd) and almost W2.1(,1,T), everywhere. Observing that IV for any p E we get the L2-convergence of the stochastic integrals of V Pr-t fj (Wt) to the stochastic integral ofVPF_tf(Wt). Since fj(Wr) f(W,) a.e. and Ef,(IV,) -+ Ef(WF), 0 we obtain (2.11.6) in the general case. Now we shall show how the Ornstein-Uhlenbeck process can be described by means of stochastic differentiable equations. Let {wt} be a standard Wiener process
in IR1 and let to be a Gaussian random variable which is defined on the same probability space as {wt} and is independent of {wt}. Let a > 0 and i3 > 0 be fixed. Then the Ornstein-Uhlenbeck process discussed in Example 2.3.15 can be obtained (with a = 1 and 0 = 1/2) as the solution of the following linear stochastic differential equation (known as the Langevin equation):
d{t = adwt -
(2.11.7)
By definition, this equation is equivalent to the integral equation tt
6 =Co+awt-/3 / C.ds. 0
Chapter 2.
88
Infinite Dimensional Distributions
If we solve (2.11.7) formally as a linear differential equation (with wf on the right), we get the following explicit representation: (2.11.8)
' d w3.
Je3+ aee
Then it is easy to see that this well-defined process is a solution to our stochastic equation (in addition, it is a unique solution). Hence lE. = e-3tlEto and e-3(t+8)IEo + a2 e-3(t+o)(a23min(t.s) _ 1 ). 2Q
(2.11.9)
In particular, cov (t r, es) = e-
(t-4-00
(E )2] +
(e2.!=n;n,t.s)
23 e
- 1)
If a = 1 and 3 = 1/2, then we get the standard Ornstein-Uhlenbeck process. If the initial distribution (i.e., the law of to) is standard Gaussian, then E£t = 0 and lEt:,t;, = e.-J'-al/2, hence we get. a stationary process. For any initial distribution. =1-e-t(IE(t
lEtt
ast
Let us now recall the concept of a Gaussian orthogonal measure and the associated stochastic integral. Let (T, B, µ) be a measurable space with a finite atom less measure it.
2.11.2. Definition. Let {w(B), B E B} be a mean zero Gaussian process indexed by B on some probability space (D, Y, P) such that
EE(w(Bi)u'(B2)) = µ(B1 0 B2),
YBi, B2 E B.
Then this process is called a Gaussian orthogonal measure on the space (T, B, µ) (or a Gaussian measure with orthogonal increments). We shall assume that. the a-field.F is generated by the random variables w(B),
B E B, and that the space H := L2(µ) is separable. The stochastic integral 1(h) for the elements h E H is defined as follows. If h is a simple function taking values
c, on disjoint measurable sets B,, i = 1.... , n, then we put n
1(h) := Ec,ur(B,). i=1
Clearly, 1c,12µ(Bi) = IhflN.
IEI.T(h) 12 =
t.1
Therefore, the mapping h - 1(h) extends uniquely to a linear isometry from H into L2(P). For any h E H, we denote by 1(h) the result of this extension. Another notation for this object is w(h). For example, let T = [0, 1], B = B([0,1[) and let µ be Lebesgue measure. Let, further, wt be the standard Wiener process. Putting 1
w(B) = IIB(t)du,,, 0
2.11.
Stochastic integrals
89
where the integral above is the Paley-Wiener-Zygmund integral, we get a Gaussian orthogonal measure on [0, 11 with Lebesgue measure. Then, for any h E H = 1
L2[0,1), the function w(h) is the integral f h(t) dwi. 0
In a similar manner one defines the multiple Ito integral 2m(fm) of a function fm E L2(Tm, p'). For example, if f2 is a simple function of two variables such that f2 takes values c, on the sets Bi = A; x C;, where A;, C, E B, then 12(f2) = F_c,w(A,)w(C,). Certainly, this can be reduced to the previous definition merely taking the space (T2, p2) instead of (T, p). However, sometimes it is useful to distinguish the number of variables of a function like f2. Let us note that the integral Zm(fm) is independent of the permutations of the variables of the function fm. This means that we can always replace the initial function fm by its symmetrization
fm(tl,... ,tm) =
m1
fm(to(i),... ,to(m)),
where the summation is taken over all permutations of {1.....m}. Then we have Zm(fm) = Zr(fm) and (2.11.10) E(zm(f)zm(g)) = m!(f,g)r.2(,,m), IE(Zm(f)Zk(g)) = 0 if m :A k. One can check that a symmetric kernel f is uniquely determined by its multiple Ito integral 2", (f ). In the case of Lebesgue measure p on [0, 1], the multiple integral of a symmetric
kernel u E L2([0,1]". p") can be written as the iterated integral
If(u)=n! if ... J u(ti....,t,)dwt,...dwt,,. 0
0
0
This is readily verified for simple integrands and then extends to arbitrary kernels. The next result can be deduced from the Wiener chaos decomposition (see Section 5.10 below; a direct proof can be found in [5701). We shall encounter multiple stochastic integrals in Section 5.10, where we discuss measurable polynomials .
2.11.3. Theorem. Suppose that F E L2(P). Then there exist symmetric kernels fn E L2(T", p") such that
x
F=IEF+ET"(fn), n=1
where the series converges in L2(P).
Using this result, one defines the Ornstein-Uhlenbeck semigroup (Tt)t>o by
x
TtF = IEF+ E e-"tIn(fn). These objects enable one to develop a large part of n=1
Gaussian analysis (including the Malliavin calculus) discussed in Chapters 5 and 6 without any reference to a "state space" X, in particular, no Gaussian measures on linear spaces are involved. In certain problems this approach might be preferable, since it avoids topological issues. On the other hand, there are a lot of problems where it is more natural and convenient to deal with Gaussian measures on linear spaces.
Chapter 2.
90
Infinite Dimensional Distributions
2.12. Complements and problems Gaussian measures on general linear spaces 2.12.1. Remark. A linear space X with a a-field E is called a measurable linear space if the operations
X x X - X, (x, y) -x + y, and 1R1 x X -. X, (t, x) tx, are measurable with respect to the a-field E on X and the Borel a-field on IR'. For example, a locally convex space X with the a-field E(X) is a measurable linear space. The property expressed in Corollary 2.2.11 can serve as a definition of a
centered Gaussian random vector with values in a measurable linear space (see [243]), which is useful for the study of random elements with values in topological
vector spaces with trivial duals (for example, L" with 0 < p < 1 or the space of all measurable functions on [0, 1] with the topology of convergence in measure). It should be noted that a Banach space with its Borel a-field may fail to be a measurable linear space (see [800, Example 1.2.5]). The following interesting example was constructed by Talagrand [753].
2.12.2. Theorem. Suppose that the continuum hypothesis (or, more generally, Martin's axiom) is true. Then: (i) There exist a centered Gaussian measure y on some Banach space X and a functional I E X' such that R,(1) l! X; (ii) There exists a Gaussian measure y on some Banach space X such that a-, does not belong to X (i.e., is not represented by an element of X). Martin's axiom (MA) can be introduced as the assumption that in every nonempty compact space satisfying the countable chain condition (i.e., every disjoint family of its open subsets is at most countable), the intersection of fewer than c open dense sets is nonempty. Note that the continuum hypothesis (CH) is equivalent to the same assumption without restricting to the compacta with the countable chain
condition. Therefore, CH implies MA. It is known that each of the axioms CH, MA, and MA-CH (Martin's axiom with the negation of the continuum hypothesis) is consistent with the usual system of axioms (ZFC), that is, if ZFC is consistent, then it remains consistent after adding any of these three axioms. 2.12.3. Remark. Let y be a Gaussian measure on a locally convex space X which is continuously and linearly embedded into a locally convex space Y. Then y can be considered as a Gaussian measure on Y. However, the reader should be warned that the Cameron-Martin spaces of ry may be different on X and on Y. For example, let us extend the measure y mentioned in Theorem 2.12.2(i) to the space Y = (X')' (the algebraic dual of X*) and equip it with the topology a(Y,X'). Then R,(1) E Y belongs to the Cameron-Martin space of the measure y on Y, although it does not belong even to X (hence neither to the CameronMartin space of y on X). Therefore, the Cameron-Martin space may increase if we increase the space (clearly, it cannot become smaller due to formula (2.4.3) of Theorem 2.4.5). This circumstance should be taken into account when using the construction of Example 2.3.10, which enables one to pass to the space 1RT.
However, it will be shown in the next chapter that for Gaussian measures on separable Frechet spaces (and, more generally, for arbitrary Radon Gaussian measures) the set H(y) does not depend on whether y is regarded on X or on Y.
2.12.
Complements and problems
91
Note also that for the validity of Theorem 2.5.2 it is not sufficient (in the case where R., (X') ¢ X) that the measurable set A satisfy the equality A+ h = A for all h E H(-y). The next example answers a question raised by H. Shimomura.
2.12.4. Example. Let -y be the centered Gaussian measure on the Banach space X from Proposition 2.12.2 assuming the continuum hypothesis. Namely, for
this measure, there exists a functional l E X' with .,(I) ¢ X. According to Proposition 2.4.6, the space H(y) is complete. Hence E = R; 1(H(y)) is a closed linear subspace in X. Let lE be the orthogonal projection of I to E in the space X; and let f = 1- IE. Put A = {x: f (x) > 0}. Then -y(A) = 1/2, although
A+h=A, VhEH(-y). PROOF. It is clear that R, (f) 54 0, hence y(A) = 1/2. In addition, A + h = A for all h E H(y), since f (h) _ (f, R; 1(h)) = 0 for h E H(y). 2.12.5. Remark. Let X be a locally convex space and let -y be any measure on £(X). The Lebesgue theorem implies the sequential continuity of the Fourier transform of -y, provided the space X' is equipped with the *-weak topology a(X', X). For a GaussianJfdv measure y, this means the sequential continuity of the linear func-
tion L: f i-
and the covariance function f '--* R.,(f)(f) on X' with the *-weak topology. In Problem 2.12.17 it is suggested to prove that, except for the
case when the measure y is concentrated on a finite dimensional subspace, the covariance function cannot be continuous in the *-weak topology on X. However, for the Radon Gaussian measures discussed in Chapter 3 both functions are continuous in the Mackey topology on X` (defined in Appendix).
The Hellinger integral and product-measures The following result enables one to get in certain cases a sufficiently precise estimate of the variation distance between equivalent Gaussian measures, since for such measures, in principle, one can evaluate the Hellinger integral. 2.12.6. Proposition. Let µ and v be two probability measures on a measurable space (11, B) and let A be a measure on B such that p. << A and v << A. Then the number
dv
H(lL, v) := f FLTdA dada does not depend on our choice of a measure \ with the aforementioned property, and the following inequalities hold true:
2[1 - H(µ, v)] < IIu - vuI < 2/l whereIIp -
dv dt,
I
H(µ, v)2,
is the variation distance. L1(A)
PROOF. Let p = pA, v = qA. Then IIµ - vJI = II p - gIIv(a) Note that (f - V, ,q)2
92
Chapter 2. Infinite Dimensional Distributions
The number H(p, v) is called the Hellinger integral of p and v. Let us recall the following Kakutani theorem 1395].
x
x
n=1
n=1
2.12.7. Theorem. For any two product-measures p = ® pn and v =
vn
vn are probability measures, the following alternative takes place: either p - v or p 1 v. In addition. M - v precisely when the following product converges: where pn
x r
HJ
Qn dvn,
n=1
where en is the density of Mn with respect to vn.
The integral involved in the formulation of the Kakutani theorem coincides with the Hellinger integral and can be written in a more symmetric form fn
gn dAn,
where A. is an arbitrary measure with An << A. vn << An, In and gn are the corresponding densities of pn and vn with respect to A. (see Proposition 2.12.6). As we have seen, for all Gaussian measures the same alternative is valid. This is not surprising, since, as it will be shown below, any Radon Gaussian measure is linearly isomorphic to a product-measure. Hence the question arises about. possible relations between Gaussian measures on 1R" and product-measures. It follows from the results given below that the existence of the aforementioned isomorphism does not mean that any Gaussian measure on IR" is equivalent to a product-measure (as it happens in case of 1Rn where every nondegenerate Gaussian measure is equivalent to the product of n one dimensional Gaussian measures).
2.12.8. Example. There exists a centered Gaussian measure p on IR" which is mutually singular with every product-measure on R. One can take for p the image of the countable product -y of the standard Gaussian measures on the real line under the mapping A: IR" R", (Ax), = x1 + n-lxn+i. PROOF. It is clear that the measure p defined above is centered Gaussian. Note
that n-lxn -. 0 y-a.e. Hence, for p-almost all y, there exists lim n-xyn. This limit
is a Gaussian random variable and is not a constant, since the coordinate functions are Gaussian random variables with respect to p and
f yn p(dy) =
J(Ax)n y(dx) = 0,
whereas
f y2 p(dy) = J(Ax)2 y(dx) = 1 + rig - 1. Suppose now that v = on 1 vn is the product of probability measures vn on 1R1. By Kolmogorov's zero-one law 1697, p. 356, Ch. IV, §1, Theorem 1 and its corollary]), for the linear space
L={xEIR": 3 lim nxxn}
one has either v(L) = 1 or v(L) = 0 (note that L E E(IR")). In the first case, the function 1: x
lira xn coincides with some constant v-almost everywhere. If n-x
v(L) = 0, then v 1 p, since p(L) = 1. If v(L) = 1, then I = c v-a.e. for some number c, hence we get again that v 1 p, since p({l = c}) = 0 as noted above. 0
2.12.
93
Complements and problems
The proof of this example (borrowed from (154]) can be simplified a bit if we use the following interesting result of Fernique [245] (see also a generalization [246]
to the products of Frtchet spaces).
2.12.9. Theorem. Let p be a Gaussian measure on R" and let v = ® v" be n
the countable product of probability measures vn on Rt equivalent to the standard Gaussian one. Then either it J. v or p v, and in the latter case the measure p is equivalent to the product ® pn of its marginals (the one dimensional projections n=l
under the coordinate mappings).
The proof is based on the fact that for a nondegenerate Gaussian measure on IR" the maximum of the Hellinger integral over all product-measures is attained on the Gaussian measure. This interesting fact was established in [245] with the aid of the central limit theorem.
2.12.10. Example. There exist smooth strictly positive probability densities x g,, on IR1 such that the product-measure v = ® ends is mutually singular with n=1
every Gaussian measure on 1R".
PROOF. We take for v the countable product of measures with some positive smooth densities en such that for some I > 0 one has
-x I<
J -x
1f(t)dt en (t)
dt < 21,
> I -`2-n. -1
It is readily verified that this is possible. According to Problem 2.12.25, we have vh - v for all h E 12. Let p be a Gaussian measure on Rx. If p is not singular with respect to v. then 12 C H(p), since otherwise ph I p for some It E 12, whence it follows that v has no component absolutely continuous with respect top. Note also that v(11) = 1, which implies p(1") = 1 by the zero-one law. Let a be the mean of p and po = p_a. Then 1LO is a centered Gaussian measure with H(po) = H(p). It is clear that µe(1") = 1 (this follows, e.g., from Theorem 2.5.8). Denote by y the countable product of the standard Gaussian measures on Rt. We know that H(y) = 12. According to Theorem 3.3.4 of the next chapter, the containment 12 C H(po) and the equality po(l'°) = I yield y(l') = 1, which is false. Instead of using the measure 7 one could apply the previous theorem.
Oscillation constants In many problems one has to deal with random variables of the form a(l:) = lim sup I
n-x
I.
where { _ (E") is a sequence of random variables. If a(l;) is a constant (possibly, +x), then it is called the oscillation constant of t;. A proof of the next result and an interesting discussion of the oscillation constants can be found in [119, Ch. 8]. 2.12.11. Theorem. Let 1; = (1;") be a sequence of centered Gaussian random variables with variances on tending to zero. Then it has oscillation constant o
Chapter 2. Infinite Dimensional Distributions
94
(possibly, infinite). If, in addition, sup 1fn1 < oe a.s., then for almost all w, the set n
of all cluster points of the sequence {1;,,(w)} coincides with [-a. a].
Miscellaneous remarks Finally, it is worth noting that in some problems it is useful to consider exten) sions of the Ornstein-Uhlenbeck semigroup and the Gaussian measures to the complex plane (respectively, in t and in A). In the first case we get the Fourier-Wiener transform, which can be written formally as y(A-112.
T-,,,/2
.F(f)(s)
Jf(ix + / y)'Y(dy)
x Note that if f is a polynomial of degree n and t is a complex number, then Tt f can
be defined as E e-ktlk(f). In particular, for t = -i7r/2, we get. k=O n
F(f)
Erklk(f) k=0
which shows that .F extends to a unitary operator on the complex space L2(-Y). The Fourier-Wiener transform on L'(-yn) is unitary isomorphic to the Fourier transform on L2 (RI) (provided the latter is defined with the normalization making it unitary). For a related discussion, see [1411, (142), [178], 1337], (339], [381], [4461, [678). In the second case the analytic continuation of the integrals
A -. I f(ix) -t(d,) .e
to the point i enables one to define the Feynman integral for some classes of functions f. For example, if f = e`1,1 E X`, then the foregoing integral equals j(\/ 'X1) exp(_Ao(1)2/2), and the result of the analytic continuation gives exp(-io(1)2/2), i.e., "Feynman measure" can be regarded as a "complex cylindrical measure" with the Fourier transform exp(-iQ), where Q is a nonnegative quadratic form. Concerning this and other approaches to Feynman integrals, see 113], [1341, (146], (1781, [256], [291], (403], [7031, 17171.
Problems 2.12.12. Prove that the convex and absolutely convex hulls of a cylindrical set with a compact base in a locally convex space are closed. Hint: use that the convex hull of a compact set in R" is compact.
2.12.13. Show that no single-point set belongs to F(RT) if T is uncountable.
2.12.14. a) Show that there exist Borel sets A and B in R', such that the set A + B is not Borel. Hint: see [718). b) Show that there exist closed sets A, B C R" such that
A + B is not Borel. Hint: take a closed set S in the hyperplane L = {x, = 0} C R" and a continuous function f : S -+ R' such that f (S) is not Borel: note that the set A = {(f(x),x), x E S} is closed and A+L is not Borel, since (A + L)nR'ei = f(S)e,, where el _ (1,0,0....). 2.12.15. Let µ be a countably additive function on e(X) such that p o l ' is a Gaussian measure for every 1 E X. Then µ is a probability (hence Gaussian) measure.
2.12.
Complements and problems
95
2.12.16. Prove that the functions mentioned in Example 2.3.15 are indeed covariance functions, i.e., correspond to nonnegative quadratic forms.
2.12.17. Show that if the covariance function of a Gaussian measure - on a locally convex space X is continuous in the s-weak topology on X', then the measure iy is concentrated on a finite dimensional subspace. Hint: note that a seminorm q bounded on the set {x: I f,(x)I < 1, f, E X*. i = 1,... , n} has the form go(fl.... , f"), where qo is a seminorm on R". 2.12.18. Let -y on RT be an uncountable product of T copies of the standard Gaussian measures on the line. Show that H(-y) = R, (X,) = 12(T) as a Hilbert space (12(T) is the space of all mappings h : T - R' such that h(t) does not vanish for at most countably many t and E, h(t)2 < oc). Note that H(y) is nonseparable.
2.12.19. Let (T, m) be a measurable space and p a Radon Gaussian measure on L2(m) with the covariance operator K. Show that VT is given by a symmetric kernel Q E L2(T2, m®m) and Ih(t)12 <- IhIH(,.)
JZ_ IQ(t, s)I2 m(om),
Yh
E H(p).
Hint: use Lemma A.2.13 in Appendix and the Cauchy inequality applied to the representation h(t) = J Q(t, s)y(s) m(ds), where IIYIIL2(m) = IhIH(µ). 1
2.12.20. Let H°(0, 11 be the space of all functions x on 10, 1] that satisfy the Holder
condition of order a > 0, i.e., Jx(t) - x(s)J < C(x)It - sl°. Show that Pw(H°(0.1]) = I if a < 1/2 and PW'(H' [0, 1]) = 0 if a > 1/2. Hint: use Theorem A.3.22 and the equality f Jx(t) - x(s)I2m Pu'(dx) = (2m - 1)!! It - sl- for any m E !N; note, in addition, that Ix(t,) - x(s,)I/ It, - s,l are independent standard Gaussian random variables for disjoint closed intervals
2.12.21. Show that two probability measures p and v on a o-field B are mutually singular precisely when JIp - vII = 2. An equivalent condition: H(p, v) = 0. 2.12.22. Prove the separability of the process q, from the proof of Proposition 2.6.5. 2.12.23. A measure p on P(X) is called continuous along a vector h (V. A. Romanov)
if lim IIMth - pII = 0. Show that if a measure p is continuous along h, then so is any measure v absolutely continuous with respect to p. Hint: note that this claim is true for the measures of the form IBM and extend it to the measures of the form f p, f E L'(p).
2.12.24. Let f be a smooth probability density on R" with the integrable partial derivatives. Prove the estimate
f J If (x + h) - f (x)I dx < 118h f (x)I dx,
Deduce the estimate
V h E R".
J lahf(x)i2dX1/2,
JIf(x+h)-f(x)Idx<`J
f(x)
J
VhER",
in the case where the last integral is finite. 2.12.25. Let p be a smooth probability density on the real line with the finite Fisher information I = f p (t)2p(t)-' dt < oo. Denote by v the countable product of the measures with density p on the real line. Prove that, for any h E 12, one has lim JJveh - vIJ = 0. If Q > 0 almost everywhere, then vh - v for all h E 12. Hint: use Problem 2.12.24; see (686).
Chapter 2.
96
Infinite Dimensional Distributions
2.12.26. Show that the product A of countably many copies of Lebesgue measure on (0, 11 considered on R" is mutually singular with every Gaussian measure on R. Hint: note that there exist probability measures v on R' with strictly positive densities such x that A << v vn; if there is a Gaussian measure p which is not mutually singular n=1
with A. then, by Theorem 2.12.9, v is equivalent to p, and, moreover. is equivalent to the product ® pn of the one dimensional projections pn of p: letting o,2, be the variance of n=1
X
A show that °n
0 (note that otherwise ®
0) and get a contradiction
using that A( 11 (an.bn]) = 0 whenever 16n - ant -- 0. n=1
2.12.27. Let { } be a sequence of the standard Gaussian random variables (possibly, dependent). Show that if series (2.3.4) converges for some > 0. then the sequence ju.t. }
is bounded a.s.; if this series converges for all c > 0, then °nt -. 0 a.s. Hint: use the Borel-Cantelli lemma and the asymptotic of large deviations for normal distribution; see (800, Ch. V. Proposition 5.6]. 2.12.28. Let (r;.) be a sequence of independent centered Gaussian random variables with variances °n 0. Show that the oscillation constant o of this sequence coincides with
inf E>0:
vnexp
c2
<x1:
n-1
construct an example where a = 1. Hint: see 1491. §7].
2.12.29. Let ry be the countable product of the standard Gaussian measures on R'. = 1 for Y-a.e. x.
Prove that limsupk, 1xk]/
2.12.30. Let I be the countable product of the standard Gaussian measures on R' and let f be a polynomial on R'. Prove that
lim nx n-' k=1 f (xk) =
J
f (t)p(t. 0.1) dt
for Y-a.e. x.
2.12.31. A centered Gaussian process f on R' is called stationary if its covariance function has the form K(s, t) = Kn(t - s). Let Ko(t) = f e:' °(dx). where a is a finite nonnegative measure on R' (called the spectral measure of the process). Prove that for any closed interval I, the distribution of process (&,)tE1. is defined on L2(I) in a natural way and evaluate its Cameron-Martin space. Hint: see (359].
2.12.32. Let P't the Wiener measure on C]0,1]. Show that a measurable linear functional h generated by an element h E W;)2'' 10, 1] coincides a.e. with a continuous linear
functional on 010,1] precisely when h' is a function of bounded variation. Hint: notice that any function h E A""'[0. 11 whose derivative has bounded variation can be written in the form h(t) = f v[0, a] ds, where v is a signed Borel measure on (0. 1]: consider the n
functional x -- - f x(t) v(dt) + x(1)v[0,1]; notice, in addition, that the restriction of h to
1[01] has the form w is a bounded measure.
f h'(t)yV'(t)dt. and verify that the generalized derivative h' 0
CHAPTER 3
Radon Gaussian Measures I believe that the attentive reader, regretting, perhaps, that the things are not so simple, will agree, however, with me that this definition is necessary and natural... It is only the ingrained habits of thinking that force us to consider it to be more complicated. H. Lebesgue. Lecons sur ('integration et la recherche des fonctions primitives Certainly, such definitions are scientific. they are necessary. they are inevitable. they have working value, but is it not peeping out from behind them the last efforts of aging reason, beyond the limits of which is the beginning of that which we cannot imagine and which has no name yet?
N. Lusin. From a letter to V. Vernadskii
3.1. Radon measures Recall that a (finite nonnegative) measure p defined on the Borel er-field B(X ) of a topological space X is called Radon if, for every B E B(X) and every e > 0, there exists a compact set Ke C B with p(B\KE) < e. A measure u is called tight if this condition is satisfied for B = X. Some additional information concerning Radon measures can be found in Appendix. Note that a Radon measure on a locally convex space is uniquely determined by its restriction to the a-field E(X) generated by X' (in particular, is uniquely determined by its Fourier transform) and that on many important for applications spaces X all measures on E(X) are Radon. This class of spaces contains all complete separable metrizable locally convex space (and, more generally, all Souslin locally convex spaces). It is easily verified that the image of a Radon measure under a continuous mapping is again a Radon measure. Recall
that if p is a measure, then p stands for the corresponding outer measure. If p is a Radon measure on a topological space X, then B(X) stands for the Lebesgue completion of the Borel o-field B(X) with respect to p. 3.1.1. Definition. A Radon measure i on a locally convex space X is called a Radon Gaussian measure if its restriction to E(X) is a Gaussian measure.
One of the principal results proven below states that any Radon Gaussian measure is in fact concentrated on a countable union of metrizable compact sets. On the other hand, there exist natural examples of Gaussian measures without Radon extensions (the product of the continuum of one dimensional normal distributions,
see Example 3.1.2). In Example 2.3.9(ii) the measure pt may fail to have Radon extensions to IRT. Now we consider a simple example of this kind. 97
Chapter 3. Radon pleasures
98
3.1.2. Example. Let a measure y on R` = RIO-11 be defined as the product of the continuum of the standard one dimensional Gaussian measures (this corresponds to independent random variables 1t in Example 2.3.9). Then y'(K) = 0 for any compact set K C R`. PROOF. Note that no nonempty compact set in lR belongs to since by virtue of Lemma 2.1.2 every nonempty set from E(W) depends on the values of the functions in this set on some countable set and for this reason contains a nontrivial linear subspace. Every compact set K C 1 is contained in a product of some collection of closed intervals [-Ct, Ct]. One can find a number N such that Ct < N for infinitely many indices t. Hence K is covered by the set Ill x 172, where nl is the product of countably many copies of the closed interval [-N. N] and 172 is the product of the real lines corresponding to the remaining coordinates. Then
II1x112EC(X)andy(I71x112)=0. 3.1.3. Remark. Let us note a delicate point relating to the measurability, which arises when we consider a Radon measure p on C(X ). Despite that, for every
set B E B(X ), there is a set C E E(X) with p(B 0 C) = 0, this does not mean that B(X) is contained in the Lebesgue completion of C(X) with respect top. The point is that, for a Borel set of p-measure zero, there might be no set from E(X) containing
it and possessing p-measure zero. A simple example of this kind is given below. Therefore, the Lebesgue completion of E(X) and B(X) with respect top may not coincide even for a Radon measure p. Everywhere below, unless the otherwise is explicitly stated, by p-measurable sets for a Radon Gaussian measure p we mean the sets from 13(X), (i.e., from the Lebesgue completion of B(X )). Fortunately, in most of the spaces encountered in applications (e.g., separable metric or Souslin) such a problem does not arise, since the Borel o-fields in such spaces coincide with the cylindrical o-fields.
3.1.4. Example. Let T be an uncountable set and let 6 be the Dirac measure at the origin in the space IRT considered on C(RT). Then this measure extends obviously to a Radon measure, however, the point 0 does not belong to the Lebesgue completion of C(RT) with respect to the measure 6.
PROOF. As noted above, the point 0 does not belong to E(X). Hence the only
set from E(R') containing RT\{0} is RT, whence 6'(Rr\{0}) = 1. Therefore, {0} does not belong to the Lebesgue completion of C(RT) with respect to 6.
3.1.5. Example. Let Pty' be the Wiener measure on Then the set C[O,1[ does not belong to the Lebesgue completion of E(lR °'`) with respect to Pa'. PROOF, The same reasoning as in the previous example applies.
We know from Chapter 2 that the product of two Radon Gaussian measures p and v on locally convex spaces X and Y, respectively, is a Gaussian measure on C(X x Y). As explained in Appendix, pov admits a unique Radon extension to B(X x Y). Below by the product of Radon Gaussian measures we mean this extension (which is a Radon Gaussian measure). In particular, if X = Y, then the convolution p * v turns out to be a Radon Gaussian measure (as the image of µov under a continuous linear mapping).
3.1.
Radon measures
99
3.1.6. Example. Let p and v be two Radon Gaussian measures on a locally convex space X. Then y = p * v is a Radon Gaussian measure on X and
ai = a + a,,. R., = Rµ + PROOF. The fact that the measure y is Radon follows from its definition as the image of the Radon measure p®,v under the continuous mapping (x,y) - x+y (see Appendix). Evaluating the Fourier transform of y by the change of variables formula, we get
y(f) = exp(iaµ(f) - 2R,.(f)(f)) exp(ia.(f) - 2& (f)(f)) for every f E X', whence the desired conclusion. Certainly, in order to get (3.1.1), it would be enough to evaluate the integrals of f and fg, for any f, g E X*, with respect to y.
x 3.1.7. Example. The product ® p, of any sequence of Radon Gaussian n-1
measures µn on locally convex spaces X. extends uniquely to a Radon Gaussian measure p on [I X,,. In addition, a, H(µ), and Xu are described in the same n=1
way as in Example 2.3.8. PROOF. Follows from Example 2.3.8 combined with the existence of a Radon extensions of countable products of Radon measures (see Appendix).
3.1.8. Remark. Two Radon measures p and v on a locally convex space X are equivalent precisely when so are their restrictions to E(X), since, for any B E 8(X),
there exists a set C E E(X) such that (p + v)(B A C) = 0. In particular, if 7 is a Gaussian Radon measure and h E X, then the equivalence of -y and yt, on 8(X) is exactly their equivalence on E(X). This obvious observation implies that Theorem 2.4.5 characterizing the Cameron-Martin space H(y) as the set {h E X : yi, - y}
remains valid if we consider Radon Gaussian measures on B(X).
We following result applies to most of the spaces encountered in applications (e.g., to all Souslin spaces, in particular, to all separable Banach spaces). Moreover, we shall see below that every Radon Gaussian measure in concentrated on a (Souslin) subspace to which this result applies.
3.1.9. Proposition. Let y be a centered Radon Gaussian measure on a locally convex space X. Suppose that there exists a sequence {ff} C X- separating the points in X. Then X. coincides with the closure of the linear span of {fn} in the space LZ(y).
PROOF. Suppose that a nonzero element f E X; does not belong to the closure
of {f,,}. Then there exists a nonzero element g E X. orthogonal to all fn, i.e., independent of the o-field generated by {f,,}. Note that there is a sequence of polynomials Fn on lRn such that the functions Gn: = Fn(f1,.... fn) converge to g in -y-measure. Indeed, given F > 0, there is gE E X' such that
y(x: I9(x) - 9F(x)I > e) <
C.
Chapter 3. Radon Measures
100
Further, there is a compact set K, C {x: I9(x) -gf(x)I < E) with y(K_) > 1 - e. The algebra of functions of the form F(f1,... , where F is a polynomial on 1R", separates the points in K,. Hence by the Stone-Weierstrass theorem, there exist n E P and a polynomial F on lR' such that I F(f l , ... , f") -g. I < c on K.. whence, letting C: = F(f1.... , f"), we get ry(x: IG(a) -g,(x)I > 2e) < e. It remains to note that, by the independence, we have "li
r
xX
dy=
Jei9d.r JeG d'Y = 7(g)2. X
X
On the other hand, the left-hand side converges to 1, whence a(g)2 = 0, which is a contradiction.
Notation. Throughout this chapter we employ the same notation as in the preceding one. In particular, H(-y) is the Cameron-Martin space of a Radon Gaussian measure -y, - IH(,) is the natural Hilbert norm in H(-y), U stands for the closed I
unit ball in H(y). The symbols X,,*, R.,, a,, yh, h, and a(f) retain their meanings as well.
3.2. Basic properties of Radon Gaussian measures 3.2.1. Lemma. Let X be a complete locally convex space and let ,7 be a Radon Gaussian measure on X. Then the functions a,: f i--+ a, (f) and f F-+ R., (9) (f ) i.e., the for any g E X are continuous on X" in the Mackey topology topology of the uniform convergence on weakly compact convex sets in X.
PROOF. Let 6 > 0. There exists a compact set K such that j (K) > 1 - 6. By virtue of Proposition A.1.6 in Appendix, the closed convex hull Q of the set K is compact (hence weakly compact). Then, the condition If I < 6 on Q implies the estimate
I1-'Y(f)15f I1-e=fIdy+26<36. Q
Put 7(f) = reie, where r > 0, 0 E [0, n[. Then I1 - rI 36 and cos 0 > (1 + r2 - 962) (2r) -'. Hence, for every e > 0, there is 6 > 0 such that Ia., (f ) I e and a(f )2 < e whenever sup4 I f I < 6. In addition, IR,(f)(g)I < a(f)o(g) < a(g)/. By definition, this
means the continuity in the topology rtit(X+,X).
It will be shown below that the conclusion of this lemma remains valid for arbitrary locally convex spaces.
3.2.2. Lemma. Let y be a Radon Gaussian measure on a locally convex space X which is continuously and linearly embedded into a locally convex space Y. Then
the set H(y) is independent of whether y is considered on X or on Y. PROOF. Denote by yb' the measure y considered on Y. Clearly, it is a Radon Gaussian measure. Then H(y) C H(yY) by virtue of equality (2.4.3). Assume that h E H(y'). Since the compact sets from X are compact in Y, the set X is measurable with respect to y1' and has full measure. Then it contains h, since it is
a linear subspace and yh - -t". Therefore, y,, - y, whence h E H(y).
3.2.
Basic properties
101
Recall that for Gaussian measures which are not Radon the situation is different (see Remark 2.12.3).
3.2.3. Theorem. Let y be a Radon Gaussian measure on a locally convex space X. Then its mean is an element of X and R, (X.;) C X. In addition. H(y) = R, (X,) = {h E X : yh r 7} = {h E X : jhlHi, < x Finally, for any f E X', one has
Rr(f)(f) _ f(f(x) -
a,(f))2
y(dx) = sup f(h)2.
(3.2.1)
hEl.'N
PROOF. If X is complete, then, according to Lemma 3.2.1, for any g C X,*, the
functions a, and R, (g) are continuous on X' in the Mackey topology r,1,(X', X), hence they are represented by elements of X (see Theorem A.1.1 in Appendix). In the general case this means that _the claims to be proved are valid for the measure y considered on the completion X of the space X (see Appendix). This measure is
Radon on X. Note that X' = X' (since X is dense in k) and X.; = X;. Let us consider the symmetrization y' of the measure y defined by the formula
y'(A)_(Y* yi)(
A),
where y1 (A) = y(-A). Recall that the convolution of two Radon measures µ and v is defined by the formula I, * v(B) =
Jti(B - x) v(dx).
In this case j v = µ`i (see Appendix). The Fourier transform of the measure as one readily verifies, has the form
y'(f) = exP(-2a(f)2) =17(f)I Therefore, y' is a Radon Gaussian measure with zero mean. Considering it on X, It follows from the we get that -1 = ya, where a E X (note that i (l) = equality y(X) = 7"(X) = 1 that a E X.
In order to prove the inclusion R,(g) E X for g E X', let us consider the measure p = exp(g
- 12o(g)2) y. According to Proposition 2.4.2, this measure is
Gaussian with mean 6 = a, + R.,(g). As shown above, b c- X, whence the claim follows. Equality (3.2.1) follows from what has already been proved by Lemma 2.4.1.
0 According to Remark 3.11.20 below, the previous theorem may be invalid for non Radon measures.
3.2.4. Corollary. Let y be a Radon Gaussian measure on a locally convex space X. Then the closed unit ball U from H(y) is compact in X. PROOF. The claim follows from Proposition 2.4.6, since, by virtue of Corol-
lary 2.4.9, for some c > 0, the set cU is contained in the compact set K - K, where K is any compact set of positive measure.
0
Chapter 3.
102
Radon Measures
3.2.5. Corollary. The Fourier transform and covariance of a Radon Caussian measure y are continuous on X` with the Mackey topology -r-11 (X', X).
PROOF. The continuity in the Mackey topology follows from (3.2.1) and the fact that. U is a convex compact set in X.
3.2.6. Corollary. Let u and v be two centered Radon Gaussian measures on for all a locally convex space X such that H(u) = H(v) and I hI H(.u) = h E H(p). Then u = v. PROOF. For every f E X', one has by (3.2.1)
RN(f)(f) _ sup f(h) 2, hEU
where U is the unit ball in the Hilbert space H = H(u) = H(v) with the norm IH(,.). The same is true for
Therefore,
whence
µ(f)=i(f)and u=v. 3.2.7. Theorem. Let - be a Radon Gaussian measure on a locally convex space X. Then the Hilbert spaces Xy and H(-y) are separable. PROOF. We may assume that the measure y is centered. Let K be a compact
set of positive 1-measure. Note that X' = Un 1 nK°, where
K° = If E X': sup If (z)] < 1}. ze K
Therefore, it suffices to prove the separability of K° in L2(y). By induction, one can find a sequence { f,,) C K° such that, setting Xn = span(f1e... ,fn), Xo = 0, we get 1
dist(fn, Xn_)) > dist(K°, Xn_)) = d,,, where dist(A, B) = SuPQEAinfbEBlla - blIL2(,)-
Since the finite dimensional spaces X. are separable, then the separability of K° follows from the relationship lim do = 0, n-.oc
which we now prove. Suppose that the reasoning is analogous). Then
0 (in the case
oc
dist(fn,X,,_1)>d>0, Vn. This means that the Gaussian vector 1.8.8. Hence we have the inequality
y(x: supIf,(x)I < 1 i
satisfies the condition in Lemma
\ 1<
(Jp(x,od2)dx),
II
whose right-hand side tends to zero as n tends to\infinity. Therefore,
(x: suIfn(x)I n
1I 111
0.
3.2.
Basic properties
103
This contradicts the relationship y(K) > 0, since, for any x from K and any n, one has 1, by virtue of the inclusion f E K°.
3.2.8. Corollary. Let y be a Radon Gaussian measure on a locally convex space X. Then: (i) The space L2(y) is separable; (ii) If y is a centered measure, then X.; has a countable orthonormal basis, consisting of continuous linear functionals (iii) Let be a sequence from (ii). Every -y-measurable set coincides with some set from (({ up to a set of measure zero. PROOF. Claim (ii) follows from the density of X' in XZ. Since the linear span of the functions exp(i f ), f E X', is dense in the complex space L2(y) by Corollary A.3.13 in Appendix, we get the separability of L2(y) and claim (iii).
It follows from what was proved that, choosing an orthonormal basis {e.) in H(-y), we get an orthonormal basis in L2(-Y), formed by the Hermite polynomials Hi,.....im: k,.....km = Hk, (ei,) ... Hk (eim ), 7t E V, m = 0, 1, .. .
3.2.9. Remark. (i) Note that for any uncountable product of the one dimensional standard Gaussian measures defined on IR7 the space H(y) coincides with R7(X.) and is nonseparable, since, according to Problem 2.12.18, it equals 12(T). (ii) It should be noted that there exists a centered Gaussian measure which is not Radon, but has the separable L2 and the separable Cameron-Martin space (see Example 3.6.3 below). In the following theorem, for the convenience of later references, we collect several results that follow from the analogous theorems of the preceding chapter (or are proved by analogous arguments), but, on the other hand, are slightly more general
from the formal point of view, due to the fact that the concept of measurability has been extended (as explained above).
3.2.10. Theorem. Let y be a Radon Gaussian measure on a locally convex space X. Then: (i) The restriction of any y-measurable seminorm q to the space H(y) with the norm I IH(,) is continuous and the conclusion of Theorem 2.8.5 is valid, i.e.,
Jexp(oq)d.v < I(c,r) < oc, where a. c, r are related as described in
the theorem cited;
(ii) If a set A E 8(X )., is such that, for any rational r and any n E IN, one has A = A + re up to a set of measure zero, is some orthonormal basis in H(y). then either -y(A) = 0 or where y(A) = 1. In particular, this is true for any affine subspace from 8(X).. (iii) Any Radon Gaussian measureµ on X is either equivalent toy or is mutually singular with y.
3.2.11. Corollary. If a -y-measurable function f is such that f (x + f (x) -y-a. e. for all rational r and all vectors e from some orthonormal basis in H(y). then f coincides a.e. with some constant.
Chapter 3.
104
Radon Measures
PROOF. It suffices to note that, for every c E III', the set A = {x: f(x) < c} satisfies the conditions of the previous theorem.
3.2.12. Corollary. Let y be the same measure as in the theorem above. Then any affine y-measurable function f has the form f = 1+c. where I E X,' and C E JR' .
In particular, f is a Gaussian random variable with mean c and variance o(()2. PROOF. It suffices to consider the case, where y is a centered measure and f is linear. Let {en) be an orthonormal basis in H(-y). By the continuity of f on H(-Y), the sequence c = f (en) is an element of 12. Hence the functional X
I=FGnenEX' n=1
is well-defined. Since the function g = f -I has the property g(x + h) = g(x) for all h E H(y), then it equals a.e. some constant c. Since g is linear and y is symmetric, we have c = 0.
3.2.13. Corollary. Let y be a Radon Gaussian measure on a locally convex space X and let f and g be two y-measurable linear functions. Then they are either almost everywhere distinct or almost everywhere coincide. The latter is equivalent to the equality f = g on H(y), provided y is centered, and f and g are proper linear.
3.2.14. Corollary. Let y be a Radon Gaussian measure on a locally convex space X and let X0 be a Souslin locally convex space continuously embedded into X and having positive measure. Then the restriction of the measure y to Xo is a Radon Gaussian measure on X0. 1 PROOF. We may assume that the measure -y is centered. Note that by virtue of the zero-one law. Since X0 is Souslin, all Borel subsets of X0 are
y-measurable (see Theorem A.3.15 in Appendix) and the restriction of -y to X0 is a Radon measure. Hence every continuous linear functional f on X0 is a -y-measurable linear functional on X. Now Corollary 3.2.12 applies.
We shall see below in Example 3.11.10 that the continuity of a linear function 1 on H(y) does not imply its measurability. According to Problem 3.11.31, any y-measurable function that is additive on a full measure additive subgroup is an element of X;. A more general fact will be established in Proposition 3.11.8.
3.3. Gaussian covariances In this section, we investigate the majorization R,,(f)(f) < cR (f) (f) for Gaussian measures v and p: this majorization will be a powerful tool in our further studies. In addition, we discuss factorization of covariance operators. Below we shall make use of the following important result giving a sufficient condition for the existence of a Gaussian measure with a given covariance.
3.3.1. Theorem. Let X be a locally convex space. let C C X' be a linear subspace separating the points in X, and let V and IF be two nonnegative quadratic
forms on G. Suppose that the function g- exp(-2 V(g)) on G coincides with the Fourier transform of a Radon Gaussian measure y on X and W(g) 5 V(g). dg E G.
3.3.
Gaussian covariances
105
Then there exists a Radon Gaussian measure v on X whose Fourier transform
exp(-2li'(9)).
coincides on G with the function 9
PROOF. On the algebra of the cylindrical sets of the form
C = {x: (91(x),... 9n(x)) E B}. g, E G. B E B(Htn), n E IN. we define the function v(C) = vn(B), where v is the Gaussian measure on Yin with the Fourier transform / //
y -. expl -214'1
yi9,))
Let us verify that the function v is well-defined and additive. Note first that if 91.... , gn E X ` are linearly independent and C = (91,... .90 _1(B,) = (91.... ,gn)-t(B2). where B1. B2 C Yin, then B, = B2. Suppose that 9n+1. - , gn+m E X' are linear combinations of g l , //... , g,, and that
C = (gi.... ,gn)-(B1) = (91.... .g,,)-1(B2), Then ( 9 1 ,
B, E B(Ytn), B2 E B(IRn
, gn+,n) = T o (9i .... , gn) for some linear operator T : Yin y En+m.
whence C = (91,... ,9n+n,)-'(B2) = (9) .....gn)-1(T-1(B2)). Hence vn(Bl) = by the change of variables formula, i.e., v(C) is well-defined in this case. Finally, let C = (91,... ,9t,) 1181) = (ft.....fk) 1182),
where f, E X', B1 E B(Rn) and B2 E B(Itk). As shown above, we may assume that 9 1 , . . .g , are linearly independent and that fl,... fk are linearly independent. If n = k, then the claim follows by the reasoning above. Suppose that k > n. Let r < k be the maximal number with the following property: there exist r elements in If,.... . f k } such that adding these elements to g, j... , gn , one gets a linearly independent. system. We may assume that these are fl.... , fr. Then, according to what has already been proved. we have (g,
g.. fl,... ,fr)_1(B1 xII') _ (gl,....gn)-(81) = (gl,... ,g., ft.... . fn)-1(B2) = (91.....gn. fl.... . fr)_1(B3)
for some B3 E
Rn+r
Hence B1 x IR' = B3, whence we conclude that two different
representations of C again lead to the equal values for V(C). The additivity of v is obvious. Let us show that it satisfies the condition in Theorem A.3.19 in Appendix. Let E > 0. Let as find a compact set K such that -r(K) > 1 - e. The set S = (K - K)/2 is compact. Hence it suffices to establish the estimate v(C) > I - 2e for any cylindrical set C containing S. Let gi E G, i = 1, .... k. We shall show that vk(T(S)) > 1-2E, where T: X Yak. Tx = (gl(x).... ,gk(x)). Put p = foT-
Then µ(y) = exp(-ZB(y)where y= (y,,.... yk) and k
B(y) = V 1
Ui9i > l["(EY.9iJ.
Chapter 3. Radon Measures
106
Hence there exists a symmetric Gaussian measure a on IRk with µ = vk * a. For some a E IRk one has
µ(T(K)) = vk * a(T(K)) < vk(T(K)
- a),
since the integral of the right-hand side in a with respect to or is the left-hand side. Now, by virtue of the easily verified relationship
(T(K)
- a) n (-T(K) + a) C
we get
[T(K)
- T(K)]
= T(S)
2
vk(T(S)) > vk{(T(K)
- a) n (-T(K) +a)}
> vk (T(K) - a) + vk (-T(K) + a) - 1 = 2vk (T(K) - a) - 1
>2µ(T(K))-1>1- 2E. Therefore, by Theorem A.3.19 in Appendix, v extends uniquely to a Radon measure.
0 3.3.2. Corollary. If the convolution of Gaussian measures p and v is tight and the measure µ is symmetric, then µ and v are tight.
PROOF. Let Q and Q be the covariances of µ and v, respectively. The convolution of the measure µ * v with its image under the mapping x -. -x is a tight centered Gaussian measure having the covariance
Hence µ is tight
by the previous theorem. Let K be a compact set such that µ' (K) > 1 - e and (µ * v)'(K) > 1 - E. Put S = K - K. Since S is compact, it suffices to be shown that v' (S) > 1 - 2E. Let C be a cylindrical set with C n S = 0. We shall show that v(C) < 2E. There exists a cylindrical set E such that K C E and C n (E - E) = 0. If v(C) > 2E, then v(E - E) < 1 - 2E, whence
µ*v(E)=fv(E -x)p(dx) e+J v(E-x)p(dx)<e+v(E-E)<1-e, X
E
which is a contradiction.
3.3.3. Corollary. Let y be a Radon Gaussian measure on a locally convex space X and let H(y) be decomposed into the direct sum of its closed orthogonal
subspaces H, and H2. Then y = y, * y2, where yi and y2 are Radon Gaussian measures with the Cameron-Martin spaces H, and H2 such that a + a-fl = a,. PROOF. We may assume that -y is centered. Let us denote by 7r, the orthogonal projection to the subspace H,. By the previous theorem, there exist centered Radon 1rryR,l1H and l Gaussian measures y, and y2 with covariances 1 17r2R.rlI2H, respectively, since these quadratic forms are majorized by R.,(l)(l) = IR.,112 . Clearly, one has y, * 12 = y.
3.3.4. Theorem. Let µ and v be two centered Radon Gaussian measures on a locally convex space X. Then the following conditions are equivalent: (i) There exists c > 0 such that V f E X';
(ii) H(v) C H(p): (iii) v(L) = 1 for every v-measurable linear subspace L of full p-measure.
3.3.
Gaussian covariances
107
PROOF. It is clear from the definition of the Cameron-Martin norm that the estimate in (i) yields the estimate clhIH(.) for all h E X and the containment in (ii). Conversely, if H(v) C H(p), then, by the closed graph theorem, there for all h E H(v). The estimate in (i) follows is c > 0 such that IhIH(,.) < from equality (3.2.1) for u and v in place of -y. i.e., sup{ f (h)2: IhIH(,,) < 1},
f EX*.
(3.3.1)
Suppose that H(v) C H(p). By the closed graph theorem, there exists a constant C such that, for any h E H(v), the estimate IhIH(,,) < holds true. Replacing the measure p by its homotetic image, we may assume that C = 1. Let L be a vmeasurable linear subspace with µ(L) = 1. We shall consider s as a Radon measure on L with the induced topology. The Fourier transform of the measure it at the point f E L' equals Let us consider the following function on the space L':
;p(f) = exp(-2li (f)).
W(f) = sup{f(h)2: IhIH(V) !5 1}.
Clearly, W is a nonnegative quadratic form on L'. By equality (3.3.1) applied to the space L, we get the estimate W(f) < R (f)(f ). According to Theorem 3.3.1, there exists a Radon Gaussian measure A on L with the Fourier transform gyp. Regarded as a measure on X, A has the same Fourier transform as v. Therefore, A = v on X. In particular, v(L) = 1. Conversely, assume that condition (iii) is fulfilled. Suppose that there exists an element h E H(v) which is not in H(µ). By Theorem 2.4.7, one can find a linear subspace L E E(X) of full p-measure, not containing h. According to the same theorem, v(L) < 1 (since otherwise H(v) C L), which is a contradiction.
3.3.5. Theorem. Let p be a centered Radon Gaussian measure on a locally convex space X. Then, for any Hilbert space E continuously embedded into H(p), there exists a centered Radon Gaussian measure A on X with H(A) = E. PROOF. It suffices to note that the same reasoning as in the proof of Theorem 3.3.4 can be applied to E replacing H(v) and L = X.
3.3.6. Theorem. Let p and v be two centered Radon Gaussian measures on a locally convex space X. Then the following conditions are equivalent:
ff2dv<_ff2d,z,
V f E X'; (ii) There exists a centered Radon Gaussian measure a such that p = v * a: (iii) For every absolutely convex Borel set A. there holds the inequality µ(A) < v(A)M
PROOF. Since
Q(f):= f f2dp- ff2dv is a nonnegative quadratic form majorized by
ff2 dp, then Theorem 3.3.1 implies
the existence of a centered Radon Gaussian measure a with the Fourier transform e-4/2, whence p = v * a. Condition (ii) implies trivially (i). In addition, (iii)
implies (i), since it suffices to take for A the strips {x: If(x)I _< t}. t > 0, and apply Problem 1.10.11.
Chapter 3. Radon Measures
108
Finally, if (ii) is fulfilled, then. by virtue of Anderson's inequality (see Theorem 2.8.10). we get (iii) for absolutely convex sets A E E(X). Therefore, if E(X) = B(X)
(as is the case for most of the reasonable spaces), then the proof is complete. Anderson's inequality will be proved below for all Boret absolutely convex sets, but now. for completing the proof, it suffices to verify this inequality for any set A which
is the absolutely convex hull of a compact set K. In turn, to this end it suffices to be able to approximate A by absolutely convex sets from E(X ),,. We shall give below a simple constructive method of finding such approximations, but now we do this in a rather primitive direct way. Since v is Radon, given e > 0, there exists
e. This set is a
a set It'. D K open in the weak topology such that
union of a family of open cylinders, hence, by virtue of the compactness of K, this family admits a finite subfamily covering K. This enables us to assume from the very beginning that Ii', is a cylinder. Recall that the absolutely convex hull of K is the union of the sequence of compact sets
K, = {A, k1 + ... + ankn, ai E IR1,
IA I < 1, k; E K}. t=1
Choosing cylinders 11',, D K with v(l('n\Kn) < e2-", one can construct a set lt' E E"(X) such that A := absconv K C IV and v(Wi'\A) < e. There exists a cylindrical set C, containing W, such that v(C\W) < E. Then v(C\A) < 2e. Let C = P-1(B),
where P: X - IR" is a continuous linear mapping and B E B(IR' ). In the same way as in Chapter 2 (see Lemma 2.1.3). we notice that P(A) is absolutely convex compact and P-1(P(A)) is an absolutely convex cylinder which contains A and is contained in C.
3.3.7. Corollary. If either of the conditions (i) - (iii) in Theorem 3.3.6 is satisfied. then
f q(x) v(dx) S f q(x) ft(dx) X
X
for any nonnegative Borel function q such that the sets {q < t} are absolutely convex.
3.3.8. Remark. Similarly to the case of a Hilbert space X, where the covariance operator R of a Gaussian measure 7 can be written in the form R = f v"R-. in the general case. for any Radon Gaussian measure ^y on a locally convex space X. one has the factorization
R, = SS'.
(3.3.2)
where S: E -- X is a continuous linear operator from some Hilbert space E and
S': X' -' E is its adjoint. Indeed, one can put E = Xy and Sf = R f. Then S' is the identical embedding of X' into X. Note that, for an arbitrary choice of E and S satisfying (3.3.2), the following equality holds:
S(E) = H(= R, (f)(f ), the operator S' extends to a continuous operator from X,; to E, hence R,(X,;) C S(E) by virtue of (3.3.2). The inverse inclusion follows from the Hahn-Banach theorem, since the set SS'(U) _ R, (V), where U is the closed unit ball in X. , is compact in X. If we had a unit vector e E E with S(e) 0 R.,(U), then we could find a functional f E X` such that y)-Indeed. by the equality (S' f, S' f) s
3.4.
The structure of Radon Gaussian measures
109
f(S(e)) > 1, f (SS' (g)) < 1 for all g E U. Then (S' f, e)E = f (S(e)) > 1, whence
IS'fIE > 1. On the other hand, (S'f,S'g)E < 1, Vg E U, whence IS'fIE < 1, which is a contradiction. This observation, which is due to Vakhania (see [796], [798], [800]), is also true for non Gaussian measures possessing covariance operators. Apart from being interesting in its own right, this result may be helpful for evaluating the Cameron-Martin space (and its analogues for non Gaussian measures).
3.3.9. Example. Let P%% be the Wiener measure considered on L2[0,1]. Then its covariance operator can be written as SS', where Sf (t) = f o f (s) ds. Indeed, in this case S' f (t) = ft' f (s) ds. It remains to use the identity , = fmin(ts)f(s)ds
JJf(u) duds,
0
0
a
which is verified by differentiating in t. Therefore, we arrive at the description of H(Pu') as the primitives of the elements of L2[0,1]. We saw in Corollary 2.3.2 that the exact analogue of Proposition 1.2.2 is not valid for infinite dimensional Hilbert spaces. However, there exist infinite dimensional locally convex spaces X such that every nonnegative continuous quadratic form on the dual space with the Mackey topology is the covariance of a Radon Gaussian measure on X and the converse is also true. The class of such spaces contains, e.g., the dual spaces to the nuclear barrelled locally convex spaces (which is a special case of the well-known Minlos theorem [549]: see also [800, Theorem VI.4.4]
and Problem 3.11.36. An important example is the space of distributions S'(lR"). In Section 3.11 we shall consider another example, where the Gaussian covariances admit an explicit description: the spaces L. 3.4.
The structure of Radon Gaussian measures
The following fundamental result is due to Tsirelson [774], [775].
3.4.1. Theorem. Let - be a Radon Gaussian measure on a locally convex space X. Then there exists a sequence of metrizable compact sets K" such that
(UK) PROOF. Let E > 0. We shall prove that there exists a compact set K such that
y(K)> 1-E, and the initial topology on K is metrizable. The main idea of the whole construction is as follows. We may assume that the measure -y has mean zero and that X is the minimal closed linear subspace of full measure (the existence of the minimal closed linear subspace of full measure follows from the zero-one law and the existence of the topological support). We shall find an absorbing set T C X' (i.e., a set whose
homotetic images cover X') and equip it with some separable metric d. Then we
shall find a compact set K C X with j(K) > 1 - e such that, for any x E K, the function F=: g -4 g(x) on (T, d) is continuous. Suppose that all this is done. Then the initial topology of X is metrizable on K. Indeed, the initial topology on K coincides with the weak topology a(X,X'), since K is compact. Let Q be a
Chapter 3.
110
Radon Measures
countable set dense in (T, d). Every functional f E T is the pointwise limit on K of some sequence of elements fn E Q. since, by virtue of our choice of T, for any x E K, the function g g(x) is continuous on (T, d). Therefore, f (x) - f (x)
for any x E K. Since the elements of T separate the points in the set K, this implies that the elements in Q separate the points in K as well. Therefore, K is a metrizable compact space (see Proposition A.1.7 in Appendix). Let us now choose T. This will be the set
T = {I E X':
I), XES
where S is an arbitrary compact subset of X of positive measure (such a set exists, since y is a Radon measure). In other words, T is the polar of S. Let us define a metric r on T by the formula r(f, 9) = III - 911V(-,)Note that r is indeed a metric, since the equality f (x) -g(x) = 0 for y-a.e. x implies that f(x) - g(x) _- 0 (otherwise there exists a closed hyperplane of full measure, which contradicts our assumption that X is the linear support of y). Let
S'={xEX:
sup ll(x)I<_I}. lET
Thus, S' is the bipolar of S, hence it coincides with the closed absolutely convex hull of S (see [670, p. 160, Theorem IV.1.5)). Denote by E the linear subspace U', nS'. Since E contains S, by virtue of the zero-one law, it has full measure.
Let us consider the random process l;(t,x) = t(x), t E T, X E E, where E is equipped with the measure y. It is clear that is a Gaussian process and that suptET IE(t, x)I < or, for all x E E. According to Theorem 3.2.7, the space (T, r) is separable. By virtue of Proposition 2.6.5, the process on T has a separable modification C°. By the It&-Nisio theorem 2.6.3, there exists the nonrandom oscillation a of e. Note that the function a is finite, since sup,ET Ir;(t,x)I < x for x E E. Now we define a new metric d on T by the formula d(f, 9) = r(f, 9) + Ia(f) - o(g)I. It was shown in Proposition 2.6.4 that (T, d) is separable and that our process has
a modification (denoted again by e) such that the path t ' °(t,x) is continuous on (T, d) for every x E E.
It remains to take a compact set K C E such that y(K) > I - t and t(z) _ V°(t,z) whenever z E K, t E T. Let us show that we can do this. On the space C = Cb(T) of bounded continuous functions on (T, d). one can introduce a metric p such that (C, p) is separable and the mappings F, : f - f (t), t E T, are continuous in the metric p. Having chosen a countable everywhere dense set To C T and a countable basis {Un} of the topology in (T, d), for such a metric we take the function
r
p(f,g) = sup2- I}!vfn(f) - Mn(9)I + n
Imn(f) - Inn(9)I]
`
where
Mn(f) = sup arctan f (t), mn(f) = inf arctan f (t). t.
To
We could also deal with kS' for k sufficiently large in place of E; then the corresponding sample paths are uniformly bounded, and there would be no need in arctan in the definition of Mn and m,,. Note that the functions F, generate the Borel
3.4.
The structure of Radon Gaussian measures
111
a-field of (C, p), since they separate the points and (C, p) is separable. The mapping T, taking x to e( -, x) E C, is Borel. By the separability of C and the Radon property of y, there exists a compact set K, C X such that y(KK) > 1 - e and the mapping ' is continuous on KE (this is proved in the same manner as the classical Lusin theorem). Recall that t;'0 is a version of C, hence -y (x: t;°(t, x) = t(x)) = 1 for any t E T. For every t E T, the set
K(t) = {x E KE: t;°(t,x) = t(x)) has measure y(KK). By virtue of our choice of Kf and the continuity of the functionals t, all these sets are compact. Since the restriction of y to KK is a Radon measure, the intersection of the compact sets K(t) has measure y(KK) as well (see Problem A.3.35 in Appendix). This intersection is taken for K. O 3.4.2. Corollary. Any sequentially continuous function is measurable with respect to every Radon Gaussian measure.
3.4.3. Corollary. Let y a Radon Gaussian measure on a sequentially complete locally convex space X. Then, for every c > 0, there exists an absolutely convex metrizable compact set K, such that y(K,) > 1 - E. PROOF. It suffices to use the fact that the absolutely convex closed hull of a metrizable compact set in a sequentially complete locally convex space is compact metrizable (see Proposition A.1.6 in Appendix). The reader is warned that a countable union of metrizable compact sets may fail to be metrizable, hence the results above do not imply the existence of a metrizable support (see Example 3.6.3 below); Tsirelson's theorem yields the following isomorphism theorem.
3.4.4. Theorem. Let µ and v be two centered Radon Gaussian measures on locally convex spaces X and Y, respectively, such that dim H(µ) = dim H(v). Then the measurable spaces (X, p) and (Y, v) are linearly isomorphic in the following sense: there exist Borel Souslin linear subspaces X° C X and Y° C Y and a
one-to-one Borel linear mapping T: X° - Y° such that µ(X°) = v(YO) = 1 and it oT-1 = v. PROOF. It suffices to prove the claim for infinite dimensional H(A). Let -y be the countable product of the standard Gaussian measures on the real line. Let us choose a metrizable compact set K C X of positive µ-measure. There exists a sequence If,,) C X' separating the points of the linear span X1 of the set K. By the aid of the process of orthogonalization we may assume that such a sequence {bf }
is an orthonormal basis in X. The mapping F: X1 -+ IR'°, Fx = (fn(x))n_1, is linear, continuous, and injective. It is clear that µ o F-1 = y. The space X1 is a countable union of metrizable compact sets, in particular, it is Souslin. All these properties are inherited by its image F(X1), which has full measure with respect to y. For the measure v, we find a space Y1 C Y and a continuous linear mapping G: Y1 - lR°° with the analogous properties. Put Z = F(X1) n G(Y1). Then Z is a a-compact linear space of full -y-measure. Put X° = F-(Z), Yo = G-1(F(X°)). The mapping T = G- IF: X° - Yo is linear and one-to-one. Note that, by Theorem A.3.15 in Appendix, both mappings F and G take Borel sets to Borel ones. Therefore, the mappings T and T-1 are Borel, and the sets X0 and
Chapter 3.
112
Radon Measures
Yo are Borel and Souslin. It is clear that p(Xo) = 1. Therefore, 7(F(Xo)) = 1, whence v(Yo) = 1. Hence p oT-1 = v. 0
An important open problem related to our preceding considerations is whether one can always find in any locally convex space with a Radon Gaussian measure a convex compact set of positive measure (which is possible, as we already know, for all sequentially complete spaces). For solving this problem, it suffices to understand whether every full measure linear subspace in 12 or 1R" with a Gaussian measure contains a convex compact set of positive measure. A similar problem remains
open for convex sets of positive measure (note that for non Gaussian measures both questions are answered negatively).
3.5. Gaussian series This section is devoted to the representations of Gaussian vectors by series of
independent one dimensional Gaussian vectors.
3.5.1. Theorem. Let -y be a centered Radon Gaussian measure on a locally convex space X. H = H(7), {en} an orthonormal basis in H, a sequence of rndependent standard Gaussian random variables on a probability space (Q, P), and A E £(H). Then the series
GMAen
(3.5.1)
n=1
converges as. in X. The distribution of its sum is a Radon Gaussian measure A with the covariance
RA(f)(f) = (A'i (f),A'R,(f))y. If A = I, then
x x =
en(x)e,,
7-a.e.
(3.5.2)
n=1
PROOF. One can reduce this theorem to the case A = 1, however, we shall consider at once the general situation. By Theorem 3.3.1, there exists a centered Radon Gaussian measure A on X with the Fourier transform 9F-+ exp
2( A' R., (g), A'R, (g))H1_
Let S be a random vector with the distribution A. defined on the same probability space as {Sn}. This can always be achieved. For example, if the space Xa is infinite dimensional, one can take for (11, P) the space (X, A), then take for any orthonormal basis in X* and put S(x) = x (if the space Xa is finite dimensional, then so is the operator A and the claim is trivial). For any g E X', the series 1:,x 1 ng(Aen) converges a.s. and its sum has the same centered Gaussian distribution as g(S). Indeed, this follows from the equality of the variances: 2
(R,(g),Aen)H
9(Aen)2 = n=1
n=1
_
2
n=1
(A'R_(g),A'R,(g)) y
1,5,
Gaussian series
113
Note that it suffices to prove the convergence of series (3.5.1) in the completion
of X, since S with probability 1 is in X. Hence we may assume from the very beginning that X is complete. Then there exists an absolutely convex metrizable compact set K such that .(K) = P(S E K) > 0. The union of the sets nK is a linear subspace of positive measure. Therefore, by the zero-one law
)(nK) = P(S E nK)
1.
The initial topology coincides with the weak one on the set K due to its compactness. Since K is metrizable, there exists a countable family of continuous linear functionals {g;}, generating the topology of K, hence the topology of every set nK as well. Unfortunately, this family may fail to generate the topology of the union of the nK's (e.g., in the space 12 with the weak topology, the coordinate functions generate the topology of every ball, but not of the whole space). Therefore, we have to deduce the convergence of our series in X from the condition that, with probability 1, the sequence {g;(Sn)} converges for every i, where (w)Ae,.
Sn(w) ;=1
The easiest way to do this is to prove that, with probability 1, the sequence Si remains in one of the sets nK. Denoting by q the Minkowski functional of K, it suffices to verify that
supq(Sj) < oc a.s. To this end, note that the sequence of the finite dimensional vectors S3 is a martinThen gale with respect to the sequence of the o-fields off, generated by the sequence q(3,) is a submartingale (see Problem A.3.38). The covariances of the vectors Sn are given by the equalities
RS., (9)(9) = (P,,A'R,(9),
where P. is the orthogonal projection in H to the linear span of the vectors e1, ... , en. These covariances are majorized by the covariance of S, since we have IhI,,, dh E H. According to Theorem 3.3.6, for any symmetric convex set C, one has
P(S, E C) > P(S E C). This means that lEq(S,,) < IEq(S), where the right-hand side is finite by Fernique's theorem. By virtue of Theorem A.3.6 in Appendix, we get the desired claim. Let us prove (3.5.2). It suffices to verify this equality for almost all x from a full measure set E that is the union of a sequence of metrizable compact sets in X. We know that there exists a sequence of continuous linear functionals g; separating the points in E. Since the series on the right in (3.5.2) converges for a.e. x E E, it suffices to show that, for every i E IN, one has 9,(x) = E
a.e. on E.
(3.5.3)
n=1
To this end, it remains to observe that g,(en) since {e;,} is an orthonormal basis in X.
which yields (3.5.3),
Chapter 3. Radon Measures
114
Note that the previous theorem yields the equality f f (x) -1 (dx) =
ff(en)
X
0
P(dj),
(3.5.4)
n-1
where {ln } is any sequence of independent standard Gaussian random variables on a probability space (0, P).
3.5.2. Corollary. Let 7 be a centered Radon Gaussian measure on a locally convex space X with the infinite dimensional Cameron-Martin space, {Cn} an or-
thonormal basis in X,;, and en = R,(f,,). Then, for any function f E LP(-y), the function (3.5.5)
fn(x) = f f (Pnx + Sny) 7(dy), X
where
x
n
C,(x)ei, Sny =
Pnx =
C,(y)e i=nt1
i=1
serves as the conditional expectation of f with respect to the afield generated by Sn In particular, {fn} -y f in LP(7) and 7-almost everywhere. The same is true for the Bochner integrable mappings with values in Banach spaces.
CI,
PROOF. For any bounded Borel function g of the form g(x) = g(Px), by Theorem 3.5.1, we have the equality
f f(x)g(Pnx)7(d) = f f
ynen) g(EY e) <<(dy), i=1
n=1
R-
X
where p is the countable product of the standard Gaussian measures p, on the real line. In a similar way,
f/n( x)g(Pnx)7(dx) X
yie. + E z,ei) g(` yiei),u(dz),u(dy) = f f f (E ,>n i=1 R- R-
3=1
It remains to be noted that p coincides with the product of the measures
x
n
®µ; and
®p,.
i=n+1
i=1
Therefore, by Fubini's theorem, the integrals on the right-hand sides of these two equalities coincide. Thus, f f(x)g(x)'Y(dx)
=
Jfn(x)(X).Y(dX),
which shows that fn equals the conditional expectation of f with respect to the aforementioned a-field. The vector case follows by [800, Theorem 11.4.2].
3.5.3. Remark. Note that if f in Corollary 3.5.2 is symmetric, i.e., f (x) = f (-x) a.e., then, by the symmetry of 7, so are the functions fn. In addition, if f is convex or concave, then the functions fn have the corresponding property. Indeed, it suffices to apply the corresponding inequality to the representation f (APnx + (1 - A)Pnz + Sny) = f (APnx + ASny + (1 - \)Pnz + (1 -'\)Sny).
3.5.
Gaussian series
115
3.5.4. Example. Let. ('p.) be an orthonormal basis in L2[0' 11 and let {Sn } be a sequence of independent standard Gaussian random variables. Put
fn(s)ds
en(t) = 0
Then, for a.e. w, the series
x(t,w) := F_G(W)en(t) n=1
converges uniformly in t E [0, 1] and its sum is a standard Wiener process. In addition, the measure on C[O, 1] induced by the mapping w'-. x( , w) is the Wiener measure PL1'.
PROOF. We can apply Theorem 3.5.1, since {en } is an orthonormal basis in the Cameron-Martin space H(P1t) = Wo1[0.1].
In terms of Gaussian random vectors the formulation of the results above is this: given a Gaussian random vector t with a Radon distribution y in a locally convex space X, the series
n= al+
Snen n=1
converges almost sure for any sequence {ln} of independent standard Gaussian random variables and any orthonormal basis {en} in the Cameron-Martin space H(y) of y and its distribution is y. If A is a bounded linear operator on H(y) and a., = 0, then its measurable linear extension is given by the almost sure convergent series
_
Art = E n Aen. n=1
3.5.5. Proposition. Let y be a centered Radon Gaussian measure on a locally convex space X, let {en} be an orthonormal basis in H(y), and let f be a y-measurable function on X such that f o r every i = 1, ... , n, there exists a version
off that is constant along every line x + 1R1 e x E X. Then there exists a Borel function g on IR' such that f (x) = glen+1(X), en+2(X), ) y-a.e. PROOF. We may assume that f is bounded, passing to the function arctanf. Let us consider the orthonormal basis in L2(y) formed by the products of the Hermite polynomials Hk(F,). Then a version of f with the desired properties is given by the expansion of f with respect to this series, since f is orthogonal to all products of the functions Hk(ei) that involve i < n. Indeed, Hk(e,) is constant along the lines x+IR1e, if j 54 i. Hence it suffices to show that Hk(ei) is orthogonal to any function g E L2(y) that is constant along the lines x + IR1e1. This property
is trivial in the one dimensional case; in the general case it follows by Fubini's theorem in view of Theorem 3.5.1.
3.5.6. Corollary. In the situation of the previous proposition, let A be a -tmeasurable set invariant under the translations along e1,... , e,. Then for any set B from the a-field generated by Fl,... ,en, one has y(A n B) = y(A)y(B),
Chapter 3.
116
Radon Measures
i.e., the events A and B are independent. Note that in the case of a separable Banach space Theorem 3.5.1 can be proved much easier. For instance, it can be deduced from the martingale convergence theorem for the martingales with values in separable Banach spaces. If X is a separable
Hilbert space, then this result becomes almost trivial. In order to get some orthonormal basis (en} with the desired property, it suffices to take the orthonormal basis { f} in X formed by the eigenvectors of the covariance operator of the measure -y. If {kn} is the sequence of the corresponding eigenvalues, then the vectors en = form an orthonormal basis in H(-f). It is clear that x = gr(e)en, n=1
where S,, (x) = x,, / ykn, xn = (x, f,,) x.
x
In the case where X is a separable Banach space, the series E tn(x)en conn=1
verges to x not only almost everywhere, but also in the norm of LP(-I, X) for every
pE [1,oc). 3.5.7. Theorem. Let y be a centered Gaussian measure on a separable Banach space X with the norm 11 11, let {e, } be an orthonornal basis in H = H(-Y), and let {{, } be a sequence of independent standard Gaussian random variables on a probability space (S2, P). Then, for any A E £(H), for all sufficiently small c > 0, the following relationship is valid:
x
f
lim / exp(cl
n
- xJJ
,-n
S1
In particular. E t;iAe;
12)
t;, (w)Ae,
1y
-+ 0 in L1(P) for every p r= (1, 0C.).
1_n
PROOF. Indeed, the nonnegative functions 2
fn(w) = exp(cl
.;(w)Ae,ll
\ =n converge almost everywhere to 1. Hence, for the proof of the convergence to 1 of their integrals, it suffices to get the estimate
sup n
P(dw) < o.
J dff
This estimate takes place if c > 0 is chosen according to Fernique's theorem in such a way that
L exp(2c1j S112) dP < oo,
where S = F;_1 6;Ae,. Indeed, this follows from Theorem 3.3.6 applied to the distribution of the Gaussian random vector
x
Y. = E ,Ae, :=n+1
and the measure PoS-1 (this measure is Gaussian, since so are the variables Since, for any 1 E X', one has
J!!
l(Ae,)2 < 1l(S)2dP,
1(Y,,)2dP = t=n+1
S2
3.5.
Gaussian series
117
then the theorem cited gives the estimate fexp(2cIYn 112)
dP < f exp(2cl{SII2) dP.
Certainly, the same is seen also from the vector martingale convergence theorem.
0
3.5.8. Corollary. For any p > 1, one has Jim
fr X -
IIP
Fi x)ei
I
y(dx) = 0.
.1
X
3.5.9. Remark. The reader should be warned that in the reasoning above one cannot replace the separable norm II II by an arbitrary measurable norm. The corresponding counterexample can be extracted from Example 3.6.3 below.
3.5.10. Theorem. Let y be a centered Gaussian measure on a separable Banach space X with the norm II ' Then one can find an orthonormal basis {en } II
in H(y) such that E Ilea II2 <.00. n=I
PROOF. Let {4pn} be any orthonormal basis in H(-y). The series n 1 1i'n1n converges in the space L2(y, X). Hence there exists an increasing sequence of natural numbers kn such that
r / II n=1
z
J (x)W.1II.r y(dx) < oc.
(3.5.6)
The whole proof is based on the observation that, for every n, there exists an orthogonal operator Un in the finite dimensional subspace H. C H(y) generated b y the vectors Oj j = k + 1, ... , kn+1, such that II UnVj II2 <
f Ij (x)w1 II
y(dx).
(3.5.7)
If this is already done, one can put ek := !'k if k < kl and ek := UtI'k if kn + 15 k < kn+1, n E IN. Clearly, one gets an orthonormal basis which, by virtue of (3.5.6) and (3.5.7), has the desired property. The existence of orthogonal operators L;, with property (3.5.7) is established by the following argument. Denote by Un the group of all orthogonal operators on H,,, where Hn is equipped with the Euclidean norm from H(-y), which for notational simplicity is denoted throughout this proof by I I. Since U,, is a compact group, it has an invariant Haar probability measure p (see, e.g., 1220, 4.18.1, 4.18.21).
Denote by S, the unit sphere in the Euclidean space H. For any e E S,,, the image of the measure µ under the mapping U ' Ue coincides with the normalized Lebesgue surface measure a on Sn, since it is a spherically invariant probability measure (which is unique). Therefore, L.. IUej112 ji(dU) = f IIeIi2o(de),
dj = kn + 1,... ,kn+1.
Chapter 3.
118
Radon Measures
Denote by 7,, the standard Gaussian measure on H and take the measure v with density (dim H,,) -'I I2 with respect to the measure yn. Then v is a probability measure. Using that the image of the measure v under the mapping y - yl yl -' is also the measure a (by virtue of the spherical invariance of this image), we get, by the change of variables formula, (dim H,,)
1f
'
Jrr
v(dy) = f ll cll2 a(dc).
Ilyll` ?n (dy) = fH^ Il l yl
Since "1n is the image of the measure I under the mapping k,,., k.,-1
we arrive at the following sequence of equalities: IIU viii ) t<(dU) = (kn-1 - kn)
J
=L
IIyIt2
f
n
hell ja(de)
n(dy) = J
I
I
't(dr.).
J=k..r1
"
This relationship implies the existence of an operator Un satisfying (3.5.7). since it is a probability measure.
3.5.11. Corollary. Let o be a centered Gaussian measure on a separable Banach space X. Then there exists an orthogonal basis {en } in H(j) such that x x
-!
Ile.,ll
<.00
and TI=1
However. not every orthonormal basis in H(-)) has the property indicated in the preceding theorem. For example, let X = co and let -) be the countable product of the one dimensional centered normal distributions with variances
on=(log(n+1))
1
Then it is straightforward to verify that the vectors e,, = (0.....a,,.0....) form an orthonormal basis in H(7), - (co) = 1 and Ilen ll2 = (log(n + 1)) j . One can construct a similar example for the classical Wiener space. Certainly, for any orthonormal basis in H(y), one has the relationship llenll -. 0, since H(t) is compactly embedded into X. A special case of expansion (3.5.2) is the so called Karhunen-Lol ve expansion of a centered Gaussian process tt on T = [a. b) with square-integrable paths in the series
GM = L: G(.)Vn(1), n=1
where 14,,j is a sequence of independent standard Gaussian random variables deis the orthonormal basis in fined on the same probability space as , and
3.6.
Supports
119
L2[a, b] formed by the eigenfunctions of the integral operator K given by b
Kf(t) =1 K:(t,s)f(s)ds, K,(t.s) := 1F.{r a
3.5.12. Example. Let (T, F, a) be a measurable space and let -y be a centered Radon Gaussian measure on the space LP(T, F, a), where p > 1. Then there exists a measurable centered Gaussian process ry,. t E T. such that y is the measure induced by this process in the sense of Example 2.3.16. PROOF. Let (en} be an orthonormal basis in the Cameron-Martin space of y
and let t;,, = e . Put tt
E1;;(x)e,(t).
Sn: LP(T,a) -. LP(T,a).
t=1
By Corollary 3.5.8, the sequence {Stt } converges in LP (y, LP(T, a)) to the identity
mapping. Hence the sequence Rn(t,x) =
is fundamental in the space LP (T x LP (T, a), a®y). Therefore, its subsequence converges a&y-a.e. Clearly,
the limit has a F ® E (LP(T, a))-measurable modification ( , ), which has the required properties.
3.6. Supports of Gaussian measures
3.6.1. Theorem. Let y be a Radon Gaussian measure on a locally convex space X. Then the topological support of y coincides with the affine subspace a., + H(y),
where H(y) stands for the closure in X. In particular, the support of y is separable.
PROOF. Without any loss of generality we may assume that y is a centered measure. Let L be the closure of H(-f). It is clear that y(L) = 1, since, by Theorem 3.5.1, -f-almost every element x is the limit of linear combinations of the vectors e E H(y). Therefore, the topological support S, of -y is contained in L. Suppose that there exists x E L not belonging to S,.. Then x is contained in some open set V of measure zero. The set V contains some element h E H(-y). Therefore, the open set W = V - h has measure zero as well (by the equivalence of the shifted measure). In addition, it contains the origin. The existence of the topological support implies that the union of all sets W + h, h E H(-y), has measure zero. It remains to note that this union covers L, which contradicts the equality y(L) = 1. O
3.6.2. Definition. A Radon Gaussian measure is called nondegenerate if its topological support is the whole space.
It is clear that a measure y is nondegenerate precisely when its Cameron-Martin space is everywhere dense.
One can get the impression that Theorem 3.4.1 enables us to find for a general Radon Gaussian measure a metrizable linear subspace of full measure. The following example shows, however, that this is not always true.
Chapter 3. Radon Measures
120
3.6.3. Example. Let X be the space I' equipped with the topology a(11. 11) and let y be the restriction to 1" of the countable product of the one dimensional centered Gaussian measures with variances a2 = log(n + 1)-1. Then: (i) -y is a Radon Gaussian measure on X, but -y(E) = 0 for any separable Frechet
(or just separable metrizable locally convex) space E embedded continuously into the space X;
(ii) The measure y extends to a centered Gaussian measure on £(I x) (where 1" is considered with the norm topology) which is not tight, hence is not Radon, even with respect to the weak topology a(l", (1x)'), but has the separable L2 and the separable Cameron-Martin space. PROOF. The proof of (i) is based on the following two facts which are easily verified with the aid of estimate (1.1.2) of Gaussian tails: 1/2 R
J p(t, 0, 2) dt = 0,
J p(t. 0. an) dt > 0 nom!-R
1/2
for R sufficiently large. Indeed,
2 (an
2,R
\
/
H
- R /I exp\(- Rn 2a
<1-
H
p(t.U. a,2,)dt <
° Pxp sir R
2
2a Rn -R hence the convergence of the product above is equivalent to the convergence of the series T_ (log(n + 1))-1/2(n + 1) R'. By the Banach-Alaoglu theorem. the sets n=I
KR = {x E 1" : JJxfl a < R} are compact in (I",a(l",I1)). The measure y. defined on 1R", is concentrated on their union. By virtue of Theorem A.3.19 in Appendix, the restriction of y to the space X equipped with the o-field generated by the coordinate functions, extends uniquely to a Radon measure on the space 1" with the *-weak topology a(I" , ll ). This extension is a Gaussian measure. Suppose that E is a separable Freshet space continuously embedded into X. Then, by the uniform boundedness principle, the
embedding of E to 1" with the norm topology is continuous as well. Therefore, there exists an open neighborhood of zero U in E that is contained in K,;2. hence has -y-measure zero. Applying theorem 3.6.1, we see that y is identically zero on E. If E is a separable metrizable locally convex space, but is not complete, then it is contained in a separable Freshet space E0 continuously embedded to 11. Indeed,
let j : E -, X be the corresponding injection and let k be the completion of E. It is readily seen that j extends to a continuous linear operator A: E -- l, which generates the embedding of the factor-space E/KerA. Claim (ii) follows from the fact that every functional from (11)' is measurable with respect to the Lebesgue completion of the a(l", l l )-cylindrical a-field with the measure y (see Corollary 3.11.4 below). Clearly, y is not tight on l" with the norm topology by assertion (i). This implies that it is not tight with respect to the weak topology a(11, (11)'). Indeed, otherwise y extends uniquely to a Radon Gaussian
measure with respect to the weak topology (see Theorem A.3.19 in Appendix). Then there exists a sequence of weakly compact metrizable subsets in lx whose
3.6.
Supports
121
union Al has full measure. Hence there is a sequence {xn} C Al that is weakly dense in Al. It remains to note that the closure Z of the linear span of in the norm topology is a norm separable full measure subspace of l" contrary to what has been proved above. Indeed, by construction, Z is separable. In addition, being convex, it is weakly closed, hence contains M.
0
3.6.4. Remark. In connection with the existence of a countably additive extension of the measurer to the o-field generated by (l")*, it is worth noting that there is no such extension to 8(l1) (see Proposition 3.11.5). Let us make some remarks about linear supports of Gaussian measures. The terminology here differs from the one used above, where the topological supports were discussed. We shall say that a measure p on a locally convex space X has a Banach (or Hilbert) support if there exists a continuously embedded separable Banach (respectively, separable Hilbert) space B C X of full p-measure. Typically, a support in this sense is not the topological support, since B need not be closed in X. For example, the space C[0, 11 is a Banach support of the Wiener measure considered on L2[0,1], and the closure of the Holder space H'I'[0, 11 in the Holder space H 114 [0, 11 is a Banach support of the Wiener measure on C[0,11 (recall that the Holder spaces themselves are nonseparable). The following theorem shows that on the Frechet spaces all Radon measures (not necessarily Gaussian) have reflexive Banach supports.
3.6.5. Theorem. Let p be a Radon measure on a Frechet space X. Then there exists a linear subspace E C X with the following properties: (i) There is a norm 11 - 11E on E, with respect to which E is a reflexive separable Banach space such that the closed unit ball in this norm is compact in X; (ii) The equality lfiI(X\E) = 0 holds true.
PROOF. We may assume that p is nonnegative. For any n, there exists a compact set K. satisfying the condition 1/n. We can find a number
cn > 0 such that the set cK is contained in the ball (with respect to a metric generating the topology of X) of radius 1 /n centered at the origin. It is easily seen
that the set U cK,, is completely bounded, hence its closure K is compact. It is n
known (see (220, Lemma 9.6.41) that there exists an absolutely convex compact set A such that K is compact as a subset of the Banach space EA (the linear span of A) with the norm equal to the Minkowski functional pA of A. Applying this lemma once again, we get a yet bigger absolutely convex compact set C D A such that A is compact as a subset of E,_ with the norm per.. According to the factorization lemma of Davis-Fiegel-Johnson-Pelchinsky (see (195. Ch. 5. §4J), there exists a Banach space Y, which is reflexive and continuously embedded into EE and, in addition,
the unit ball of EA is bounded in Y. Note that K is compact in Y by virtue of the compactness of K in EA combined with continuity of the embedding EA Y. Let E be the closure of the linear span of K in Y. Then E is a reflexive separable (by virtue of the compactness of K in Y) Banach space, which has full measure.
The closed unit ball UE from E is precompact in X, since C is compact and the embedding Y -- E, is continuous. Moreover, U. is compact in X, since, by virtue of reflexivity of E, the set U& is weakly compact in E, hence it is weakly compact in X as well, whence it follows that it is closed in X. 0
Chapter 3.
122
Radon Measures
Hilbert supports do not always exist.
3.6.6. Example. Let y be a Radon Gaussian measure on the space X = L2(P). where P is a probability measure. Suppose that H(y) = T(X), where the x kn = oc, where operator T E G(X) is not nuclear (which is equivalent to n=1
the k .'s are the eigenvalues of the covariance operator of the measure y). Then y(E) = 0 for any Hilbert space E continuously embedded into L2(P) and contained
in L-(P). PROOF. We shall assume that -y is symmetric. Let us choose a closed separable subspace X0 of full -y-measure. It suffices to prove that y(E n Xo) = 0. Note that
E0 = E n Xo is a closed subspace in E. By Lemma A.2.16 in Appendix, the space ED with the norm from E is separable. Then every Borel set in Eo (with its Hilbert norm) is Borel also in X. Therefore, the restriction of y to E0 is a Borel measure on E0. If y(Eo) > 0. then y(Eo) = 1 by the zero-one law, hence y on E0 is a Gaussian measure. The Cameron-Martin space H(y) does not depend Eo is on whether we consider y on X or on Eo. Hence the embedding H(y) a Hilbert-Schmidt operator. By Lemma A.2.13, the embedding Eo - L2(P) is a Hilbert-Schmidt operator as well. Therefore, the embedding H(y) -+ L2(P) is a nuclear operator, which contradicts the condition. 3.6.7. Example. Let Pll be the classical Wiener measure on the space C[0,1]. Then Pt (E) = 0 for any Hilbert space E which is continuously linearly embedded into C[0. 1].
PROOF. It suffices to recall that the covariance operator of the Wiener measure on L2[0,1] has the eigenvalues c(n - 1/2)-2.
3.7. Measurable linear operators
3.7.1. Definition. Let y be a Radon Gaussian measure on a locally convex space X and let Y be a locally convex space. A mapping F: X -+ (Y, E(Y)) is called a -y-measurable linear operator (or a -y-measurable linear mapping) if there
exists a linear mapping Fo: X -+ Y measurable with respect to the pair of the o-fields (8(X).,, E(Y)) such that F0 = F y-a.e. The mapping Fo is called a proper linear version of F. The simplest example of a measurable linear operator is any continuous linear operator. The following corollary of Theorem 3.5.1 enables one to weaken the condition of continuity (for most of the spaces encountered in the applications this
does not lead to a real weakening, though). Clearly, if Y = IR', then we get the previously defined notion of a measurable linear functional.
3.7.2. Proposition. Let y be a centered Radon Gaussian measure on a locally convex space X and let A: X Y be a sequentially continuous linear mapping with values in a locally convex space Y. Then the mapping A is a 1-measurable linear operator. In addition, the measure yoA-1 is a centered Radon Gaussian measure on the space Y. PROOF. Both claims follow from the fact that the measure y is concentrated on a countable union of metrizable compact sets and that a sequentially continuous mapping of a metrizable compact set K to Y is continuous, hence y-measurable (in addition, is a Radon measure).
3.7.
bten8urable linear operators
123
Let us now apply Theorem 3.5.1 for investigating the structure of measurable linear mappings on a locally convex space X with a centered Radon Gaussian measure y. The following result is one of the classical ones in this circle of problems.
3.7.3. Theorem. Let y be a centered Radon Gaussian measure on a locally convex space X.
(i) Let F: X -. X be a -y-measurable linear mapping and u = y o F-'. Then. denoting by F0 any proper linear version of F, we get
H(p) = Fo(H(y)). In addition, the mapping Fo: H(y) -» H(p) is a continuous operator with unit norm between these Hilbert spaces.
(ii) Let p be an arbitrary centered Radon Gaussian measure on X. If the space H(y) is infinite dimensional. then there exists a -y-measurable linear mapping
F: X - X such that p = y o F'. If H(p) is infinite dimensional as well, then the mapping F can be chosen in such a way that its proper linear version
is an isometry of the spaces H(y) and H(p). PROOF. (i) It is clear that p is a centered Gaussian measure. The inclusion
Fo(H(y)) C H(p) follows from the characterization of H(y) as the set of all vectors h such that yr,
y
and the equality Fa 1(B+Fo(h)) = Fo'(B)+h, B E E(X), h E H(-y). The inverse inclusion is seen from the fact that any vector h = R,,g with g E Xµ equals Fov, where v = R,, f , f = go Fo (note that g o Fo E X.; ). Indeed, for every I E X', one has
l(RN9) =
f l(x)g(x) p(dx) = f l(Fo(x)) g(Fo(x)) y(dx),
which, by Lemma 2.10.5, coincides with
f loFo(x) f (x) y(dx) = loFo(R, f) = l(Fov). Hence Fov = R,,g = h. Clearly, IhIH(P)
= II9IIL=(v) = IIfIIL=(,) = It'IH(,)
If u E H(y) is an another vector with Fou = h, then u = R.,. for some t E X.. For any l E X' with IIIIIL2(,,) < 1 we get as above
f
whence
lgdp = 1(h) = l(Fou) = l o Fo(u) = Jl o Fo t; dy S IIeIIL2(,) = I uI H(,), II9IIL=(,.) <_ ItIH(,)-
(ii) This is a direct corollary of Theorem 3.5.1. Indeed, let it,,) be an orthonormal basis in X1 and an orthonormal basis in H(p). Then the mapping
x F(x) = Ebn(x)e. n=1
is well-defined (in the case, where the basis is finite, we add a countable sequence of zero vectors), is a measurable linear operator, and y o F-' = p. If the basis {en } is infinite, then it is the image of the orthonormal basis in H(y) corresponding to {in }.
0
Chapter 3.
124
Radon Measures
3.7.4. Remark. In connection with claim (i) of the preceding theorem, it is worth noting that, as the next example shows, the image of a Radon Gaussian measure under a measurable linear mapping T may fail to be a Radon measure. 3.7.5. Example. Let -y be the measure mentioned in Example 3.6.3 and let T be the identity mapping of l" with the topology a(I",ll) to I" with the topology a As noted in Example 3.6.3, this mapping is measurable with respect to the pair of a-fields B(l", a(l",11)) and E(l'°), but the measure yoT-1 (i.e., the extension of the measure -y to the a-field generated by (I'°)') is not tight, hence is not Radon.
3.7.6. Theorem. Let 1 be a centered Radon Gaussian measure on a locally convex space X and let H = H(-y) be its Cameron-Martin space with the natural Hilbert norm I 1,, = I IH(.,). Then every operator A E G(H) extends to a linear mapping A which is measurable from (X, B(X) ,) to (X, B(X )) such that the image of 7 under this extension is a Radon Gaussian measure. In addition,
Ax = E en(x)Ae
(3.7.1)
y-a.e.,
n=1
where {en} is an arbitrary orthonormal basis in H(y). Finally, any two extensions of A that are (13(X),, B(X ))-measurable and linear on full measure linear subspaces and induce Radon measures coincide -y-a.e. PROOF. Let us replace from the very beginning the space X by the linear span X0 of any fixed metrizable compact set of positive -y-measure. This linear span, as we know, is a countable union of metrizable compact sets (in particular, is a Souslin space), has full measure and contains H(y). This simplifies significantly the measurability issues. Let {e } be an orthonormal basis in H(-y). Let us choose proper linear versions of e;,; certainly, we can always take {en } such that en E X. Then en(ek) = 5nk. According to Theorem 3.5.1, the series
x E en(x)Aen n=1
converges -y-a.e. in our subspace Xo. Let us denote its sum defined on the set X1 of all those x E Xo, where the series converges, by Ax. Since we deal with a countable union of metrizable compact sets, it is clear that X1 is a full measure Borel linear
subspace containing H(-y) and A is a linear mapping, which is measurable with respect to the pair of the a-fields (B(X) ,, B(X )). It is worth mentioning that in the case of an arbitrary locally convex space X, the limit of a sequence of Borel mappings may fail to be Borel. Now we can extend A to all of X by taking a linear subspace Z C X such that X is the direct algebraic sum of X1 and Z and putting
A(x+z)=Ax forxE X1 andzEZ. ForanyhEH,onehashEX1 and x x fn(h)Aen =
Ah = n=1
Ah. n=1
Since Ax E Xo for every x E X1 and every Borel measure on the space Xo is Radon, we conclude that -y o A'1 is a Radon Gaussian measure.
3.7.
Measurable linear operators
125
Suppose that T is a (B(X) B(X)) -measurable mapping such that T is linear on a full measure linear subspace Y C X, T J H = A and -(o T` is a Radon measure.
Let us prove that T = A y-a.e. For every I E X', one has
loT(h)=loA(h)=loA(h). VhEH(y). By Theorem 2.10.7. 1 o T = l o A y-a.e., which implies that y o T-' = y o A-', since both measures are Gaussian. Using that both measures are Radon, we get a sequence of metrizable compact sets It,, C X such that, letting K = U3 c, K,,, we have yoT -' (K) = yoA-' (K) = 1. Let us take a sequence (1.1 C X' separating the points of the set K. As noted above, the sets L. :_ {x E X : 1,, oT(x) =1 oA(x)} L. nT-'(K) f A-1(K) has full have measure 1. Therefore, the set L := 0 measure. By construction, Tx = Ax for every x E L, whence the claim. To simplify our terminology. the extensions obtained in the previous theorem will be called measurable linear extensions.
Note that if X is a Souslin (e.g., separable Banach or I; chet) space, then B(X) = E(X) and every measure on E(X) is Radon. Therefore, in this case a linear extension is unique up to a modification.
3.7.7. Remark. Let y be a centered Radon Gaussian measure on a locally convex space X with the Cameron-Martin space H. Given two operators A, B E C(H) such that A + B = I, we get A + B = I a.e. In particular, if we have two orthogonal projectors ir) and 72 on H such that I = 1r1 + ir2, then we get a "stochastic- decomposition I = aI +F' a.e. It is worth mentioning that there exist infinite dimensional separable Banach spaces X such that X cannot be represented as the direct sum of closed infinite dimensional subspaces (see [3091).
3.7.8. Lemma. Let A E C(H) and let f E X'. Then
R,(foA)=A'R,f. PROOF. We may deal with a proper linear version of A. Let h E H(-y). Then, by Lemma 2.10.5. one has
(R,(f o A). h)H( I = f (Ah) = f (Ah) = (R, f, Ah)H(,) = (A*R,f, h)H(,), whence the desired equality.
3.7.9. Lemma. In the situation of Theorem 3.7.6, let T E C(X, X') be a finite dimensional operator. Then one has
trace (R o TI H) =
fTxx)dx).
PROOF. It suffices to prove this claim for every one dimensional operator of the
form Tx = f (x)g, f, g E X'. Then Th = f (h) R,9. The integral on the right-hand side of the formula to be proved equals (f,g)Lz(1) = f(R,g), which coincides with the trace of R,oTIH.
3.7.10. Proposition. Let -y be a centered Radon Gaussian measure on a locally convex space X and let H be its Cameron-Martin space.
Chapter 3.
126
Radon Measures
(i) Suppose that A: X -. H is a y-measurable proper linear mapping. Then AI H E 1-1(H) and the corresponding Hilbert-Schmidt norm equals I AxI2 -y(dx).
V Conversely, every operator A E 7{(H) admits an extension to a measurable
linear mapping A: X - H. (ii) Suppose that A: X -. X' is a linear mapping such that the function (x, y) (Ax, y) is sequentially continuous in every argument separately. Then the operator R., o A111: H - H is nuclear. PROOF. (i) As shown in Theorem 2.8.3, the operator C = AI y is continuous on the Hilbert space H. By Fernique's theorem 2.8.5, the function I AxiH is exponentially integrable. According to Lemma A.2.10(ii), it suffices to verify that C' E 1-1(H) and prove the corresponding equality. Let {en} be an orthonormal basis in H. Then I IAxIH 'y(dx) =
f.
(Ax, en)2 7(dX)-
X n=1
X
It remains to note that J(Ax,efl)2H
(dx) = IC-enIN.
X
Indeed, (Ax, en)H is a 'y-measurable proper linear functional. Its restriction to H coincides with the functional
h'- (Ch,en)H = (h,C'en)H, whose norm equals I C' en I H .
In order to prove the converse, let us take an orthonormal basis {en } in H and note that the functionals
h - (Ah,en)H = (h,A'en)H have measurable linear extensions gn to X with the L2-norms IA ' en IH . Since A' is a Hilbert-Schmidt operator, we get
x E IIgntIL2(,) < 00n=1
Thus, n 1 gn(x)2 < oo -y-a.e. and we can put
x gn(x)en.
AX =
(3.7.2)
n=1
(ii) Let us find a compact set Qo C X of positive measure. Let Q be the closed absolutely convex hull of Qo, let EQ be the linear span of Q, and let pQ be the Minkowski functional of the set Q. Note that, for some C > 0, the following inequality is valid: (3.7.3) I (Ax,y)I 5 CpQ(x)pQ(y), dx. y E Eq. Indeed, the bilinear function (x, y) ' -. (Ax, y) on the normed space (EQ,pQ) is
continuous in every argument separately, since if x,, - 0 in the norm pQ, then
3.7.
Measurable linear operators
127
x -a 0 in X. By Theorem 2.8.5, one has pQ E L2(-y). Let K E C(H) be a finite dimensional operator with IIKIIc(H) 5 1, having the following form:
Kh = f, (h)a1 + + f (h)a,,, f, E X', a, E H. In other words, K is the restriction to H of the continuous finite dimensional operator from X to H, denoted also by K. Then the function x
(AKx,x)
f;(x)(Aa,.x)
is integrable with respect to the measure -y and I (AKx, x) I < C pQ(Kx) pQ(x), whence
f I(AKx,x)I (dx) 5 C f PQ(Kx)PQ(x)7(dx) C f PQ(Kx)2 Y(dX) f PQ(x)2'Y(dx) < C f PQ(x)27(dx). since
f PQ(Kx)2 Y(dx) 5 f PQ(x)2-Y(dx)
by virtue of the condition IIKIIc(H) < 1 and Theorem 3.3.6. Therefore,
J(AKxxr(dx) l < C f PQ(x)2 7(d,) On the other hand, according to Lemma 3.7.9, one has the equality f (AKx, x) ry(dx) = traceH (R, o A o KI H).
(3.7.4)
Thus, we arrive at the following estimate which is valid for all continuous linear operators K on X with finite dimensional ranges contained in H: ItraceH (R, o A o KI H) 15 MII KII c(H).
Since the restrictions of the elements from X' separate the points in H, the set of such operators K on H is dense in the space of all finite dimensional continuous operators on H with the operator norm. Hence this set is dense in the space of compact operators 1C(H). According to Theorem A.2.11 in Appendix, R., oAIH is a nuclear operator. where Al = CIIPQII;,=(,)
3.7.11. Corollary. Let y be a centered Radon Gaussian measure on a Banach space X, H its Cameron-Martin space, A EC(X) and A(X) C R., (X' ). Then AI H is a nuclear operator. PROOF. We may assume that H is dense in X, passing to the closure Y of the space H in X (which, as we know, has full measure). Note that R.,(Y') = R..(X'). Hence A(Y) C R,(Y'). The operator R, is injective on the space Y. It remains to note that, by the closed graph theorem (see Theorem A.2.1 in Appendix) , the mapping Ao:= R1'oA: Y- Y'
is continuous and AIH = R,oAoIH.
Chapter 3. Radon Measures
128
3.7.12. Corollary. Let X be a centered Gaussian measure on a separable Banach space X. Then, for any continuous linear operator A on H, one has J
f Ilxllx y(dx)
IIAxIIX y(dx) <_
Vp > 1.
(3.7.5)
In addition, for any A E C(X, X'), one has (3.7.6)
I IlxllX y(dx).
IIR, o AIIL(,)(H) <_
x PROOF. Estimate (3.7.5) follows by Corollary 3.3.7, since for any A E £(H) with IIAIIC(H) < 1, the covariance of the measure v = y o A-1 is majorized by that of y. Indeed, IR.,(f oA)IH = IA'Rfl < IR.fI for any f E X'. Now (3.7.6) follows from identity (3.7.4) noted in the proof of Theorem 3.7.10, since the supremum of its right side over the finite dimensional operators K of the form considered in that proof (with II KII c(H) 5 1) equals IIR, o Allc,,,(y), while the left side is majorized by
(f IlKxll r y(dx)) (f 1/2
IlAllc(x.x-)
1/2
IIxIIX
y(dx))
x
X
which is estimated by the right-hand side of (3.7.6) due to (3.7.5).
0
3.7.13. Remark. (i) Note that the alternate representation (3.7.2) of A for a Hilbert-Schmidt operator A corresponds to the expansion
x
x
n=1
n=1
Ah =
h E H.
If X is a Hilbert space, then for every A E C(H), the series in (3.7.2) converges in X to Ax for y-a.e. x, since 5 n=1
n=1
llenllx < .
(ii) Note that in the case of a Banach space, the mapping
£(X, H) - 11(H),
A
Al H.
is a continuous linear operator with the norm estimated by X11 = VI-f 11x112 y(dx), i.e.. IIAII%(H) 5 MMIIAIIc(x.H).
3.7.14. Example. Let y be a centered Radon Gaussian measure on a locally convex space X and let A be a Hilbert-Schmidt operator on H = H(-Y). Then AA' is the covariance operator of the Gaussian measure y o A-1 on H. be an orPROOF. Denote by S the covariance operator of y o A-l. Let thonormal basis in H and let {l:,,} be a sequence of independent standard Gaussian
3.8.
Weak convergence
129
random variables. Then, for any h E H, one has
(Sh,h) = f (y, h)2 1.0A-1(dy) =IE. n=1
H DC
_
(en, A`h)H = (A`h, A`h)N, n=1
which yields our claim.
3.8. Weak convergence of Gaussian measures 3.8.1. Definition. A sequence 1") of Radon measures on a topological space X is called weakly convergent to a Radon measure p if, for each bounded continuous
function f on X, one has
r lim J nix f(x)pn(d,) = 11(x) p(dx).
(3.8.1)
It would be more natural to discuss the weak convergence for Baire measures (i.e., measures on the Baire o-field Bo(X) generated by all continuous functions) rather than for Borel measures, which, in general, are not uniquely determined by their integrals of continuous functions. However, any Radon measure on a locally convex space is uniquely determined by its restriction to B0(X) (and even to C(X )), hence a sequence of such measures cannot have more than one weak limit. The weak convergence of measures plays an essential role both in the theory and applications. It is clear that the weak convergence is generated by the locally convex topology on the linear space of all signed Radon measures on X defined by the seminorms
v-
JfdiII,
fECb(X)
Having defined this topology, we may talk, e.g., about the weak compactness and the relative weak compactness of families of measures. In particular, a family of Radon measures is called relatively weakly sequentially compact if any sequence of its elements has a weakly convergent subsequence. This somewhat cumbersome term has to emphasize that the weak convergence may fail to be metrizable unlike the convergence in the variation norm. The weak convergence of probability measures can be characterized without use of continuous functions. The proof of the following theorem of A. D. Alexandrov can be found in [800, Ch. 1J.
3.8.2. Theorem. A sequence {pn } of Radon probability measures on a completely regular topological space X converges weakly to a Radon probability measure p precisely when either (and then every) of the following conditions is satisfied:
(i) p(U) < lim inf M (U) for any open set U; nix (ii) A(Z) > limsupµn(Z) for any closed set Z;
n-x
(iii) p(B) = Iim pn(B) for any Borel set B whose boundary hasp-measure zero.
3.8.3. Definition. A family of nonnegative Radon measures M on a topological space X is called uniformly tight if, for every e > 0, there exists a compact set
K, such thatµ(X\Kf) <e for allpE M.
Chapter 3. Radon Measures
130
One of the principal results concerning the weak convergence of measures is the Prohorov theorem which we state without proof (see [800, Ch. I, §3]).
3.8.4. Theorem. Let M be a family of Radon probability measures on a completely regular topological space X.
(i) If M is uniformly tight, then M is relatively weakly compact. (ii) If X is a complete separable metric space, then the converse is true: the relative weak compactness of M implies the uniform tightness. In particular. every weakly convergent sequence of Borel probability measures on any complete separable metric space is uniformly tight.
3.8.5. Corollary. Let {pn} be a uniformly tight sequence of Radon probability measures on a locally convex space X. Suppose that the sequence {µn (f )} converges
for every f E X' (or for every f from some set r c X' separating the points in X). Then the sequence {pn} converges weakly to a Radon measure.
PROOF. The set (.a,,} is relatively compact in the weak topology, hence it has a cluster point p. The condition implies that its cluster point is unique (see converges to p (otherwise one Corollary A.3.13 in Appendix). Therefore, would get another cluster point). Note that if a sequence of random elements C. on a probability space (1l, P) with values in a topological space X converges almost sure to a random element , then the measures P o £n 1 converge weakly to P o -1. The following important theorem of Skorohod [707] shows that this example has a universal character.
3.8.6. Theorem. Suppose that a sequence of Borel probability measures pn on a complete separable metric space X converges weakly to a Borel measure p. Then there exists a probability space (tt, P) (in fact, [0, 1] with Lebesgue measure) f2 --* X such that An = P o (n , p = P o t` and and measurable mappings {n
l: a. e.
N. Krylov [431] proved that to every Borel probability measure p on 1Rn, one [0,1] -+ En, where [0, 1] is equipped with can associate a measurable mapping
, a.e., provided pn ' p Lebesgue measure A, such that A o £u = p and f, weakly. It is possible to extend Krylov's theorem to complete separable metric spaces (see also Problem 3.11.46).
It is clear that the weak convergence of measures may be not enough for the convergence of the integrals of unbounded continuous functions. Let us give a simple sufficient condition.
3.8.7. Lemma. Let {p,,} be a uniformly tight sequence of Radon probability measures on a completely regular topological space which converges weakly to a
Radon measure p. Then, for any continuous p-integrable function f on X, satisfying the condition
lim sup
R-- c
one has Jfdn
^
f
IfI On = 0,
IfI>R
Jfd.
PROOF. Let e > 0. Let us choose R > 0 such that
J Il>R If I dpn
+If1>R
IfIdp < e,
do E IN.
3.8.
Weak convergence
131
Put A = (if I < R} and then find a compact set K C A such that
pn(A\K) + p(A\K) < eR-',
`dn E 1V.
Since X is completely regular, the continuous function f extends from K to all of X to a continuous function g with Ig{ < R (see Appendix). Note that Ig(x)I < R < If (x) I whenever If (x) I > R. Therefore,
JfI?R For all n such that
Iffdn
<
IgP dpn +
f
IgI dp < e.
III?R
JgdPn - / g dpl < e, we get
_Jfdp
+e
ffdPn_ffdI+eR_1R+efgdlzn_jgdp+2e <
If gdpn x
- Ix gdp +,-'R+2e < 4e, 0
whence the desired claim.
3.8.8. Corollary. The statement of Lemma 3.8.7 is valid for any sequence of probability Borel measures A. on a complete separable metric space X weakly convergent to a Borel measure p if the continuous function f satisfies the condition in this lemma.
3.8.9. Proposition. Let r = {yQ} be a uniformly tight family of Radon Gaussian measures on a locally convex space X. Then the family of their means {a.,, } is contained in a compact set. In particular, for any weakly convergent sequence from {yj, the corresponding sequence of means converges.
PROOF. It follows from the condition that the sets of measures r1 and r2 obtained from r under the mappings x i-. x/ v12 and x i -. -x1-'f2_, respectively, are uniformly tight. Therefore, the set
ro={p*v, uEr1, vEr2} is uniformly tight, too. Indeed, if p(K) > 1 - e and v(S) > 1 - e, where K and S are compact sets, then Q = K + S is compact as well and, in addition, Jiz(Q - x) v(dx) > (I - e)2,
p * v(Q) > S
since K C Q - x whenever x E S. By construction, one has E ro for y c- r. Indeed, the centered measure y_Q, coincides with the convolution of the images of
the measure y under the mappings x " x/f and x'-. -x/f. Let us now choose a compact set K such that y(K) > 1/2 and y(K +a,) > 1/2 for ally E r. Then
a,EK-K,since (K+a,)f1K#0.
To prove the last claim, let us note that the formula for the Fourier transform of a Gaussian measure shows that the weak convergence of yn toy implies the weak convergence of a,,, to a, in X. Since on the compact sets in X the weak topology coincides with the initial one, the last claim follows. 0
Chapter 3. Radon Measures
132
3.8.10. Corollary. A family of Radon Gaussian measures is uniformly tight precisely when the set of their means is relatively compact and the set of the corresponding centered measures is uniformly tight.
3.8.11. Theorem. Let yn be a uniformly tight sequence of centered Gaussian Radon measures on a locally convex space X. Then, for any continuous seminorm
q on X, there exists a > 0 such that sup f exp(aq(x)2)'Yn(dx) < 0c. n
X
If, in addition, the sequence {yn } converges weakly to a measure yo, then, for any w < a. one has lim f exp(Kq(x)2) 'Yn(dx) = f exp(Kq(x)2) -Yo(dx). nx X
X
In particular, for any positive r, one has
lim f nx X
q(X)r Yn(dx)
= Jq(x)'o(dx). X
PROOF. Let Q be a compact set such that ry,,(Q) > 3/4 for all n. Denote by K the closed absolutely convex hull of Q and by gK the Minkowski functional of the set K, defined on the linear span of K. Then yn(K) > 3/4 for all n E IN. Since q is a continuous seminorm, we get sup q(x) = S < oo.
K
Therefore, q(x) < SgK (x) for all x from the linear span of K, which has full measure by virtue of the zero-one law. According to Fernique's theorem 2.8.5. there exists
a positive number k such that sup f exp(kgK(x)2) yn(dx) < 00. n
X
This gives the first claim. The remaining claims follow from Lemma 3.8.7.
The next results gives a simple and natural approximation of a Gaussian measure by its finite dimensional projections.
3.8.12. Proposition. Let -y be a centered Radon Gaussian measure on a locally convex space X and {e,) an orthonormal basis in H(y). Put Pnx = e, (x)e,. =1
Then the sequence of Gaussian measures y o Pn ' converges weakly to y.
PROOF. Let f be a bounded continuous function on X. We know that P,,x - x in X for y-a.e. x. Hence, by Lebesgue's theorem, we get
lim -oc f .f(x)'Y0Pn'(dx)=rim f f(Pnx)7(dx)= f f(x)'Y(dx), X
which is our claim.
X
X
0
For some spaces, constructive conditions for the weak compactness of families of Gaussian measures are known.
3.8.
Weak convergence
133
3.8.13. Example. Let X be a separable Hilbert space. (i) A family I' of centered Gaussian measures on X with the weak topology is uniformly tight precisely when
sup 1 II=II2 y(dx) < x.
'Er
(ii) A family r of centered Gaussian measures on X with its Hilbert norm is uniformly tight (equivalently, relatively weakly compact) precisely when there exists an injective nonnegative compact operator T on X such that y(T(X)) = 1
for all yerand
supl IIT-'xll2 y(dx) < x. 'Er
An equivalent condition: sup trace (T-' K,T-') < x, where K, is the covariance ,Er operator of the measure y. (iii) A sequence of centered Gaussian measures yn with covariance operators Kn converges weakly to the centered Gaussian measure y with covariance operator K if and only if { Kn } converges to K in the Hilbert-Schmidt norm. (iv) A family I' of centered Gaussian measures on X is uniformly tight (equivalently, relatively weakly compact) if and only if the corresponding covariance op-
erators K, satisfy the following conditions: sup-,Ertrace K, < x and for some x
(and then for any) orthonormal basis {e,) in X the series F_ (K,e,,, en) converges n=1
uniformly in y E r.
PROOF. (i) The existence of a weakly compact set K of measure greater than
3/4 for all -y E I' implies the existence of a ball with this property. Hence the desired uniform estimate follows from Fernique's theorem. The converse follows from Chebyshev's inequality i ( I I x I I > R) < _ R-2 f IIxII2 y(d-r')
and the weak compactness of the closed balls in any Hilbert space. (ii) Let K be a compact set such that y(K) > 3/4 for all y from IF. Choosing an arbitrary orthonormal basis in X, we may assume that we deal with 12. Note that K is contained in some compact ellipsoid Q of the form
/ Q =
{(xn):
-I
t,2x2 < 1}.
where t, -» 0, t, > 0. Indeed, by virtue of compactness of K, one has
lim sup F x? = 0.
"x rEK,=n
Hence we can pick an increasing sequence of natural numbers N. such that
x
E x2<4-" VxEK. Put tl = 1. to+l := 2-" if Nn _< Nn+i. Then K C Q and the operator T E C(12) with the eigenvalues t" is compact. By virtue of the zero-one law, one has y(T(X)) = 1 for all y E r. The uniform boundedness of the integrals
Chapter 3. Radon Measures
134
IIT-1x1 2y(dr)
as in case (i), from Fernique's theorem. It is clear that f K,(X) C T(X), follows, since the Cameron-Martin space is contained in every linear
subspace of full measure. By the closed graph theorem (see Theorem A.2.1 in Appendix), the operators A, := T-1 K1 are continuous. The operators KT-1, defined on T(X). have continuous extensions A;, since
(VT-1Tz,y) = (z, K7 y) _ (Tz.A,y). Hence the operators T-' KT-' are continuous and nonnegative. It remains to note that, taking the eigenbasis {e,} of the operator T, we get
f IIT-'xIIl7(dx) _F'J(x,Te)27(dx) x
= E(K,T-1 ei,T-'ei) = where the trace does not depend on a basis. (iii) The convergence in the Hilbert-Schmidt norm implies the pointwise convergence. Further, since the operators of the form e,.,: x r-. (x, e, )e J form an orthonormal basis in ?{(X), we conclude that on any compact set A in f(X), the 2
series
(e,.3. A)
converges uniformly in A E A. Taking into account the
2
equality (c. , . A)
= (es, Aei )2, this yields the uniform convergence of the series
IIAei1I2. Hence, reasoning in the same manner as above, we pick numbers ti > 0
such that t, -. 0 and supt_ 2I{
2
e, ll
(3.8.2)
Therefore, the sequence is uniformly tight, hence is relatively compact in the weak topology. Since its weak limit may be only the measure -y, we get the weak convergence. Conversely, if the measures -y,, converge weakly to y, then, according to (ii), (3.8.2) holds true. It follows from the arguments given above that is completely bounded in ?{(X). Together with the pointwise the sequence converges to OK_ in ?{(X ). convergence of K to K, this means that { Indeed, if lim II K7 ,, - AIIW(x) = 0, then ,lim II K,,, - AIlc(x) = 0. Hence
-x
-.x
Jim IIKR, - A2IIc(x) = 0, whence A = fT. Claim (iv) follows from (iii), since the 1-x condition mentioned in (iv) is equivalent to the relative compactness of the family
K I E I'. with respect to the Hilbert-Schmidt norm.
0
3.8.14. Remark. Note that the convergence of the covariance operators in the uniform operator norm does not imply the weak convergence of the corresponding Gaussian measures. For instance, it suffices to consider the measures y on 12 with the covariance operators of the form n-'P,,, where P is the orthogonal projection onto the linear span of the first n vectors of the standard basis. In this case, IIn-'P,,ILc(1=) - 0, but the measures y,, do not converge weakly to the Dirac measure bo.
3.8.
135
Weak convergence
3.8.15. Example. A sequence of Gaussian measures p with covariance operators K,,,, and means a,,, on a separable Hilbert space X converges weakly to a Gaussian measure p with covariance operator K, and mean a if and only if the following conditions are satisfied: (i) lim Ila.., - aMII. = 0; n- ac
(ii) lim
KMIIc(x) = 0;
II
(nl) lM f IIXIIa ,n(dx) = f Ilxll p(ox) X
X
PROOF. Condition (iii) can be written as lim
II
Ki,,, Ilat(x) = II
K,.II
x(X)
Therefore, the necessity part follows from the results above. The sufficiency reduces
easily to the case where aµ = a = 0. Condition (ii) yields the relationship
-
KNIIc(x) = 0. Together with (iii) this shows that the sequence { K,,,, } converges to K,, in the weak topology of the space 11(X ). By the convergence of the norms, we get the convergence in the Hilbert-Schmidt norm (recall that if vectors vn in a Hilbert space converge to v weakly and IIvnII -. IIvII, then IIv - vnll ---' 0). Hence we get the claim. lim
K,,,,
II
3.8.16. Remark. Let (St, o) be a measurable space and let p E (1, oo). As shown in [41], a sequence of Gaussian measures pn on X = Lp(1l, v) with covariance
operators K,,,, and means an,, converges weakly to a Gaussian measure p with covariance operator KM and mean a if and only if the following conditions are satisfied:
(i) }li
mIla,,., -avllx =0;
(ii) lim II K,,,, - KHII R
0C
(iii) liini
n-a
f
IIxIIX pn(dx) =
f
0;
IIxIIX p(dx) for some (and then for all) positive r.
X
X
3.8.17. Example. Let X be a separable Hilbert space embedded into a Hilbert space Y by means of a Hilbert-Schmidt operator. Suppose that {pn} is a sequence of centered Gaussian measures on X with the covariance operators K such that the sequence (Knx,x)x converges for every x E X. Then the sequence considered on Y converges weakly.
PROOF. We may assume that X is dense in Y. This reduces the proof to the case where Y is the completion of X with respect to the inner product (Ax, Ay) x, where A is an injective nonnegative Hilbert-Schmidt operator on X. By hypothesis and the Banach-Steinhaus theorem, the operators v - ,,, hence also K,,, are uniformly bounded. Let (en} be the eigenbasis of A corresponding to eigenvalues an.
We can write A = TAO, where T is a diagonal operator in the basis {en} with 0 and A0 is a nonnegative Hilbert-Schmidt operator. Then the eigenvalues t,, ellipsoid
V={xEX: IIAoxIIxSR}={xEX: (Aox.x)x
tight on Y, which, together with convergence of (Knx, x), yields the claim.
Chapter 3.
136
Radon Measures
3.9. Abstract Wiener spaces In this section we discuss the concept of an abstract Wiener space introduced by L. Gross [314]. Our starting point is a separable Hilbert space H with the canonical cylindrical Gaussian measure v, whose Fourier transform is x exp(-I(x,x)H).
As we already know, this measure is not countably additive. The main idea of Gross's method is to find a suitable extension of the space H on which the measure v becomes countably additive. Certainly, we have to precise in what sense "extensions" are understood, since a measure that is not countably additive on a smaller space, neither is on a bigger one. A correct interpretation can be given in the language of linear images of cylindrical measures. We shall say that v is a cylindrical measure on a locally convex space X if v is a nonnegative additive function on the algebra R(X) of the cylindrical sets such
that, for every continuous linear operator P: X -. R", the set function
voP-1: B
v(P-1(B))
is countably additive. The Fourier transform of a cylindrical measure v is defined by the formula
1(f)= fexp(it)vof(dt), fEX'. IRL
Cylindrical measures differ from measures in the usual sense not only by a smaller domain of definition, but also by a possible lack of the countable additivity.
3.9.1. Example. Let X be an infinite dimensional separable Hilbert space. For every cylindrical set C of the form C = P-'(Co), where P is the orthogonal projection onto an n-dimensional subspace X" c X and Co E B(X"), we put v(C) = ' Y- (CG),
where 7 is the standard Gaussian measure on X,, (identified with R"). Then v is a cylindrical measure that is not countably additive. This measure is called the canonical cylindrical Gaussian measure on X. Its Fourier transform has the form y ,-. exp (- 2 (y, y)). PROOF. It is straightforward to verify that the function v is well-defined. The countable additivity of the finite dimensional projections of v is obvious, since they are Gaussian measures. The fact that the function v itself is not countably additive is known from Corollary 2.3.2. 0
Let v be a cylindrical measure on a locally convex space E and let F: E -. X be a continuous linear mapping with values in a locally convex space X. Then
voF-1: C -v(F-1(C)) is a cylindrical measure on X. Suppose now that E is continuously embedded into
X and that v is a cylindrical measure on E. Denote by j the embedding E - X. We shall say that v is countably additive on X if the measure vo j-' is countably additive.
For example, let E = 12 and let X = {(x"): E n-2x2 < 00}. Then the n-l
canonical cylindrical Gaussian measure v on 12 is countably additive on X (in
3.9.
Abstract Wiener spaces
137
fact, v on X is the restriction of the countable product of the standard Gaussian measures).
3.9.2. Definition. Let H be a separable Hilbert space and let P(H) be the set of all orthogonal projections in H with finite dimensional ranges. A seminorm q on H is called measurable in the sense of Gross (or Gross measurable) if, for every e > 0. there exists a finite dimensional orthogonal projection P E P(H) such that
v(xEH: q(Px)>e)<e, v'PEP(H), P1 P,
(3.9.1)
where v is the canonical cylindrical Gaussian measure on H.
Note that the set on the left in (3.9.1) is v-measurable, since the function q is continuous on all finite dimensional subspaces, being a seminorm. The following lemma shows that in fact the seminorm q is continuous on all of H.
3.9.3. Lemma. Any seminorm measurable in the sense of Gross is continuous.
PROOF. It suffices to show that q is bounded on the unit ball Ut, in H. Suppose that suphECH q(h) = oc. It is easily seen that in this case one can find an orthonormal sequence {hn} C U such that q(hn) = C -. oc. Let Pnx = (x, hn)hn. Then
v(x:
e)
=I-J
p(t) dt,
-e/Cn
where p is the standard Gaussian density on IR1. Moreover, given Pe E P(H), one
can find such a sequence with the additional property P 1 P. It is clear that the right-hand side does not tend to zero as n oo. 0
3.9.4. Definition. A triple (i, H, B) is called an abstract Wiener space if B is a separable Banach space, H a separable Hilbert space, i : H B a continuous linear embedding with dense range, and the norm q of B is measurable on H (more precisely, q o i) in the sense of Gross.
3.9.5. Theorem. Let (i, H, B) be an abstract Wiener space. Then the canonical cylindrical Gaussian measure v on H is countably additive on B. In addition. H coincides with the Cameron-Martin space of v on B. PROOF. Denote by q the norm of B. By definition, there exist an increasing sequence of finite dimensional projections P -. I and a sequence c -. 0 such that
v(x: q(Pmx) >
c,,,
'dm > n.
(3.9.2)
Let {en} be an orthonormal basis in H chosen in the set Un 1 Pn(H). Let us consider the Hilbert space E, obtained as the completion of H with respect to the norm
x x ,-+
1/2
{n_2(x.en)}n=1
Denote by jE the embedding H -. E. Applying Theorem 2.3.1, we conclude that the measure v1 = vo jE 1 is countably additive on our auxiliary space E. Clearly,
Chapter 3.
138
Radon Measures
we have H = H(v1). We get the sequence n
S,(x) = E k=1
of B-valued random vectors on (E, v1). Relationship (3.9.2) implies that the sequence Sn is fundamental in measure vi. Since B is complete, this sequence converges in measure to a random vector S in B (this fact is proved in the same manner as for real random variables, see Problem 3.11.38). Let us now verify that the distribution of S. denoted by v2, coincides with the Gaussian measure v o i1 on B. Indeed, for every f E B", there exists a unique vector v E H with f(h) = V h E H. The measure v2 of - 1 coincides with the distribution of the random variable f (S) on the space (E, vi). By virtue of convergence of Sn to S in measure, we get that
x x f(S) = Ek(x,ek)Ef(ek) = Ek(x,ek)e(vek)x k=1
k=1
Note that the functionals G(x) = k(x,ek)e are independent standard Gaussian random variables on (E, v1), since their restrictions to H = H(vi) are given by the vectors ek of an orthonormal basis. Therefore, the variance of f (S) equals Ivl = Ek 1(v,ek)H . o f-1 is also centered On the other hand, by definition, the measure Gaussian with variance o, 2 = Ivy, since the functional foi on H is given by the vector v. This yields that for any g E B', one has (g, f )L2f,,, _ (u, v),,, where u E H is such that g(h) _ (u, h),, , V h E H. Since g(v) = (g, f)L2(,,) = g(R,,, f ), we get the equality v = R,, f, which implies that H = H(v2). 0 voi-1
3.9.6. Theorem. Let y be a centered Gaussian measure on a separable Banach
space X such that H = H(y) is dense in X and let is H
X be the natural
embedding. Then (i, H. X) is an abstract Wiener space.
PROOF. Let {1:,,} be an orthonormal basis in X:, such that { E X. It is clear that y coincides with the image of the standard cylindrical measure v on H under
the natural embedding is H - X, since the variance of every functional f E X with respect to the measure y equals the square of the norm of f I,,. Let us verify the measurability of the norm of X in the sense of Gross. Put n
Pnx=EE,,(x)e,,
e,=R.
,).
,=1
Note that, for any e > 0, one has the equality
lim ry(xEX: IIx - Pxilx > E) =0.
n
Indeed, by Theorem 3.5.1, Pnx -+ x in X y-a.e. Therefore, there exists N such that y(xEX: JIPnx - Pmxi'x > r) <e, Vn,m>N. By definition, one has
y(xE X: ItPnx - PmxIlx > e) =v(xE H: I,Pnx - PmxIIx > e). For any orthogonal projection P E P(H) with P 1 PN, by virtue of the relationships P,,vP = 0 and limk-x P,g+kPx = Px, one has
3.9.
Abstract Wiener spaces
139
+ E H: IIPXIL, > e) = lim v(x E H: {I(PvT,i - P, )Pxll, > F). A
Finally, by Theorem 3.3.6, we get
+ E H: II(Pv+k - Pv)Pxli., > e) < v(.t E H: II(P.,-+x - P,)xJJ.r > e) <
e+
whence our claim.
The following observation is obvious from the reasoning above.
3.9.7. Example. Suppose that a norm q on H is given by q(x) = ITxl r,, where T E £(H). Then q is measurable in the sense of Gross if and only if T is a Hilbert-Schmidt operator.
3.9.8. Example. Let q be a Gross measurable norm on H. Then there exists a symmetric compact operator A on H such that
q(h) < Ahl,,,
Vh E H.
(3.9.3)
PROOF. By Theorem 3.9.5. the unit ball U, of H is compact with respect to the norm q. Now the claim follows by a simple duality argument. To this end, let us put U, = {h E H: q(h) < 1} and let V be the polar of Lq, i.e.,
V = {zr E H: sup I(v°.h),, 15 1}. hEU,
Then V is compact in H. Indeed, it is closed and bounded in H, since Uq contains a neighborhood of the origin in H. Hence any sequence v E V contains a subsequence
w, that is fundamental in the norm q. Then {w,,} is fundamental in H by the relationship
Iwn -
12
= 2q(w,r - u'..)
u're - u'm 2
urn - I'm < 2&-, - U'rez)+ Q(? bn - w n) H -
which follows by the inclusion (wn - wm )/2 E V. By Problem A.3.32. there exists a
symmetric compact operator A on H such that V C A(U ). It remains to note that {h E H: IAhIH < 1} C Uq, which yields (3.9.3). Indeed, let IAhJH < 1. If h ¢ Uq, then, by the Hahn Banach theorem, there exists y E H such that (y.h) > 1 and (y, x) < 1 for all x E Uq, i.e.. Y E V C A(U ). This is a contradiction. since
y = Au. where u E U,,. and (y. h),, _ (A-'y,Ah) < JAhJ < 1.
0
3.9.9. Remark. It is not always possible to find a Hilbert-Schmidt operator A in (3.9.3). For example, this is the case for the classical Wiener measure P" on C[0, l] with the sup-norm. Indeed, otherwise P"' would have a Hilbert support (see Example 3.6.7).
The reader should be warned that in most of the papers dealing with "abstract Wiener space", this term is used just as a synonym of a nondegenerate centered Gaussian measure on a separable Banach space. Sometimes one uses the fact that a norm on H that is majorized by a Gross measurable norm is Gross measurable as well and yields a tight Gaussian measure. Clearly, this follows immediately from the much more general result in Theorem 3.3.1.
Chapter 3. Radon Measures
140
3.10. Conditional measures and conditional expectations The following result is an infinite dimensional version of the classical theorem about normal correlation. Let t; be a random vector defined on a probability space (ft, F, P) and taking values in a locally convex space X. Suppose that l: is Pettis integrable (i.e., weakly integrable: see Appendix). We say that a Pettis integrable random vector 0 in X is the conditional expectation of i; with respect to a a-field A C F if, for every I E X' and every bounded A-measurable function p, one has
3.10.1. Theorem. Let C and q be two Gaussian random vectors with values in a locally convex space X such that the vector (C, q) has a Radon Gaussian distribution in X x X. Then the conditional expectation E(glq) of the vector f with respect to the a-field generated by q exists and is a Gaussian vector. In addition, = where ( is a Gaussian random vector independent of q and hating a Radon distribution.
PROOF. We may assume that X is Souslin, since both l; and q belong with probability I to some Souslin subspaces. It suffices to prove the claim for centered vectors. Let us embed X into 1R" by a continuous linear injection generated by C X' separating the points in X. Then and q as 1R"-valued a sequence vectors are represented by two centered Gaussian sequences {t;,,} and (in fact, trt = qrt = f.(q))- Let. E(t;,lq) be the orthogonal projection of C, to the
x
in L2(P). Then E(C,lq) = E c,q°, 1=1 x where {q°} is the orthogonalization of {rl,} and E c' < oc. Hence E(t1 q) is closure of the linear span of the sequence.
,=1
measurable with respect to the o-field generated by q. Writing
E(£,lq) + C.
i E IN, we get a centered Gaussian vector C = {(i) independent of q. Indeed, for rt
any reals o i ..... a., the random variable E a,(, is in the closure of the linear span i=1
of tj and q,, j E IN, hence is centered Gaussian. In addition, it is independent of q, since it is orthogonal to the linear span of (%J and has Gaussian joint distribution with q. Clearly, the vector {E(lilq)}x 1 can be taken for E(l;lq) in the space IR". Let us show that and ( with probability 1 belong to X. It suffices to do this for E(t;Irl), since 1; = E(t;lq) + S. Denote by p the distribution of l; in X (and
also in lRx) and by ry the distribution of E(t;lq) in I. Let a be the distribution of ( in 111'. By construction, p = 7 * a. Hence we can apply Theorem 3.3.6, which gives 7(X) = 1. Now we conclude that E(l;Iq) is the conditional expectation of t;
in X. Indeed. let I E X. It remains to be shown that 1(() is independent of q. To this end, it suffices to note that there exists a sequence {g } of finite linear combinations of the functionals f, such that 1(() a.e. In order to get such a sequence, we just orthogonalize {f,} in V(a) (where a is the distribution of () and obtain a basis in X;, since X was assumed to be Souslin and {f,} separates the points.
The following theorem deals with conditional measures - an object related to the conditional expectation.
3.10.
Conditional measures
141
3.10.2. Theorem. Let y be a centered Radon Gaussian measure on a locally convex space X, let h E H(y) be a vector with unit norm, and let Y be a closed hyperplane such that X = Y G IR1 h. Denote by v the image of the measure -y under
the natural projection to X -. Y. Then, for any y E Y, one can choose a Gaussian measure yY on y + R' h in such a way that
(B) = I yV(B)v(dy), dB E B(X).
(3.10.1)
This choice can be made with the additional property that the measure ye has mean
y - h(y)h and covariance f - f (h)2, i.e.. is the image of the standard Gaussian measure on R' under the mapping t - th + y - h(y)h and, for the natural parametrization t +-» y + th, the measure yY has density p( , -h(y), 1), where h is a proper linear version. PROOF. We may assume from the very beginning that X is a Souslin space that
is a union of countably many metrizable compact sets, since, replacing X by the corresponding linear subspace Xo of full measure, we have X0 = (Y n X0) ® R1h, Y n X0 is closed in X0, h E X0 and v(Xo) = 1. Note that Y = g-1(0) for some g E X. We may assume that g(h) = 1. Then the natural projection X --+ Y has
the form x - x - g(x)h. Recall that h(h) = (h, h)H(,) = 1. It is not difficult to verify that, with the aforementioned choice of y11, the set function on the right-hand side of (3.10.1) is well-defined (i.e., the function y - y1'(B) is v-measurable) and is a Radon measure (since every Borel measure on a Souslin space is Radon). Indeed,
yv(B) = yi (t: th + y - h(y)h E B), where yl is the standard Gaussian measure on R.'. Hence the measurability of yV(B), B E B(X), with respect to v follows from the measurability of the function yY(B) = yi (t : th + x - g(x)h - h(x)h + g(x)h(h)h E B)
= yi
(t: th + x - h(x)h E B).
Now it suffices to note that the function (t, x)
IB (tit + x - h(x)h)
is yl®y-measurable for any B E 8(X) = E(X), since this is true if h E X`, and in the general case there exists a sequence {In} C X' such that In -+ h y-a.e. Hence, in order to prove equality (3.10.1), it suffices to check that the Fourier
transforms of both sides coincide. Let f E X*. Then 7(f) = exp(- 20(f )2) . On the other hand,
v(f) = exp[if (y - h(y)h) - 2'f
(h)2].
Chapter 3. Radon Measures
142
Let us integrate this expression in v. By using the change of variables formula and the equality h(h) = 1, we get
fex[i( y - h(y)h) exp (
2f
11
f (h)2J v(dy)
(h)2) fexp[ii(x - g(x)h - h(x)h + g(x)h)] 7(dx) .r
=exp -If(h)2 exp
-2a(f
-f(h)h)'21.
It remains to note that f (h)2 + a(f - f (h)h) 2 = f (h)2 + a(f)2 - 2f(h)(f. h)L2(,) + f (h)2, which equals or(f)2, since (f, h)L2(,) = f (h) by Lemma 2.10.5.
3.10.3. Corollary. Let E C H(y) be a k-dimensional linear subspace, let Y be a closed linear subspace in X such that X = Y r E, and let v be the image of y under the natural projection X -. Y. Then, for every y E Y. one can choose a Gaussian measure yy on E + y such that equality (3.10.1) is valid. In addition. we can choose these measures in such a way that they have equal covariances and yy has a Gaussian density (a shift of pk) with respect to the natural Lebesgue measure on E + y induced by the inner product from H(y).
3.10.4. Remark. For the existence of the conditional Gaussian measures on E + y it is not necessary that E be contained in H(y) or be finite dimensional. However, in the general case, conditional measures may be concentrated at single
points. For example, this happens if E n H(y) = 0 due to the singularity of the shifts to the vectors from such E.
3.10.5. Remark. We know that for any set A with y(A) = 1 and any h E H(-y), the sections {t E IR' : x + th E A } have full Lebesgue measures for almost every x. However, even in the two dimensional case the corresponding measure zero set can depend essentially on h. Recall that by a classical example of Nikodym (see, e.g., [323, Ch. V, §31), there exists a full measure set A C IR2 such that for every x E A, there exists a unit vector v= with the property (x + 1R' vs) n A = {x}.
3.11. Complements and problems More about measurable linear functions Note that, by Theorem 3.4.1 and Lusin's theorem, for any linear functional f that is measurable with respect to a Radon Gaussian measure and any positive E. there exists a metrizable compact set of measure greater than 1 - e, on which the functional f is continuous. A simple observation yields the following result.
3.11.1. Proposition. Let y be a centered Radon Gaussian measure on a locally convex space X and f E X.. Then one can find a proper linear modification off (denoted by the same symbol) with the following properties: (i) There exists a Borel Souslin linear subspace X0 C X of full -r-measure such that the functional f is Borel on Xi,:
3.11.
Complements and problems
143
(ii) If X is sequentially complete. then there exists a metrizable absolutely convex
compact set K such that the functional f is continuous on all sets nK and one has 7(nK) -. 1.
PROOF. Let us take a sequence If,} E X' convergent to f y-a.e. By the classical Egorov theorem and Theorem 3.4.1, there exists a metrizable compact set
S of positive measure such that fn -+ f uniformly on S. Let K be the closed absolutely convex hull of S. The Hahn-Banach theorem implies that, for every I E X', one has SUR EK 1l(4 = sup=ES J1(x)j. Hence the sequence {f"} converges uniformly on K. The same is true for mK replacing K, whence the continuity of
the limit on mK. Setting
f'(x) = n-w lim fn (x) on the set L of all the points where the limit exists (note that L is linear and has full measure), we get a measurable linear functional, which coincides with f on a set of positive measure, hence almost everywhere. Now f- can be extended to all of X by linearity using an arbitrary Hamel basis in X. Since the functionals fn are continuous, L is a Borel set (moreover, L E E(X)) and the functional f' is Borel measurable on L (and even belongs to the first Baire class). In order to prove claim (i), note that the linear span X0 of the set S is contained in L, and coincides with the union of the sets Dn,m = StisS +
+toSnl Et,J <m,
,E
S}.
It is clear that these sets are metrizable compact as continuous images of the metriz-
able compact sets [-m, m]" x Si'. The subspace X0 has full measure by virtue of the zero-one law. In order to prove claim (ii), it remains to note that y(nK) -+ 1 0 and that n K is a metrizable compact set by Proposition A.1.7 in Appendix.
3.11.2. Proposition. Let y be a centered Radon Gaussian measure on a locally convex space X (or a centered Gaussian measure on l; (X )) and let f : X - IR' be a y-measurable function such that
f( f)= x + y
f(x) + f(y)
2
for y$y-a.e. pair (x, y) E X xX. Then f E X'. PROOF. It follows from the condition that f is a centered Gaussian random x variable (see Theorem 1.9.5). In particular, f E L2(y). Let f = 2 In(f) be n=o
the decomposition of f in the Wiener chaos (see (2.9.2)). Let us define t by the equality e-' = 1//. Then the condition and the equality f dy = 0 imply that
J x Tt f = f / f a.e. Since Ttf = E e- "' In (f ), one has I.(f) = 0 whenever n # 1. n=o
Thus, one has f = 11(f ), whence f E X,*.
0
The following theorem contains some precisions of Theorem 2.10.9, obtained partially by Tsirelson [774] (whose arguments we use) and proved in the form presented below by Talagrand [753[. Recall that a Borel measure p on a topological space X is called r-additive if, for any increasing net of open sets U. C X, one has
Chapter 3.
144
Radon Measures
the equality
limµ(U0) =µ Any Radon measure is r-additive, but there exist non r-additive Borel measures (see [800, Ch. I]). Note that if X is a complete metric space, then a Borel measure p on X is r-additive if and only if it is Radon (see [800, Ch. I]).
3.11.3. Theorem. Let X be a locally convex space and let ) be a Gaussian measure on E(X) (or a r-additive Borel Gaussian measure). Then, for any affine function f on X. the following conditions are equivalent: (i) There exists a set A E E(X) (respectively, A E 8(X)) of positive measure such that f (A) does not intersect some compact set of positive Lebesgue measure;
(ii) In claim (i) one can take an interval for such a compact set: (iii) f is a-) -measurable function: (iv) For some real number r. the function f - r belongs to the closure of X' in the space L2(-y).
In either case f is a Gaussian random variable. PROOF. Clearly, one may assume that f is linear. We shall consider the case where - is either a centered Gaussian measure on .6(X) or a centered Radon Gauss-
ian measure and the function f is bounded on some set A of positive measure. In the case of a measure on E(X) the set A can be taken in E(X ), and it can be taken compact for a Radon measure. By virtue of linearity, the function f is bounded on the absolutely convex hull B of A. Hence the set MI of all measurable functions V, for which f < a.e., is nonempty. Indeed, one can take O = (SUPB f) p,,, where p is the Minkowski functional of B, which is -y-measurable by Problem A.3.41. According to a well-known result from measure theory (see 1214, p. 335, Theorem IV.12.6]), the set Mf has infimum f, in the space Lo()) of all --measurable functions with its natural ordering V < 71, a fi(x) < V'(x) a.e. This means that
f < fi a.e. and fl is the minimal function from Lo(y) with such a property. In a similar way, there exists the maximal function fo E Lo(7), for which fo < f a.e. The functions fo and ft are called the measurable minorant and measurable majorant of f, respectively. Note that, for a function with the measurable majorant rl. the function n(x)+q(y) on (X xX,-yery) is the measurable majorant for The function l(x, y) = f (x) on XxX has the measurable majorant I1(x. y) = f, (x), and the function f(x+y) = f(x)+f(y) 9(x y) =
f
v"2-
has the measurable majorant gl (x, y) = f1 (x) + f1 (Y). The transformation
v2
T: (x,y)
x -yr +y
v2 v/'2preserves the measure 7 $ ti and transforms I into g (i.e., g o T = 1) and 1 into g2(x, y) = fl (xv+12-y). By the uniqueness of measurable majorants, we get gl = g2
7®y-a.e. According to Proposition 3.11.2, one has fi E X.. In a similar manner, fo E X. Since fo < f < ft and f, - fo is a Gaussian random variable with mean 0 zero, we get fo = fi a.e. Hence f = f, a.e.
3.11.
Complements and problems
145
3.11.4. Corollary. Any Gaussian measure -1 on the space 1°C' with the *-weak topology a(11,11) extends to a Gaussian measure on 1" with the norm topology, i.e.. all functionals from (I")' are -y-measurable.
PROOF. It suffices to note that any ball in 1" is an element of the a-field generated by the coordinate functions. Hence any functional I E (l")' is bounded on a positive measure set from E((l".a(lx l1))) In connection with the last corollary the following fact is worth noting (which is valid, as shown in [119, Ch. 8], under the weaker set-theoretical assumption that every uncountable cardinal not exceeding the continuum is accessible).
3.11.5. Proposition. Let p be a centered Gaussian measure on 1" such that
an = f xn µ(dx) - 0. If p(co) = 0, then, assuming the continuum hypothesis. the measure p cannot be extended to a measure on the Borel a-field of 1". In particular, this is the case if p is the product of centered one dimensional Gaussian measures with variances i. a'2 = (log(n + 1)) PROOF. Suppose that it extends to 8(l'). It follows from our assumption that the measure it is concentrated on some separable closed subspace Z C 1" (see [64, Appendix III] or [119, Appendix 2]). Then every centered ball of positive radius from Z, hence also every centered ball from 1', has positive measure. This means that the oscillation constant of the sequence of the coordinate functions equals zero (see Theorem 2.12.11). Therefore, p(c0) = 1, which is a contradiction. The last claim follows from Example 2.3.7.
3.11.6. Remark. Even if X is a separable Hilbert space, a 7-measurable linear functional f may fail to have a Borel linear modification on the whole space. Moreover, according to Proposition 3.11.12 below, if a linear function on a Frtchet space (or on a barrelled space, see [670, Ch. II]) is measurable with respect to every Radon Gaussian measure on X, then it is continuous. Let us give a one more characterization of measurable affine functions.
3.11.7. Lemma. Let f be a measurable real function on IR' such that there exists a function g with the following property: for every fixed h, one has
f (x + h) - f (x) = g(h) a.e. Then g is linear and f coincides a. e. with an affine function x - cx + a. PROOF. First of all note that g(h + k) = g(h) + g(k). To see this, let us put Q2 = {x: f (x + m) = f (x) + g(m) }. Then there exists z in the intersection of the full measure sets 12h+k, Ilk, and fIh -k, whence
g(h + k) = f(z + h + k) - f(z) = f(z + h + k) - f(z + k) + f(z + k) - f(z). Further, g(0) = 0 and g is continuous at zero. Indeed, given e > 0, let us take a positive measure set B such that f (x) E (y - e, y + e), V x E B, for some real y. Then B - B contains some interval (-r, r). Hence g(h) < (-2e, 2e) if h E (-r, r),
Chapter 3.
146
Radon Measures
because g(h) = f (x + h) - f (x) for some x E B such that x + h E B. This implies that g(h) = ch for some real c. By Fubini's theorem, there exists xo such that f (xo + h) = f (xo) + ch for a.e. h. This yields an affine version of f. 0
3.11.8. Proposition. Let y be a centered Radon Gaussian measure on a locally convex space X and let f be a 7-measurable function such that, for every fixed h E H(7), one has f (x + h) - f (x) = g(h) y-a.e., where g(h) is some number.
Then f = c+ fo y-a.e., where c E R' and fo E X. . PROOF. Let {e;) be an orthonormal basis in H(-y). According to Lemma 3.11.7, for every i, the function f admits a version gyp; that is affine along the lines x+Rlej. = c; a.e. for some number c;. By Proposition 3.5.5, By the zero-one law, f(x)=ciet(x)+...+caen(x)+i0n(x)
where vG (x) = gn
a.e.,
(x), en+2(X).... ) for some Borel function g on R. Note
that E c? < oo. Indeed, otherwise -l
JexP(itf)d.r = exp(-2t2
c?1 Jex(itibn)d-t--0, bt94 0,
as n -- oo, which is a contradiction, since y o f -1 is a probability measure on 1111.
Hence the series E c,e, converges almost everywhere, whence the convergence of the sequence {76n}. By the zero-one law, {t1,,} converges a.e. to some constant, 0 whence our claim.
3.11.9. Corollary. Let y be a centered Radon Gaussian measure on a locally convex space X and let T be a y-measurable mapping with values in a separable FYechet space Y such that, for every fixed h E H(-y), one has T(x + h) - T(x) = g(h) 7-a.e., where g(h) is some element of Y. Then T has an gain modification. The following example shows that the continuity of a linear function l: X Rt on the Hilbert space H(y) does not imply its measurability with respect to y. This example was constructed by H. von Weizsi cker assuming the continuum hypothesis;
N. Nedikov constructed an analogous example assuming Martin's axiom and the negation of the continuum hypothesis. We do not know whether such an example can be constructed without extra set-theoretical assumptions.
3.11.10. Example. Let y be a centered Gaussian measure on a separable Banach space X (or a centered Radon Gaussian measure on a locally convex space) such that H(y) is infinite dimensional. Suppose that the continuum hypothesis is true. Then there exists a linear function I on X, equal zero on H(7), but nonmeasurable with respect to the measure 1. PROOF. Let X be a separable Banach space. Since measurable proper linear functionals vanishing on H(y) equal zero almost everywhere, it suffices to extend the identically zero functional from H(y) to a linear function I on X in such a way that 1(x,) = 1, where {xa} is a family of vectors linearly independent of H(y) with 0.
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Such a family is constructed by the aid of the transfinite induction. To this end, we establish a one-to-one correspondence between the family K of all compact sets in X of positive y-measure (this family has cardinality c) and the interval of ordinals I = [0, w1), where w1 is the first uncountable ordinal, which corresponds, by virtue of the continuum hypothesis, to the cardinality of K equal c. Let 3 E I and assume that, for all a <,3. some vectors xa E K0, Ka E K = {Kr}TE1, are chosen in such a way that the family {xr}r<e is linearly independent of H(y). The set of all ordinals not exceeding ,3 is at most countable. Hence the algebraic sum of the linear span
of xa, a < ,3, with the set H(y) has measure zero (this follows from the fact that H(y) is infinite dimensional, see Problem 3.11.30). Therefore, the compact set Kp of positive measure contains an element x,3 that does not belong to this algebraic sum. By the transfinite induction principle, we get a family of vectors with the desired properties (note that y'({xa}) = 1, since {x0} intersects every compact of positive measure). If X is a locally convex space, we apply our construction to the linear span of an arbitrary metrizable compact set of positive measure and then extend the functional obtained to the whole space by linearity.
3.11.11. Example. Assume that the continuum hypothesis is true. Then, on every infinite dimensional separable Frechet space X (e.g., on X = IR" and X = 12), there exists a linear functional I which is nonmeasurable with respect to every Gaussian measure µ on X which is not concentrated on a finite dimensional subspace. Moreover, u. (Ker 1) = 0 and p (Ker l) = 1.
PROOF. It is easily verified that the family K of all compact sets in X whose linear spans have uncountable algebraic dimensions has cardinality continuum. According to the continuum hypothesis, K can be set in a one-to-one correspondence with the interval of ordinals I = [0, wl ), where w1 is the first uncountable ordinal. By the transfinite induction principle, there exists a linTherefore, K = early independent set A, which contains exactly two elements u0, t', from every Ka. Indeed, for any 3 < wl. the interval [0,;3) is at most countable. Since the linear span of K,1 has uncountable algebraic dimension, there exist two vectors uo, va E Ka
such that adding them to {un. va, a < 3}, we get a linearly independent system. Put l(u0) = 0, 1(va) = 1, and complement A to a Hamel basis. Let I be zero on all additional elements of this basis. Extending I by linearity to the whole space, we get a functional which is nonmeasurable with respect to all measures, vanishing on the finite dimensional subspaces. Indeed, if the set l`' (c - 1/2, c. + 1/2) were measurable with respect to such a measure µ and had positive measure for some c E IR' , it would contain some K E K. which is impossible, since l assumes the
values 0 and 1 on every Kb. In fact, µ.(l-'(0)) = 0 and u*(1-1(0)) = 1, since 1-1(0) meets every compact set of positive u-measure. Note that the previous example (borrowed from [1691) works for all (not necessarily Gaussian) measures vanishing on the finite dimensional subspaces. The next result is due to Borell [96] (see also [408]).
3.11.12. Proposition. Suppose that a linear functional I on a F1-echet space X is measurable with respect to every Radon Gaussian measure on X. Then I is continuous.
PROOF. Suppose that I is discontinuous. Then there exists a sequence a" 0 in X such that l(a") > 2". Let V be the closed absolutely convex hull of {a,,}. Then
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148
V is a bounded set (see [670, p. 39]). Clearly, {an} has the infinite dimensional linear span (any linear function on a finite dimensional space is continuous). Hence we may assume that the vectors a are linearly independent. Let ry be the Gaussian measure on 12 that is the countable product of the one dimensional Gaussian measures with variances n-2. The mapping n-2xnan
F: 12 -+ X, (x,,) a-+
n=l
is well-defined and continuous, since the unit ball of 12 is mapped into 2V. The measure µ = yoF-l is Gaussian on X. The sequence {n-'a,, } is contained in the unit ball of the Hilbert space H(µ) = F(H(y)). Indeed,
x H(7) = {(hn): E n2h2 < x }. nsl
The unit ball of H(-y) is U = {(hn): F_n t n2h,2, < 1}, and the unit ball of H(p) equals F(U). Therefore, the functional I cannot be IA-measurable, since otherwise it would be continuous on H(µ) and bounded on the sequence W4 a.). El
3.11.13. Corollary. Let X and Y be JWchet spaces and let T : X - Y be a linear mapping that is (B(X),,,E(Y))-measurable for every Radon Gaussian measure 7 on X. Then T is continuous. In particular, this is true if T is Borel. PROOF. By the previous proposition, t oT is continuous on X for every 1 E Y
This means the continuity of T in the weak topologies of X and Y. whence the continuity in the initial topologies which coincide with the Mackey topologies (see [670, p. 158. Ch. IV. Theorem 4.7]). In a similar manner one proves (see Problem 3.11.40) that any convex function on X that is measurable with respect to all Gaussian measures is continuous. It is seen from the proof that in the case of an arbitrary sequentially complete locally convex space we get the sequential continuity of 1. Obviously, one cannot drop the sequential completeness assumption (it suffices to take the linear subspace of all finite sequences in 12 and note that every Gaussian measure on it is concentrated on a finite dimensional subspace). An example of a discontinuous sequentially continuous linear functional shows that a linear function measurable with respect to all Radon Gaussian measures may fail to be continuous in the case of an arbitrary (even complete) locally convex space.
Gaussian measures on LP \Ve already know that there is a natural correspondence between Gaussian measures on LP and Gaussian processes with the paths in LP (see Example 3.5.12 and Example 2.3.16). Let us consider Gaussian covariance operators on V.
3.11.14. Example. Let (T, T, a) be a measurable space with a nonnegative (finite or a-finite) measure a and let l; = (Et)teT be a measurable centered Gaussian process. Suppose that t tt(w) belongs to LP(a) for a.e. :,:, where p E 11. x).
Then the covariance operator R,,,: L9(a) -. L"(a), q = p(p - I)-', of the Gaussian measure u induced by 1; on LP(a) according to Example 2.3.16 is given by the
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formula Kt(s,t)f(t) a(ds), f E L9(a),
(3.11.1)
T
where k. is the covariance function of PROOF. By definition, h = R,(f) is the function in LP(a) such that, for every 9 E L9 (a), one has
f h(t)g(t) a(dt) = E [f f (t).t(,a) a(dt) f g(s)t:a(w) a(ds)] T
1'
.
T
It remains to rewrite the right-hand side as Jf KK (t. s) f (t)g(s) a(dt) a(ds). T 7'
which is possible by Fubini's theorem (Kf is measurable by the measurability oft;).
Note that Fubini's theorem is applicable due to the estimate EI&I < Kf(t.t)112 and the integrability of the function Kf(t, t)P'22 with respect to or established in Example 2.3.16. The fact that the integral on the right in (3.11.1) is well-defined D is seen from the estimate IKt(t,s)l < Kf(t,t)112Kf(s,s)1/2. Now we can find a characterization of the covariance operators of Gaussian measures on LP(a).
3.11.15. Proposition. Let LP(a), where a is a nonnegative measure (finite or a-finite) on some measurable space (T.T), be separable, let 1 < p < oc. and let µ be a centered Gaussian measure on LP(a). Put q = p(p -- 1)-1. Then the LP(a) of p is given by the formula covariance operator Rµ: L9(a)
R (f)(s) = f K(s, t) f (t) a(dt), f E L9(a),
(3.11.2)
where K is a symmetric nonnegative definite measurable function on T2 satisfying
f K(t. t)P12 a(dt)
< x.
(3.11.3)
'r
Conversely, for any symmetric nonnegative definite measurable function K on T2 satisfying (3.11.3), the operator defined by (3.11.2) coincides with the covariance operator of some centered Gaussian measure on LP(a). PROOF. According to Example 3.5.12, there is a measurable centered Gaussian
process t; having the sample paths in LP(a) and generating the measure p. Let K be the covariance function of t;. By Example 3.11.14, the covariance operator of p is given by (3.11.2). Since t is centered Gaussian, there exists a constant c such that K(t, t)p12 = (1:t:? )p12 < cEjt"tjP for all t. According to Example 2.3.16, the right-hand side of this estimate is integrable with respect to or. Let us prove the converse. According to Example 2.3.16 and Example 3.11.14, it suffices to show that there exists a measurable centered Gaussian process with covariance function K. To this end, note that the claim holds true if p = 2. since the integral operator generated by K is nuclear. Indeed, by condition we have
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150
Radon Measures
K(t. s)2 < K(t, t)K(s, s). Hence the integral operator Af (t) = 1 K (t. s) f (s) a(ds) on L2(a) is Hilbert-Schmidt. Since this operator is nonnegative, it has an eigenbasis {e,, } corresponding to nonnegative eigenvalues {k,,}. Then
k_ n=1
/
K(t,
J
a(ds) a(dt).
n=] T T
By the identity K(t, s) =
and FLbini's theorem, the right-hand side
coincides with fi
rs-1T
a(dt)K(t,t)a(dt),
kt(W) I
Ill'
since, for a_e. W. one has
fr(w) _
J
(w)e,,(s)a(ds)e,,(t).
rs=I T
where the series converges in L2(a).
In the general case, let us take a positive function e such that K(t. t) is integrable against the measure u: = ea and let l; be the corresponding measurable centered Gaussian process with distribution in L2(v). It remains to note that by virtue of (3.11.3) one has ft(w) E LP(o) a.s. Indeed, by the Gaussian property, there exists C > 0 such that EIStf P < C(IE 2)P/2 = CKt(t, t)P/2.
Supports and r-additivity of Gaussian measures Let us mention the following nice result conjectured by A. Tortrat and proved by Talagrand 17521. Note that the r-additivity can be defined also for a measure p on the or-field £(X) in a locally convex space X as the property that, for every increasing net of open sets UQ E £(X) with U. U, E £(X), there holds the equality p(UQ UQ) = limo p(UQ). It is known (see [800. Theorem 1.3.21), that in this case p extends uniquely to a Borel measure which is r-additive on 8(X) (i.e., the corresponding property is preserved for all open sets U,).
3.11.16. Theorem. Every Gaussian measure on IRT is r-additive. 3.11.17. Remark. The property stated in the previous theorem may be useful, since any Gaussian measure on £(X) extends naturally to a Gaussian measure on the space IRT with T = X* (see Example 2.3.10). A useful property of r-additive measures is that they have topological supports, since any union of measure zero open sets has measure zero in the case of a r-additive measure.
3.11.18. Remark. Let X be a Banach space and let p be a probability measure on £(X) which is r-additive (e.g., Radon) for the weak topology of X. Then, by a result of R. Phillips (see [800, Ch. 1, §5, Corollary 41), it admits an extension to a Radon measure in the norm topology of X. In particular, if X is a reflexive Banach space (e.g., LP(a), p E (1, oo)), then every measure p on £(X) has a Radon extension in the norm topology (see [800, Ch. I, §5, Corollary 5)).
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The reader should be warned that there exist Borel Gaussian measure, which are not r-additive (see [753, p. 183]). Example 3.11.11 yields non Radon Gaussian measures on separable normed spaces.
3.11.19. Example. Assume that the continuum hypothesis is true. Then there exists a Borel Gaussian measure p on a separable Euclidean space X such that p(K) = 0 for every compact set K C X. PROOF. Let -y be an arbitrary nondegenerate centered Gaussian measure on 12. As shown in the proof of Example 3.11.11, there exists a linear functional I on 12
such that y.(1-I(0)) = 0 and y'(1-'(0)) = 1. Let us put X = I'(0). Clearly, X is a separable Euclidean space (with the inner product from 12). We shall take for p the restriction of y to X. That is, for every B E 8(12), we put p(X nB) = y(B). Note that if X n Bl = X n B2, then y(B,) = y(B2), since X has outer measure 1. Hence p is well-defined. Clearly, it is a Borel measure, since every set C E B(X) has the form C = X n B for some B E 8(12) (note that this is true for all open sets by the definition of the topology in X). Every continuous linear functional on X is a restriction of a (unique) continuous linear functional on 12, which shows that p is centered Gaussian on X. Finally, p(K) = 0 for every compact set K C X, since K is compact in 12 as well.
3.11.20. Remark. Let X be any separable Banach space and let I be the linear functional constructed in Example 3.11.11. The previous example shows that, for every Gaussian measure y on X not concentrated on a finite dimensional subspace, the corresponding measure p on Xo = Ker 1, has the following properties: its Cameron-Martin space H(p) is not Hilbert (i.e., is not complete) and there exists f E Xo such that R- f ¢ Xo. Indeed, by construction, Xo contains no continuously embedded infinite dimensional Hilbert spaces (since it contains no compact sets with linear spans of uncountable dimension). As a result, X0 cannot contain since there exists an infinite dimensional Hilbert space E continuously and densely embedded into X', which would give a continuously embedded infinite dimensional Hilbert space C H(p) C Xo. It is interesting to note that this is true for every Gaussian measure on X0 not concentrated on a finite dimensional subspace, since every such measure can be obtained as the restriction of some Gaussian measure on X as described in the previous example.
3.11.21. Remark. Note that Lemma 2.7.1 stating that, for a Gaussian meap(A)_lp[.4 sure p, the measure may be Gaussian only in the case p(A) = 1, remains valid (with the same proof) also for any Borel set A. provided p is Radon.
In connection with convergence of Gaussian series, let us mention the following result which is a special case of a general theorem due to J. Hoffmann-Jorgensen and S. Kwapien. Its proof can be found in [800, Ch. V, §61.
3.11.22. Theorem. Let X be a separable Banach space not containing closed linear subspaces linearly homeomorphic to co. Suppose that {Xn} is a sequence of independent centered Gaussian vectors in X such that D
supEIlX`II
(3.11.4)
Chapter 3.
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Radon 14feasures
for some p > 0. Then the series E X, converges almost sure in X and (3.11.4) += t
is true for every p > 0. In particular, the series in (3.11.4) converges almost sure if SUP 11 E X, Jf < oc a.s. Conversely, the almost sure convergence of this series *+
implies (3.11.4).
Note that this result applies to all reflexive separable Banach spaces. e.g., to LP [a, b], 1 < p < oo. Simple examples show that it is not valid for the space co (see, e.g., 1800, Ch. V, §5]).
Recall that a Banach space X is said to be a space of cotype 2 if E II rn II2 < o0 X
n=1
for any sequence {xn } C X such that the series E enx,, converges a.s.. where the
-1
en's are independent random variables with P(sn = 1) = P(en = -1) = 1/2. If the 11X'112` < Cc implies the a.e. convergence of the series L`,rn, then condition n=1
[' n
X is said to be a space of type 2. For example, the spaces LP have cotype max(p. 2) and type min(p, 2). The proof of the following result can be found in (164], [499], [557], [800].
3.11.23. Theorem. If ry is a Radon Gaussian measure on a Banach space X of cotype 2. then y has a Hilbert support. In particular, this is true for X = 1" with 1 < p :S 2. Conversely, if every Radon Gaussian measure on X has a Hilbert support, then X has cotype 2.
3.11.24. Remark. Let y be a centered Gaussian measure on a separable Banach space X. Recall that an operator S E G(X', X) is called nuclear if x x Sf = E Anf(an)bn, where IIa,, 11 = IIbnII = I and E IA,I < oo. We know that the n=1
n=1
covariance operator R., is nuclear. In addition, it is symmetric and nonnegative, i.e., (g. R, f) = (f, R,g) and (f, R., f) > 0 for all f, g E X*. However, not every nuclear, symmetric and nonnegative operator S E £(X', X) is the covariance of a Gaussian measure on X. It is known that the class of all nuclear, symmetric and nonnegative operators between X' and X coincides with the class of the covariance operators of Gaussian measures on X precisely when X is a space of type 2 (see, e.g., [164], [472, Ch. 9], [501], and the references therein).
Measurable linear extensions and the second quantization 3.11.25. Lemma. Let y be a centered Radon Gaussian measure on a locally convex space X and let A and B be two bounded linear operators on the Hilbert space H = H(ry) such that AA' + BB* = I. Then the image of the measure ry y under the mapping (x, y) -' Ax + By, X x X -» X.. is y. PROOF. Clearly, the mapping above is (B(X x X), E(X) )-measurable and, for
every f E X', the functional g: (x, y) f (Ax) + f (By) is linear ry:&y-measurable. In addition, the functional g is a Gaussian variable with mean zero and variance
f f(Ax)2y(dx)+ f .f(by)21'(dy)=IR-,(foA)] +JR (foB)IN.
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The latter coincides with I A' R f I + I B' R- f I , since, according to Lemma 3.7.8, H B. It remains H R,(f o A) = A'R, f and similarly for to note that, for every h E H, one has by condition
IA*hI2
2 + IB'hIH = (AA'h,h)H +(BB'h,h), = (h, h),.
Thus, g has variance IRY f I2 = a(f )2, whence the desired conclusion.
Now, for any operator T EC(H) with IIT(y) <_ 1, we can introduce its second quantization r(T) E C(LP(y)), p > 1, by the relation
F(T)f (x) =
f (T`x + Sy) - (dy),
f E LP(-y),
J x Note that r(T) equals TT if T = e-tI. The construction of
where S = I the second quantization originates in the quantum field theory.
3.11.26. Proposition. Let T E G(H) and IITIlr(H) <- 1. Then, for every p > 1, one has IIr(T)II[(LP(7)) = 1 PROOF. According to Lemma 3.11.25, for -y-a.e. x, the function
y'-' f If (T'x+Sy)IDy(dy) is y-integrable and its integral equals II f III,(,). It remains to apply Holder's inequality to Ir(T) f (x)I9.
In relation with measurable extensions of linear mappings on the CameronMartin space H of a centered Radon Gaussian measure y one might ask about continuous extensions. Clearly, even for linear functionals no continuous extension may exist. However, a more natural question here is as follows. Given an operator
A E C(H), is it possible to find some space E supporting y in such a way that A extends to a continuous operator on E? For example, if X is a Hilbert space, then, for any given f E X:,, there is a Hilbert space E continuously embedded into X such that y(E) = 1 and f has a version continuous on E. Indeed, we may assume that X = l2 and that the covariance operator of y is diagonal with x x the eigenvalues kn. Then f is given by E tax,,, where E knt2, < oo. We can n=1
n=1
take for E the completion of the space of finite sequences with respect to the norm -+fx)(x)2. The following result in this direction can be deduced from (x, IIxII E = the constructions in [921. Its proof is suggested as Problem 3.11.48. In Chapter 7 we shall mention another result of the same sort.
3.11.27. Proposition. Let H be a separable Hilbert space and let A E G(H). Then there exists an injective nonnegative Hilbert-Schmidt operator T on H such that A is continuous with respect to the norm IIxII E := ITxI H . In particular, A extends to a continuous linear operator on the completion E of the space (H, II - IIE) and the standard cylindrical Gaussian measure of H is countably additive on E. Another possibility to get continuity is to use a stronger topology.
3.11.28. Proposition. Let y be a centered Radon Gaussian measure on a Banach space X (or, more generally, on a Fl tfchet space) and let A: X -+ Y be a -f-measurable linear mapping with values in a separable nonmed space Y. Then
Chapter 3. Radon Measures
154
there exist a full measure linear subspace E C X and a linear operator A0 on E such that: (i) E is separable with respect to some norm II [I, the natural embedding (E. II II z) X is continuous and the Borel sets with respect to the norm iI II e are y-measurable; (ii) IIAo(x)[I,. < IIxI[, for every x E E and A = Ao a.e. Paoor. We may assume that A is a proper linear version (which we take for AO). Let. us take for E any separable Banach space (B. 11 [Id) continuously embedded into X and having full measure (we know that such a space always exists). Finally, let us equip E with the norm IIxII,: = IIx1I5 + IIAxI[,.. It remains to be noted that E is separable with respect to this norm, which follows from the separability of the set ((x, Ax) : x E B} in the separable normed space B x Y. In order to see the measurability of the Borel sets with respect to the new norm, note that the open balls with respect to this norm are measurable by the measurability [I9 and A. By the separability of E, this implies the measurability of all open subsets of E, hence of all Borel subsets. of II
Tensor products It is worth noting that abstract Wiener spaces admit the usual multiplication. It is less obvious that, as stated in the following result from {160[, they admit the tensor multiplication. A proof can be found also in [150), [472]. Recall that the tensor product H1 ®2 H2 of two Hilbert spaces H1 and H2 is defined as the completion of their algebraic tensor product H, 0 H2 with respect to the inner product (a0b,cod)2 = (a, 0H, (b,d)H,. The tensor product X1 $£ X2 of two Banach spaces X, and X2 is defined as the completion of their algebraic tensor product with respect to the norm I[xflL := sup{I(y1gy2)(x)1. IIy,Nx,- < 11-
3.11.29. Proposition. If (i 1, Ht, X,) and (i2. H2, X2) are abstract li'iener spaces. then (i1 0 i2, H, 02 H2, X, 0, X2) is also an abstract Wiener space. If y, is a Gaussian measure on X, with the Cameron-Alartin space H,, i = 1. 2. then the corresponding Gaussian measure on X1'9. X2 is denoted by yl fit. %. In addition, one has R,, r<. (y1 g'y2,Z1 0 22) =
Problems 3.11.30. Let y be a Radon Gaussian measure with the infinite dimensional CameronMartin space H. Prove that ry(H + L) = 0 for any linear subspace L having a countable Hamel basis. Hint: consider the case dim L < oc.
3.11.31. Let y be a centered Radon Gaussian measure on a locally convex space X. Show that any -y-measurable function that is additive on a full measure additive subgroup is an element of X.. Hint: see (98] or Proposition 3.11.8.
3.11.32. (a) Show that a Borel linear subspace in a locally convex space X has measure zero with respect to every nondegenerate Radon Gaussian measure on X precisely when it contains no continuously and densely embedded into X separable Hilbert space. (b) Give an example of a complete separable locally convex space, on which there is no nondegenerate Gaussian Radon measures. Hint: consider the space P(R') of compactly supported smooth functions with the topology of the inductive limit of the spaces Cu ([-n, rz])3.11.33. Show that on every separable Ft chet space (in particular. on every separable Banach space). there exist nondegenerate Gaussian Radon measures. Hint: note that such a space contains a continuously linearly embedded separable Hilbert space.
3.11.
Complements and problems
155
3.11.34. Show that there exists a linear subspace E C C[O.1] with the following properties: its Wiener measure equals 1 and there is a Hilbert norm II IIxIIE <- I]xlic[o,i[ for all x E E. Hint: see [781).
IIE on E such that
3.11.35. Prove that the support of the classical Wiener measure on C[O,1] is the hyperplane defined by the equality x(0) = 0.
3.11.36. Let X be a nuclear locally convex space, i.e., the topology of X is defined by a family of seminorms of the form p. = q7, where qa, are nonnegative quadratic forms such that, for every seminorm pt there exists a seminorm p3 with pa < c(a, 8)p3 and the natural embedding of the Euclidean spaces (X/Kerp3,p3) - (X/Kerp,.p0) is a Hilbert-Schmidt operator. Let us consider on X' its strong topology p(X', X) (i.e.. the topology of the uniform convergence on the bounded subsets of X). Prove that, for any continuous nonnegative quadratic form Q on X, there exists a Gaussian measure ry, which is Radon for the topology p(X', X), such that y(x) = exp(-;Q(x)) for all x E X.
where X is considered as a subset of the dual to (X'.,3(X',X)). In particular, if X is complete, then 7 is a Radon measure for the Mackey topology r(X', X) (see [670, p. 144, Ch. IV, 5.5]). This result applies, e.g., to the space X = S(lRd). Hint: see [800, p. 409, Ch. VI, Theorem 4.2].
3.11.37. ([607]). Let u be the measure on L2[0.1] generated by a centered measurable Gaussian process C with the continuous covariance function Kf. Show that for every signed measure m on [0,1], the function h(t) = fo KK(t, s) m(ds) is in H(p). Hint: take a sequence of continuous functions f,, such that the measures m = f dx converge to m weakly; observe that the corresponding vectors h belong to H(µ) and converge to h in
ff
L2[0,1] whereas the sequence Ih
is bounded.
00
3.11.38. Prove that if a sequence of measurable mappings X.: (11, P) - X with values in a separable Banach space X is fundamental in measure P, i.e.,
lim P(IIX,,-XkII>c)=0, Ve>0. .,k-x then this sequence converges in measure to some measurable mapping.
3.11.39. Let t and q be two independent centered Gaussian random vectors in a locally convex space X inducing Radon measures Pf and P,,. Suppose that PC_., « PP.
Construct an example showing that the equality P, (H(Pt)) = 0 may be true. Prove that the following conditions are equivalent: (i) P,, (H(Pt)) = 1; (ii) the vector ({ + q, q) in X x X induces a measure that is absolutely continuous with respect to the measure induced by ({, q). Hint: see [609], where a more general case is considered, when q may be non Gaussian.
3.11.40. Show that if a convex function f on a Frechet space X is measurable with respect to every Radon Gaussian measure, then it is continuous. In the case of an arbitrary locally convex space show that the function f is bounded on all bounded sets. 3.11.41. Let y be a centered Gaussian measure on a separable Hilbert space X with covariance operator K and let A E £(X). Prove the following relationships: J(Ax, x) 'y(dx) = trace AK, X
J(Ax, x)2 - (dx) = [trace AKJ2 + 2 trace (AK)2. X
If, in addition, A > 0, use Chebyshev's inequality to prove the estimates
y{x: (Ax, x) > 1) < trace AK, Hint: use Problem 1.10.17.
y{x: I (Ax, x) - trace AKl > 1 } < 2 trace (AK )2.
Chapter 3. Radon Measures
156
3.11.42. In the situation of Problem 3.11.41. assuming that 2vrK-AvrK- < 1, prove the equality
f
1/2
e(Ax.x) -y(dx) = [det(I -
= [det(I
- 2AK)]
i; z.
x Hint: use Problem 1.10.18.
3.11.43. In the situation of Problem 3.11.41, for any y,,... ,y2k E X, show that f(x,y,)...
(x,l/2k)7 (dx) = G12k1(0)(y1.....112k).
where G(x) =exp[-1(Kx,x)]
x
3.11.44. Let X be a metric space. Show that a sequence of Radon probability measures µ., converges weakly to a probability measure µ precisely when (3.8.1) holds true for every bounded uniformly continuous function f on X. Moreover, show that it suffices to have (3.8.1) for every bounded Lipschitzian function f. Hint: use Theorem 3.8.2 for the first claim and then use the fact that every bounded uniformly continuous function f on X can be uniformly approximated by bounded Lipschitzian functions.
3.11.45. Construct an example of a sequence of Radon Gaussian measures on a locally convex space, which converges weakly to a Gaussian measure, which is not tight (hence is not Radon). Hint: take the measure y constructed in Example 3.6.3 and consider
it on 1" with the topology o(1", (1'°)'); use Theorem A.3.9 in Appendix to show that the finite dimensional projections µ" of -y converge weakly to y.
3.11.46. (A.V. Kolesnikov). Let y be the measure on Ui" that is the countable product of the standard Gaussian measures on the real line. Show that to every Gaussian
measure p on 12. one can associate a Borel mapping &,: 111" . 12 such that s,,,, - f, in measure -y provided a sequence of Gaussian measures v" converges weakly to v. Hint: letting K, be the covariance operator of p and a its mean, consider the mapping 1;,, :=
G +Q,,, where Q = K is a Hilbert-Schmidt operator on l2 and the measurable linear extension corresponds to the measure y. 3.11.47. Let p and v be two centered Radon Gaussian measures on a locally convex
space X such that H(p) C H(v). Prove that H(µ s v) = H(v). Hint: use Theorem 3.3.4 f EX',forsome c>0. to show that 3.11.48. Prove Proposition 3.11.27. Hint: consider the case where IIAII < 1. take an injective nonnegative Hilbert-Schmidt operator S and consider the norm 1/2
114.
l
ISA"xIH)
and note that E (A")'S2A" is a trace class operator.
CHAPTER 4
Convexity of Gaussian Measures What has been said should be enough to make clear that in the terrain of analysis a rich vein of gold had been struck, comparatively easy to exploit and not soon to be exhausted.
H. Weyi. David Hilbert and his mathematical work
4.1. Gaussian symmetrization In this section we briefly describe the operation of symmetrization for Gaussian measures on R". Denote by ry" the standard Gaussian measure on R"; the symbol ryk will be also used to denote its orthogonal projections onto k-dimensional linear subspaces in R" (not necessarily equal Rk). Recall that 4 stands for the standard Gaussian distribution function. Let 1 < k < n, L a linear subspace in R' of dimension n - k and let e I L be a unit vector. These objects generate the mapping S(L, e), which to every closed set A C R" associates the set S(L. e)(A) defined as follows: for every x E L, the section of the set S(L, e)(A) by the k-dimensional affine subspace x + L1 is
{y: (y,e) > r}n(z+L1), where r = r(x) is such that
ryk({y: (y,e)>r}n(x+L1)) =7k(An(x+L1)),
(4.1.1)
and in the case where the measure on the right-hand side of (4.1.1) is 0, we put S(L, e)(A)f1(x+L1) = 0. and in the case where it is 1, we put S(L.e)(A) = x+L1 (this corresponds to r = +oo and r = -oo in the formula for the section). Thus,
S(L, e)(A) = U ({y: (y, e) > r(x)} n (x + L1)). xEL
The mapping S(L, e) is called a Gaussian k-symmetrtzation With respect to L in the direction of e. Gaussian k-symmetrizations for open sets are defined analogously
with the open half-spaces {x: (x,e) > r} instead of the closed ones. Clearly, it is also possible to define k-symmetrizations for all sets (using again open half-spaces), but we do not need this.
4.1.1. Example. Let k = n. Then L = {0} and
S({0},e)(A) _ {x: (x,e) > 4i-1(1 is a half-space for every closed set A. It is straightforward to verify the following properties of Gaussian symmetrizations. 157
Chapter 4. Convexity
158
4.1.2. Lemma. For arbitrary closed (or open) sets A and B, one has: (i) If A C B, then S(L,e)(A) C S(L,e)(B); (ii) R"\S(L,-e)(A) = S(L,e)(R"\A); (iii) S(L, c) (A + v) = S(L, e)(A) + v for all v E L. In particular, if Lo C L is a linear space such that A + Lo = A, then S(L, e) (A) + Lo = S(L, e) (A);
(iv) S(L, e)(A) = S(L,e)(A)+(L+Ht'e)' and S(L, e)(A) + Ae C S(L, e)(A) for
all)>0;
(v) If {A,} is an increasing sequence of open sets and A = U°° 1 Ai, then
S(L,e)(A) = U S(L,e)(Ai); =1
In particular, S(L,e)(A) is open for open A and closed for closed A. (vi) If B = B + L', then 7'n (A n B) = yn (S(L, e)(A) n B). In particular,
y" (S(L, e)(A)) = -y. (A).
(4.1.2)
Note that in order to show in (v) that S(L, e)(A) is open for open A, it suffices to verify this for open cubes with the edges parallel to the coordinate axis assuming
that L = R"-k and e = en_k}1. For such a cube Q, the set S(L,e)(Q) is an open half-cylinder whose base is the projection of Q to L. Now, for closed A, the set S(L, e) (A) is closed by (ii). Note that the stability of the classes of open and closed sets under the operation S(L, e) enables one to consider compositions of Gaussian symmetrizations on closed or on open sets.
4.1.3. Lemma. Let L1 and L2 be two linear subspaces in R" such that the orthogonal complements of L, n L2 in L1 and in L2 are mutually orthogonal. Then
S(L1ie)oS(L2,e) = S(L2ie)oS(L1,e) = S(L1 n L2,e).
PROOF. Put L1 = L + M1i L2 = L + M2, where L = L1 n L2, and L, M1, and AI2 are mutually orthogonal. Denote by E the orthogonal complement to L + All + Aft + Rte. Let S1 = S(L1, e), S2 = S(L2, e). By Lemma 4.1.2(iv), for any closed A, one has S1(A) = S1(A) + (L1 + R1e)1 = S1(A) + M2 + E,
S2(A) = S2(A) + (L2+R1e)1 = S2(A)+M, +E. Since M1 C L1, the second equality implies (due to properties (iii) and (iv) in Lemma 4.1.2) that S1 oS2(A) = S1 oS2(A) + M1. Therefore, applying the first equality to S2(A) replacing A, we arrive at the representation
S1oS2(A)=S1oS2(A)+M2+E=S1oS2(A)+MI +M2+E = S1 oS2(A) + (L + Rle)1. Thus, the set S1 oS2 (A) is invariant with respect to (L + R1 e)1. By definition, the same is true for S(L, e)(A). Both sets are sent into themselves by the translations x --. x + )e, ) > 0. Therefore, the intersection of each of them with the affine
subspace F. = x + Ll is a half-space in FZ with the inner normal e. Note that,
4.1.
Gaussian symmetrization
159
letting k = dim Ll, we have that the k-dimensional standard Gaussian measure 7k of the set S1 o S2 (A) n F= equals 1k
(S2(A) n F=) = -yk(A n F=),
since Fr is invariant under (MI +L)' and (AI2+L)1, being invariant under L. The right-hand side of the last equality coincides with 1k(S(L, e)(A) n Fr). Therefore, we get the equality of the two half-subspaces:
S,oS2(A)nF==S(L,e)(A)nF., Taking the union in x, we obtain S1oS2(A) = S(L,e)(A). Hence S2oS1(A) is the same set. The next lemma enables one to reduce k-symmetrizations of high orders to the two dimensional ones.
4.1.4. Lemma. Let n > 3 and k > 2. For every k-symmetrization S = S(L,e), there exist 2-symmetrizations Sl,... , Sk_1 such that S = S1 o.. oSk_1. PROOF. We may assume that k > 2. Let h E (L+IRle)1 be a unit vector. Put L, = (IR' h+,R' e)1, L2 = L+1111 h. Then the conditions of Lemma 4.1.3 are fulfilled
(note that L1 n L2 = L) and we get S = S(L1, e)oS(L2, e). It remains to be noted that S(L1,e) is a 2-symmetrization and that S(L2,e) is a k - 1-symmetrization. Hence, repeating this k-2 times, we get the composition of k-1 2-symmetrizations.
0 In turn, the 2-symmetrizations are obtained as the limits (in a sense to be precised) of compositions of 1-symmetrizations. Let {ej }, j = 0, 1, ... , be a sequence of unit vectors in 1R.2 defined inductively as follows: eo = (0,1), e, = (1,0), and
the angle 0j+, between ej+, and -eo is 0,/2. Then ej - -eo, (ej,e,_1 +eo) = 0, and (ej,eo) < 0.
4.1.5. Lemma. Let Sj = S(el, ej) and Tj = Sj o . . . o So. Suppose A is a closed set in 1R2. If x belongs to Tj(A), then the whole cone
C(x,eo,ej) = {x+teo+sej, t, s> 0} is contained in T,(A). PROOF. We shall use induction in j. For j = 0 the claim follows from Lemma 4.1.2. Suppose it is proved for some j > 0. Let us consider the closed interval
Ij ={Aej-Aeo, -1
AI:=U{C(x,eo,ej): xET,(A)nE0}. If Q > a, then one can verify that
AlnE3 = (T,(A)nE.)+rIj+(Q-a)(eo+ej), where r = r(j, a, Q). In the proof of Theorem 4.1.7 below (see Step 1), we shall establish the following fact (without reference to this lemma, of course): if F C IRS
Chapter 4. Convexity
160
is a closed set, then 4)-' (1 - n (F + [-r, r])) < 41-'(1 - -yi (F)) - r for any r > 0. Applying this fact to the set Af n E3, we get
S,+1(MnE3) J Si+1(Ti(A)nE.+(,3-a)(eo+ej)) +r13
=U{C(x,eo.ei): xES,+1(T,(A)nEa)}nE3. Taking the union in ;3 > a, we arrive at the inclusion
S,+1(M) D U{C(x,eo,e,): x E S,+1(T,(A) n EQ)}. Therefore, if x E S,+1(T;(A) n E0) = T,+1(A) n E0, one has
C(x, eo, e,) C S,+1(M) C SJ+1 o T; (A) = Ti(A),
0
whence the claim.
4.1.6. Corollary. Put S = S(0,e1). Let A be a compact set in R2 such that 0 < 12(A) < 1/2. Then, for every R > 0, one has slim T, (A) n {x: IxI < R} = S(A) n {x: Ixi < R},
where the convergence of sets B, B is understood in the sense of the Hausdorff metric, i.e., given e > 0, for all j sufficiently large, the sets B, and B are contained in the e-neighborhoods of each other.
PROOF. The claim is deduced from the previous lemma making use of the fact that the angle between co and ej tends to 7r so that the corresponding cones approach half-planes with the boundary parallel to R'eo, whereas the set S(A) is a half-plane with the boundary line orthogonal to el. See [222], [491] for details. 0
4.1.7. Theorem. Let U be the closed unit ball in R" and let S(L, e) be any k-symmetrization in R". Then: (i) For every closed set A C R', one has
S(L, e)(A) + rU C S(L, e)(A + rU),
t1 r > 0:
(4.1.3)
(ii) For any convex open or closed set A C R", the set S(L, e)(A) is convex.
PROOF. (i) Throughout this proof we put A' = A + rU for every set A. Step 1. First we prove that relationship (4.1.3) holds true in the case n = I. Let us consider the special case where n = 1, L = {0}, e = 1, and A is a closed interval. Put a = yl (A). Note that even this case is not completely obvious. The set S(A)'
is the ray (45-1(1 - a) - r,oo), whereas the set S(A') is the ray (4+-'(1 - 3).oo), where ,3 = yl (A'). We shall show that -' (1 - 3) < 44`(I - a) - r. To this end, holding a and r fixed, we shall maximize 0). Clearly, this is equivalent to minimization of 3. Any interval of y1-measure a can be written as Ax = [x, y(x)],
where x E [-oo,-' (1 - a)] and y(x) = 4+-' (a + 4)(x)). Then -y1-measure 3(x)
of [x - r. y(x) + r] equals 1(y(x) + r) -'1(x - r). Differentiating in x. we get 3'(x) = p(x)p(y(x) + r)/p(y(x)) - p(x - r), where p is the standard Gaussian density. An easy evaluation yields ,3'(x) = 2ap(x)p(r)(e-y' - e='). Clearly, the function 3 attains its maximum at xo determined from the equation xo = -y(xo) (this corresponds to the symmetric interval). The minimum of 3 is attained at the points -oc and 4i'' (1 - a), which correspond to the rays Ax = [-oo, 4i-'(a)] and
4.1.
Gaussian symmetrization
161
!4-1(1 - a),oo]. In this case, by virtue of the identity (D'i(1 - a) = __i(), (4.1.3) turns into an equality. Let us continue by induction. Suppose that (4.1.3) is true for any n intervals
(possibly, unbounded) and both possible choices of unit vectors e and -e. Let A = u;±i A., where the A,'s are n + 1 disjoint intervals numbered in the natural order. We may assume that the sets A' have disjoint interiors. Otherwise we can make these intervals shorter without changing (U;+1 Aj)''. Put J = UJ_2A,. We shall denote S(0,1) by S and S(0, -1) by S_. Let us replace Al and An+1 by the rays S_ (A1) and S(An:.1), which gives n+1
S_ (A) = S_ (U A,) = S_ (S_ (A1) U J U S(An+1)). -1
S_(X)=S_ (S_(Ai)uJruS(An+i)). Using that S_ (A1)r C S_ (Ar) and S(An+1)r C S(Ar.1), we get S_ (S- (AI )r U Jr U S(An+1 )r) C S_ (Ar).
Let us consider the set I := S_ (A1)r U Jr U S(An+l )r. Its complement D is a union
of at most n intervals, hence the inductive assumption yields S(D)r C S(Dr), whence S_ (S-(A1) U J U S(An+1)) = S_(1R'\Dr)
= IR'\S(Dr) C JR'\S(D)r = R'\(]R'\S-(I))r. Hence,
S_ (S_ (A1) U J U S(An+1)) r C S_
(I).
Therefore,
S-(A)r = S_(S_(A1)UJUS(Ant1)) C S_(I) C S_(Ar). The inclusion for S is verified in a similar manner. Obviously, the claim is also true for open intervals. Then it remains to be true for all open sets in R1, which, in turn, implies its validity for all closed sets, since every closed set A is the intersection of its open c-neighborhoods. Step 2. Suppose that every k-symmetrization in lRk satisfies (4.1.3). Then the same is true for every k-symmetrization in iRn, whenever n > k. Indeed, for every
x E L, we put E. := x + L1. It suffices to show that
(S(A) + rU) n Er C S(A + rU) n E_,
d x E L.
(4.1.4)
The right-hand side equals S((A+rU)nEi). The left-hand side can be represented as
(S(A) + rU) n E_
= U (S(A n Ey) + rU) n E. yE L
Every nonempty member in this union is the shift to x - y of the k-dimensional 6-neighborhood of the set S(A n Ey) with 6 = r2 - _Ix- yi2. By assumption, this 6-neighborhood in the k-dimensional affine subspace Ey is contained in the set S((A n Ey) + 6U,.), where Uk = U n L1 is the k-dimensional unit ball in L1. In
Chapter 4. Convexity
162
turn, the set S ((A n El,) + bU) + x - y is contained in S ([(A n Ed) + rU] n E_ ) since
nE.
(AnEr,)+bUk+x-y c and
=S((AnEE)+bUk)+x-y by the inclusion x - y E L and Lemma 4.1.2(iii). Therefore,
(S(AnEy)+rU) nEx CS([(AnEy)+rU]nE_). Taking the union in y E L, we arrive at (4.1.4). Step 3. Using Corollary 4.1.6 and Step 1, one verifies that every 2-symmetrization on R2 satisfies (4.1.3). Applying Step 2 and Lemma 4.1.4, we get the general result (see [222] and [491] for more details).
Let us prove statement (ii). We shall consider first a 1-symmetrization S =
S(L, e), where a is a unit, vector in R' and L = e1. For any x E L, put Ez = x + R' e. The set A n E= is either a ray or an interval, and the set S(A) n Es is a ray. We have to verify that, for any x, y E L and A E [0, 11, one has A(S(A) n E.,) + (1
- \) (S(A) n El,) c S(A) n
This inclusion is equivalent to convexity of S(A). We have A n ET = x + [as, b=]e,
A n E = y + [a,,, b ]e, S(A) n E_ = x + [e., co)e, S(A) n El, = y + (c,, oo)e. By the definition of symmetrization, one has sl#(b1) - 0(a1) = 1 - 4i(c1) and the same for y. Hence the inclusion to be proved reduces to the estimate
-ac= - (1 - A)c = A4i-1(t(b=) - 4,(a:)) + (1 - \)4,-' (s(b5) -g(ay))
«_1(4(Abr+(1-A)by) -4i(Aa=+(1 -\)a,)). This estimate follows from Problem 4.10.14. The convexity is preserved by the compositions of 1-symmetrizations, hence the claim is true for all 2-symmetrizations (which follows by a construction analogous to that of Lemma 4.1.5). Applying Lemma 4.1.4, we get the general case. 0
4.1.8. Corollary. For every closed set A in IEt", we have In (S(L, e)(A) + rU) S I" (S(L, e)(A + rU)),
d r > 0.
(4.1.5)
In particular, applying this to the n-symmetrization, we get that among all sets of equal I.-measure, the half-spaces have the minimal measure of the r-neighborhood.
4.2. Ehrhard's inequality Convexity of Gaussian measures plays an important role in diverse applications. Quantitative theorems of this sort state that for some function the inequality
y(,\A+ (1 - A)B) > i,(A,Y(A),7(B)),
V,\ E [0, 11,
holds true for all A and B from a certain class of sets. Theorem 1.8.4 is an example of this kind. The following fundamental result - Ehrhard's inequality - is derived by using the properties of Gaussian symmetrizations. We shall employ our usual
convention 4t-1(0) = -oc and t-1(1) = +oo.
4.2.
Ehrhard's inequality
163
4.2.1. Theorem. Let A and B be two convex sets in R". Then one has for all A E [0,1]:
'6-1 1ry.,(AA+(I -.1)B)} > A4 1{-,n(A)I+(1
(4.2.1)
PROOF. As it was noted above, the interior of a convex set has the same mea-
sure as the set itself. Therefore, it suffices to prove our claim for open convex
sets A and B. We shall consider R" as a hyperplane in Rn+'. Then A and B are convex subsets of R"+', whence the convexity of the set A + en+1 , where en+1 = (0, 0, ... , 0, 1). Denote by C the convex hull of the sets A + en+, and B in the space R'+' Then, for every A E [0, 11, we get
Ca:=R"n (C-Aen+1)=AA +(1-A)B. Let us consider the function
f(A) = 44-'{7 "(CA)}
\)B)
on the closed interval [0, 1]. Let us take a unit vector e I en+1 and consider the n-symmetrization S in R"+' with respect to the one dimensional subspace R'en+1 in the direction of e. The set S(C) is convex. By the definition of symmetrization, (ry"(Ca)) is the number r = r,\, for which (Aen+i + R") n S(C) = {x E R"+1 : (x. e) > r} n (Aen+l + R"), since yn(CA) = 1 4)(-ra). By the convexity of S(C), one has
(1 - A) (R" n S(C)) + A ((eni.l + R") n S(C)) C (Aen+1 + R") n S(C), whence (1 - A)(ro, oo) + A(rl, oo) C (ra, oo), i.e., ra < Arl + (1 - A)ro. Therefore,
0
f (A) > A f (1) + (1 - A) f (0), which is our claim.
It is shown in [4651 that for the validity of Ehrhard's inequality it is enough to require the convexity of only one of the two sets A and B, but it remains open whether this inequality holds true for arbitrary pairs of measurable sets. As the reader could have noticed, the theorem above (as well as Theorem 1.8.4) remains valid for an arbitrary Gaussian measure if we replace U by the corresponding ellipsoid of concentration. However, we pass directly to the infinite dimensional generalizations.
4.2.2. Theorem. Let y be a centered Radon Gaussian measure on a locally convex space X. Then, for arbitrary Borel sets A and B and all A E [0, 11, one has
y. (AA + (1
- A)B) > -y(A)ay(B)'-
.
(4.2.2)
If, in addition, A and B are convex, then
4'-'{y.(.1A+(1-.1)B)}>A4'-'{7(A)}+(1-A)4 1{-y(B)}.
(4.2.3)
PROOF. First we show how to get (4.2.2) from the finite dimensional estimate (4.2.2). Note that both inequalities in question are valid for every Gaussian measure
on R", since any such measure is the image of the standard Gaussian measure under an affine mapping, and the affine mappings transform linear combinations of sets into the linear combinations of their images. Since the ry-measure of every measurable set equals the supremum of the measures of enclosed metrizable compact
sets, it suffices to verify (4.2.2) in the case where A and B are metrizable and
Chapter 4. Convexity
164
compact. In this case C = AA + (1 - A)B is also compact metrizable. We may assume that it least one of the two sets A or B has positive measure. The linear span E of the sets A and B contains all the sets we are interested in and is a Souslin space of full measure. As it was pointed out above, E is the union of a sequence of metrizable compact sets, and there is a sequence (f,,} C E' separating the points of E. From now on we consider the measure y on E. Using the orthogonalization, we may assume that the sequence { fn} is an orthonormal basis in X.. If this sequence is finite, the whole thing reduces to the finite dimensional case discussed above.
If it is infinite, then the injective continuous linear mapping T : x ' (fn (x)) n=1 , E lR", transforms the measure y into the countable product of the standard Gaussian measures on ff(1, which reduces the claim to this product. Therefore, we assume further that E = lRx and that -r is the product of the standard one dimensional Gaussian distributions.
Let us put Pnx = (xt, x2, ... , xn, 0, 0, ... ). Note that, for every compact set K C IR'°, there holds the equality
K= n P,-' (P,(K)).
(4.2.4)
n=i
Indeed, K is contained in the right-hand side of (4.2.4). Let x it K. Then there is n such that Px ¢ P,, (K). Otherwise, for each n, there is a point kn E K with Pnkn = Px. Since K is a metrizable compact set, the sequence {k"} contains a subsequence convergent to some element k E K. Then ki = x, for all i E IN (recall
that we deal with the coordinate convergence in It'), whence k = x, which is a contradiction. Hence equality (4.2.4) is established.
Let e > 0. Applying (4.2.4) to the sets A, B, and C, and taking into account decrease as n Increases, we get that for some n the measures of the cylindrical sets An = P,,- ' (P,, (A)), Bn = P,,- ' (P,, (B)), and Cn = P» 1(Pn(C)) differ from the measures of A, B, and C, respectively, not greater than in E. In addition, C = AAn + (1 - \)B,,. Applying the finite dimensional
that the sets
inequality to the measure y,, = ry o P,,- 1, we get
-t(C)
y'n(Cn) - e
e > (y(A) -
f)'- - E,
which yields (4.2.2), since a is arbitrary. Now let us turn to (4.2.3). The previous reasoning does not work here, since the corresponding finite dimensional inequality was proved for convex sets, whereas (4.2.4) is not true, in general, for noncompact sets, and the convex hull of a compact set in IR" need not be compact. In order to bypass this difficulty, we shall reduce
everything to the case of Rx as we did above. For every e > 0, there exist metrizable compact sets Al C A and B1 C B such that y(A) < ry(A1) + e and y(B) < y(B1) + e. In view of this, it suffices to prove inequality (4.2.3) for convex
hulls of AI and BI instead of A and B (note that Aconv Al + (1 - \)conv Bl C AA + (1 - \) B). Therefore, we may assume that A = cony AI and B = cony B1. Put C = \A+ (1 -.\)B. The set C1 = AA1 + (1 - A)B1 is compact metrizable as well as the union Z = A, U B1 U C1. If y(Z) = 0, then (4.2.3) is true. Hence we may assume that y(Z) > 0. Then the linear span E of the set Z contains all the sets we are interested in and is a Souslin subspace of full measure. Now we consider
the measure y on E. We find again a sequence { fn} C E' separating the points of E. As above, this sequence gives an injeetive linear embedding of E into 1R",
4.2.
Ehrhard's inequality
165
which reduces everything to the case where E = 1R" and y is the product of the standard one dimensional Gaussian distributions. Put Pnx = (x1, x2, ... , xn, 0, 0, ...) and Qax = x - Pnx. For every Borel set D, let us define the functions I. (D) (x) = f ID(Pnx + Qny) y(dy) = Iza(D - Pnx),
where µn = y o Qn 1. Inequality (4.2.2) (that has already been proved) implies th inclusion
A{In(A) > 1/2} + (1 - \){I,,(B) > 1/2} C {In (\A + (1 - \)B) > 1/2}. Indeed, if µn(A-Pnx) > 1/2 and
1/2, then estimate (4.2.2), applied
to the measure tsn, gives
logpn (AA + (1 - A)B - Pn(Ax + (1- A)y)) = log A,, (A(A
- Px) + (1 - A)(B - Pny))
>Alogµn(A-Pnx)+(1-A)logµn(B-Pny)> log 2, whence µn (AA + (1 - A) B - P (Ax + (1 - A)y)) > 1/2. Therefore, y(A{I,, (A) > 1/2} + (1 - A){In(B) > 1/2}) <
({I,, (AA + (1 - A)B) > 1/2}).
Note that the set {In(D) > 1/2} is convex, provided so is D. Indeed, if µ n(D - Pnx) > 2
and
un(D - Pnz) > 2
then, by convexity of D, we get for all A E [0, 1] that
A(D - Px) + (1 - A)(D - Paz) = D - APnx - (1 - A)Pnz, whence in the same manner as above with the aid of (4.2.2) we arrive at the estimate
p,(D-\Pnx-(1-A)Pnz) > 2 It remains to note that the convex sets {I (A) > 1/2} and {In(B) > 1/2} are cylindrical, and, in addition, by Theorem 3.5.2, for any measurable set D, the sequence {Ia(D)} converges to ID in L2(y), hence in measure y. Applying this theorem to A and B and using the finite dimensional Ehrhard's inequality, we get estimate (4.2.3). This theorem implies, in particular, the infinite dimensional Anderson inequal-
ity already familiar to us. In Problem 4.10.15 it is suggested to derive it from Ehrhard's inequality. 4.2.3. Corollary. In the situation of the previous theorem, for every symmetric convex set A and every vector a such that A and A + a are -y-measurable, one has -y(A + a) < y(A). More generally, if A + to is -f-measurable for any t E [0' 1], then
y(A + a) < y(A + ta),
`d t E [0, 1].
(4.2.5)
Chapter 4. Convexity
166
Further, the function t '-. j f (x + ta) y(dx) is n.ondecreasing on (0, +oo), provided
f is such that the sets If G c}. c E lR 1, are symmetric and convex, and f ( + ta) is y-integrable for any t > 0.
4.2.4. Corollary. Let A be a y-measurable convex set of positive measure. Then the topological support of the restriction of y to A is convex. PROOF. First of all note that the Radon measure 71A has topological support S.
Suppose that x, y E S and A E (0, 1). We have to show that for every convex neighborhood of zero V, the set B = (Ax + (1 - A)y + V) n A has positive''measure. The sets C = (x+V)fA and D = (y+V)f1A are convex and have positive 1-measures. Since AC + (1 - A)D C B. the claim follows by Theorem 4.2.2.
0
4.2.5. Remark. Note that if A and B belong to E(X ), then their linear combinations are automatically measurable (see Chapter 2), hence there is no need to use the inner measure. In particular, this is true if X is a separable Frechet space. The same remains in force for those pairs of Borel sets, at least one of which is contained in a Souslin linear subspace of full -t-measure. In the proof of Theorem 4.2.2, the convexity was employed for approximating convex sets by convex cylinders: such approximations have been manufactured for
E(X),-measurable sets in Chapter 2. This method (unlike the construction from Chapter 2) is simple and constructive: for this reason, we formulate the corresponding statement as a separate result.
4.2.6. Proposition. Let y be a Radon Gaussian measure on a locally convex space X and let A E S(X)., be a convex set. Let us take any orthonormal basis tit H(-.) and put r (I f,,(x) := J 'A y(dy). X
where
(A L
,_i
F;(x)e;. Then the sets A := {f,, > 1/21 are convex cylinders and A is absolutely convex, then so are the sets A,,.
PROOF. We may assume that I is centered. In addition, we may assume that A is a Bore] convex set (the convex hull of some sequence of compact sets). Since
fimageI,,ofinI under L2(y) by Corollary 3.5.2, we have y(A A A,,) - 0. Denote by p the the mapping I - P,,. Then P,,x). Therefore, the condition x. z E A. can be written as p, (A -
1/2, a,, (A - P,,z) > 1/2. By
the convexity of A, we have
A E (0. 11. By virtue of inequality (4.2.2) applied to the measure p we conclude that p" (A (1 - A)P,,z) > 1/2, that is, Ax + (1 - A)z E A,,,
i.e.. A,, is convex. Finally. if A is absolutely convex, then, due to the symmetry of
p,,. the functions f above are even, hence the sets A. are symmetric about the origin.
0
4.3.
Isoperimetric inequalities
167
4.3. Isoperimetric inequalities Our next application of Gaussian symmetrizations is a proof of the following isoperimetric inequality (which can be proved without symmetrizations, but not as simply). Recall that we put 45-'(0) = -oc and 45-1(1) = +oo.
4.3.1. Theorem. Let yn be the standard Gaussian measure on 1R" and let U be the closed unit ball in 1R" centered at the origin. For every measurable set A C 1R", the following inequality holds true:
-1(yn(A+rU)) >- tin(A) +r,
`dr > 0.
(4.3.1)
PROOF. Clearly, it suffices to prove this estimate for compact sets A. Let us apply to A an arbitrary n-symmetrization S. Then the sets S(A) and S(A) + rU are half-spaces. Since the Gaussian measure is preserved by the symmetrizations, by virtue of Theorem 4.1.7, we get
yn(A + rU) = yn (S(A+ rU)) > y,, (S(A) + rU) =.(4i-'{7"(S(A))1 + r)
1{1n(A)}+r),
since the half-space S(A) + rU is obtained from the half-space S(A) by shifting to a vector of length r orthogonal to the boundary. 0
Note that inequality (4.3.1) can be written as yn(A + rU) > 40(a + r),
where a = 4i-i (yn(A)). Therefore, 4i(a + r) is the measure of the set n + rU, where II is a half-space having the same measure as A. If we define the surface measure of A as the limit of the ratio r-' (yn(A + rU) - yn(A)) as r 0, then (4.3.1) shows that the half-spaces possess the minimal surface measures among the sets of given positive measure. For the formulation of a one more infinite dimensional extension we need the following technical result.
4.3.2. Lemma. Let y be a Radon Gaussian measure on a locally convex space X and let Q be a Borel set in the Hilbert space H(y). Then, for every Borel set B C X. the set B+ Q is -r-measurable. PROOF. Observe that B + Q is a Souslin set if X is a separable Banach space, hence is -y-measurable. We know that in the general case there is a metrizable compact subset K in X of positive y-measure. Let Tn be the cube in 1R" given by Ix; I < n, i = 1,... , n. Denote by K,, the image of the metrizable compact set Tn x K" under the mapping
((t..... .tn),(kl,... .kn)) - klti +...+kntn Then Kn is also compact metrizable. Recall that H(y) is a separable Hilbert space. Therefore, the set (B n Kn) + Q is a Souslin subset of X as a continuous image of the Borel set (B fl Kn) xQ in the complete separable metric space Kn x H(y) (see Appendix). The corresponding continuous mapping has the form (k, h) '-. k + h. Therefore, for every n, the set (B n Kn) + Q is -t-measurable (see Theorem A.3.15).
The union L = U' i Kn is a linear subspace. In addition, y(L) > 0. By the
Chapter 4. Convexity
168
zero-one law, one has y(L) = 1. In particular, L contains H(7), hence Q C L. It remains to note that, letting
x
C= U((BnKn)+Q), n=1
we have CCB+Q and (B + Q)\C C X\L. Indeed, let x= b+ q, where b E B, q E Q. If x E L, then b E L, since Q C Land L is a linear subspace. Hence b E B n Kn for some n, whence x E (B n Kn) + Q C C and x ¢ (B + Q) \C. Therefore, y ((B + Q)\C) = 0.
4.3.3. Theorem. Let y be a Radon Gaussian measure on a locally convex space X, A a -y-measurable set and let Uy be the closed unit ball in the Hilbert space H = H(-y). Then
Vt>0, where a is chosen in such a way that 4'(a) = y(A). If y(A) > 1/2, then
y(A+
4'(r) > 1 - I exp(-2r2).
(4.3.2)
PROOF. We use the same reasoning as in the proof of inequality (4.2.2) and reduce everything to the case of product-measures on IR" and then apply the finite dimensional inequality proved in Theorem 4.3.1. Note that in (4.3.2) we use the estimate 4'(t) > 1 - exp (- t2) , V t > 0, which is easy to be verified. 2
4.3.4. Corollary. For every positive a, there exist ro(a) > 0 and a real number c(a) such that, for all r > ro(a), the following inequality holds true:
y(A+rU,,) > i - exp(- 2 +c(a)r) r2
(4.3.3)
provided 7(A) = at > 0.
Now, following Borell (991, we shall apply the isoperimetric inequality to the estimates of the distribution functions of certain nonlinear functionals. Let y be a Radon Gaussian measure on a locally convex space X with the Cameron-Martin space H = H(-y). For any given d > 0, let us denote by Sd(y) the class of all -ymeasurable functions f satisfying the following condition: there exist a real function Al on the closed unit ball U,, in H and a sequence of y-measurable functions gn such that one has a.e. (i) (ii) lim gn = 0 a.e.
(x+th) - Al(h)I
n-x
Note that if f is Borel, this is equivalent to lim sup jt-d f (x + th) - Ad(h) nix IhINn
0
a.e.
Clearly, if f, g E Sd(y) and f = g a.e., then Ad = Ad, since, for every fixed h E H, one has f (x + nh) = g(x + nh) a.e. for all n E IN.
169
Isoperimetric inequalities
4.3.
4.3.5. Theorem. Let f E Sd(y) Then the function Ad is continuous and bounded on U with the noun I
IH. In addition, one has the equality
lim t-21dlogy(f > t) _ -1( sup IAf(h)1) t+x 2 hEUH
2/d
(4.3.4)
Finally,
exp(.Ifl21d) E L'(-y),
IAd(h)I)-2,d
V. < 1 ( sup
2 hEUH
.
f
(4.3.5)
PROOF. It suffices to prove our claims for centered measures. We may assume
that the functions gn are Borel. By Egorov's theorem, there exists a Borel set A C X with y(A) > 1/2 such that lim
sup
n...x IhIH <1. t>n, xEA
It-df(x+th)-Ad(h)I=0.
(4.3.6)
In particular, there exist no E IN, C > 0 and a positive measure Borel set B C A such that I f (x) I < C for all x E B and I f (x + nh) - Ad (h) I < 1 for all x E B, h E U and n > no. By Theorem 2.4.8, there exists 6 E (0, 1) such that
Jf(x+noh)v(dx) <
J
f (x) y(dx) + 1 < C + 1,
V h E 6U,.
B
B
Hence IAd(h)Iy(B) < C+2 for every h E 6U,,. Since there exists some x such that t-d f (x + th) - Ad (h) -. 0 for all h E U,,, we see that Al(ah) = \dAd (h) for any
h E U and a E (0,1]. Therefore, IAJ(h)I < b'dy(B)-1(C+2) for all h E U,,. In order to see that Al is continuous, let us consider the functions fn (h) = ndµ(B)
f
f (x + nh) y(dx),
n > no.
B
By virtue of (4.3.6), these functions converge to Ad uniformly on U,,. It remains to note that fn is continuous on H. Let us put Al = suphEL,H Ad (h) and fix e > 0. Since
f(x + th) = td [t-d f (x + th) - Ad(h) + Ad(h)] we conclude that there exists nE E IN such that
A+ tUH C {x: f (X) < (M + E)td},
`dt > nE.
By Theorem 4.3.3, we have
y(x: f (X) < (M + e)td) > -y(A + tUH) > $(a + t),
V t > nE,
(4.3.7)
where 4i(a) = y(A). Since y(A) > 1/2, we get
y(x: f(x) > (M+e)td) < 2 exp(-2t2J,
`dt > nE.
(4.3.8)
The same estimate holds true for -f replacing f and Mo = supuH (-f) replacing M. Therefore, letting R = max(M, M0), we arrive at the estimate
y(x: If(x)I>t) <exp
(_2( 1
t
R+e t )
Zed
Chapter 4. Convexity
170
for all sufficiently large t. Let us put
a = 2a (sup JA(h)I l + e)
2/d
U!,
where a is chosen in such a way that x < 1. Then, for all sufficiently large t, we have
y{x: exp(aif(x)I2/d) > t} <
t-11'c.
This estimate implies the integrability of exp(ajf Iz/d). Since e > 0 was arbitrary, we get (4.3.5). Clearly, (4.3.8) implies that lim sup t-21d log y(f > t) < - 2111-z/d. t-- x It remains to be shown that
-1
lim inf t-2/d log 7(f > t)
t-x
(4.3.9)
M-21d.
We shall assume that M > 0 (otherwise the claim is trivial). Let us fix e E (0, M/2). Since Al is continuous and positively homogeneous, there exists a vector h E U. such that Jhi = 1, h E X' and A j(h) > M - e. There exists a full measure Borel set A such that conditions (i) and (ii) from the definition of Sd(y) are satisfied pointwise on A. Let us put ir(x) = x - h(x)h. Denote by y1 the image of y under the mapping x h(x)h. Note that y = y1 * y o rr-' (if we write Xas the product of the one dimensional space lR'h and the hyperplane sr(X) = h-1(0), then -y becomes the product of y1 and y o it 1). Hence there exists t1 E IR' such
that y o a'' (A - t1h) = 1. Since the sequence of Borel functions g (zr(x) + t,h) converges to zero pointwise on w-1 (A - t1 h), Egorov's theorem yields the existence
of a Borel set B C A - t1h (in fact, we can choose B C zr(X) n (A - tIh), since y o rr't is concentrated on the hyperplane x(X )) such that y o ,r-' (B) > 0 and lim sup gn (a(x) + tlh) = 0. In particular, there exists of E IN such that
n"xEB
suplt-d f (a(x) + tlh + th) t>n
-
E,
dx E
'(B), d n > n,.
Noting that h(h) = 1 and hence ir(x) + t1h + (h(x) - t1)h = x, we arrive at the following relationship taking t = h(x) - t1 in the previous estimate:
{x E a-'(B): h(x) > n,t + t1} C {x: f(a) > (M - E)I h(x) - t1I Therefore, for any t > nE (M - e), we obtain
y(x: f(x) >t)
>y
l/d
t xErr-'(B):t1+ (M-E) ).
Note that for any r E lR1 one has
y(rr-'(B) n {h > r}) = y(a-1(B))y({h > r}). Indeed,
/
l
y1([rr-'(B)n{h>r}, -y) =y1({h>r}) =y({h>r})
d}.
4.4.
Convex functions
171
if y E 7r-'(B) and this measure is zero if y ¢ 7r (B), since y, is concentrated on the line IR' h and ir(h) = 0, so th + y E a-(B) precisely when y E 7r-' (B). Hence,
y(x: f(x) > t) > (x: h(x) > t, + (,tlt
E)'/d)yoa-I(B),
whence (4.3.9) follows at once, since E > 0 can be taken arbitrarily small.
4.3.6. Example. Let f be a -y-measurable function such that f (Ah) = Ad f (h) for some d > 0 and all h E H, A > 0. Assume, in addition, that there exists a full measure linear subspace (or an additive subgroup) Xo C X such that f (x + y) < f (x) + f (y) for all x, y E X0. Then f E Sd(y) and A f(h) = f (h).
PROOF. Since H C Xo, we have f(x+h) - f(x) < f(h) and f(x+h) - f(h) > -f(-x) for all x E X0 and all hEH. Hence, for all t > 0 and h E H. one has
It-df(x+th) - f(h)I = t-dlf(x+th) - f(th)I <- t-d[If(x)I + If(-x)I]. Now we can take g. (x) = n-d [If(x)1 + If(-x)I].
4.3.7. Example. Suppose that f is a continuous function on X such that f(Ax) = Adf(x) for some d > 0 and all x E X. A > 0. Then f E Sd(y) and A1(h)= f(h) for allhEH. PROOF. Set
g. (x) =
sup hE('H. t>n
If(t-'x+h) - f(h)I
Using the compactness of UH in X and the continuity of f, it is straightforward to verify that lim _ Xg, (x) = 0 for every x E X.
We shall discuss other examples in Chapter 5 in the section on measurable polynomials.
4.4. Convex functions
In this section we discuss distributions of nonlinear functions on Gaussian spaces. The material covered here is a link between the preceding chapters which dealt with linear objects, and Chapters 5 and 6 devoted to nonlinear problems. First we formulate a general result concerning the structure of the distributions of convex functionals. A measurable function f on a locally convex space with a Radon Gaussian measure y is said to be a y-measurable convex function if it has a
modification fo: X - ff' which is convex in the usual sense, i.e., fo(Ax + (1
- A)y) < Afo(x) + (1 - A)fo(y),
VA E [0, 1], Vx, y E X.
Concave functions are defined by analogy with ">" replacing "<".
4.4.1. Theorem. Let F(t) = -1 {x: f(x) < t}, where y is a Radon Gaussian measure on a locally convex space X and f is a y-measurable convex function on X. Then the function G: t 4 ' (F(t)) is concave.
PROOF. Lett,,t2EIR',AE[0,11. Set t3=At,+(1-A)t2and A,={f
' (y(A3)) > A4i-' (y(A1)) + (1 - A)4 ' (y(A2)),
Chapter 4. Convexity
172
which yields G(t3) > AG(t1) + (1 - \)G(t2).
4.4.2. Corollary. Suppose that the conditions in Theorem 4.4.1 are satisfied. Then:
(i) The function F is continuous everywhere apart from the point
to = inf{t: If f < t} > 0}; (ii) The function F is absolutely continuous on the ray (to,+oo) and has the positive derivative F' at all points of this ray excepting, possibly, at most countable set S. where F has the one-sided limits and jumps down; moreover. F is continuous on the set (to, +oo)\S; (iii) For every t1 > to, the function F' has bounded variation on ft 1, +00) if it is defined on S as the left-hand limit, in particular. F is bounded on (ti, +oo). The proof is left as Problem 4.10.17. There exist examples showing that F may have a jump at the point to and F' may be unbounded in a neighborhood of to and have jumps at the points of a countable set (see, e.g., [1891). An important example of a measurable convex functional is a measurable seminorm. 4.4.3. Proposition. Let -y be a centered Gaussian Radon measure on X. For every -y-measurable seminorm q on X. there exist a sequence {fn} C X. and a sequence of numbers an > 0 such that
q(x) = sup[fn(x) +an]
1-a.e.
(4.4.1)
PROOF. The function v(t) = y(q < t) is nondecreasing, hence it has at most countably many points of discontinuity. Denote by Al an arbitrary countable set which is dense in the set of the points of continuity of V. Write M as M = {ti}. Put V = {q < 1}. Clearly, tV = {q < t} for all t > 0. By Proposition 4.2.6, for each k E IN, there exists an absolutely convex cylindrical set Ck with a compact base such that y(t.Ck 0 tiV) < 2-i-k Vi < k. Note that the proposition cited enables one to construct such cylindrical approximations for finitely many measures y,(B) = y(tiB), i = 1,... , k, simultaneously. The Nfinkowski functional qck of the set Ck is a E(X)-measurable seminorm, which,
for some countable collection ft,, of elements of X', can be represented as
gck(x) = supfk.i(x)Let us denote by D the countable collection of the functionals fi, obtained in this way. Let us show that q(x) = limsup f(x) a.e. (4.4.2) JED
Denote by qo the random variable on the right-hand side in (4.4.2), and by ' its distribution function. By the monotonicity and right-continuity of either functions, it suffices to verify that W(t) = t'(t) for all t E M. According to the Borel-Cantelli lemma, the condition En 1 P(A,,) < oo, where An are sets measurable with respect to a probability measure P, implies that the P-probability of the set A of all the points that belong to infinitely many sets A,,, is zero (see, e.g., [697, p. 272J). For any fixed r E Al, by virtue of the Borel-Cantelli lemma, almost every point x from rV belongs to infinitely many sets rCk (since y(rV A rC)L) < 2-k) and therefore
4.4.
Convex functions
173
belongs to the set {qo < r}. This means that the set rV is contained in the set A := {qo < r} up to a set of measure zero. Hence y(rV) < y(A). Let us fix e > 0 and pick s E Al such that s > r and y(A,) < y(A) + e, y(sV) < y(rV) + e, where A, :_ {qo < s}. For every x E A, there is m such that x E sCk, Vk > m. It follows from the construction of Ck that x E sV for almost every x E A. Therefore,
y(A) < y(rV)+e. Since e was arbitrary, we get the estimate y(A) < y(rV). Thus, one has cp(r) = V)(r). It remains to be shown that (4.4.2) can be represented as sup. To this end, let us consider D as a subset of the Hilbert space X.. Let K be the closure of D in X; and
let KO be the set of its cluster points. Then C(f, x) := f (x), f E K, is a Gaussian random process on K. Since X,* is separable, there is a separable modification of this process denoted by the same symbol {(f, x). Taking into account that y is symmetric and that qo(x) < oo a.e., we get oo
sup If(f,x)I = sup
JEK
a.e.
fED
By the It&-Nisio theorem 2.6.3, this process a.s. has a nonrandom finite oscillation b. We have seen in Theorem 3.4.1 that the process C has a modification (again denoted by the same symbol) that is continuous on K with the metric
d(f,9) = If - 9IIL2(,) + Ib(f) - 09)I and that K is separable with respect to this metric. Let us choose an at most countable set S = {f,, } C Ko which is dense in KO with respect to d. According to Problem 4.10.18, the sets K and KO are compact in X.*,. Therefore, one can write
qo(x) = lim sup f(x) = sup limsup g(x), JEKu9ED.9-J
fED
which equals a.e.
sup lim sup (g, x). fES 9ED. g-f
By the Ito-Nisio theorem 2.6.3, we arrive at the representation qo(x) = sup JES
x) + 16(f)] = sup [f (x) + '6(f)] 2
This yields the desired representation.
JES
a.e.
2
O
Note that the seminorm defined by equality (4.4.1) with f,, E X' is lower semicontinuous, i.e., the sets {q < r} are closed. According to the Hahn-Banach theorem, any lower semicontinuous convex function f can be written in the form f(x) = sup. [fa(x)+cQ] with some fa E X' and c,, E IR'. If X is a countable union of metrizable compact sets, then it is easily seen that the corresponding collections of functionals and numbers can be taken to be countable. In particular, every lower semicontinuous convex function on a locally convex space X with a Radon Gaussian measure -y has a modification of the form (4.4.1), although it may fail to have this form pointwise. For example, let us take for X the space of all bounded functions on (0,11 with the topology of pointwise convergence and endow it with the Wiener measure. Let A be the set of all Lebesgue measurable functions on (0,11 bounded by 1 in the absolute value. Then A is an absolutely convex sequentially closed, but not closed set. The linear span of A is a Banach space with
Chapter 4.
174
Convexity
respect to the norm p.,, which has measure 1 with respect to the Radon extension
of the Wiener measure to X. Hence pA is a measurable seminorm that is not lower semicontinuous. Another example with the same property is if I, where f is a discontinuous sequentially continuous linear functional (recall that such a functional is measurable with respect to every Radon Gaussian measure: see Problem A.3.26 in Appendix concerning existence of such functionals). In relation to Proposition 4.4.3, see Problem 4.10.26.
4.5. H-Lipschitalan functions
4.5.1. Definition. Let ry be a Radon Gaussian measure on a locally convex space X, let H = H(7) be its Cameron-Martin space with its natural norm I I x, and let Y be a normed space. A mapping F: X Y is said to be H-Lipschitzian with constant C if it is and there exists a number C such that for -y-a. e. x one has IIF(x + h) - F(x)ll, S CIhI, Vh E H.
4.5.2. Lemma. Assume that Y is a normed space and that F: X -. Y is an H-Lipschitzian mapping with constant C. Then there is a full measure set X0 such that X0 + H = X0 and, for every x E X0, one has
IIF(x+h)-F(x+k)II,.
(4.5.2)
In particular, F has a version F0 such that (4.5.2) is fulfilled for every x E X X.
PROOF. Let it be the set of all points x such that (4.5.1) holds true. By Proposition 2.4.10, the set & of all those x Cr H such that. H f1 (52 - x) is dense in H has full measure. Then (4.5.2) holds true for every x E Xo. Indeed, let x E Xo,
h, k E H, and let e > 0. There exists v E H such that Iv - kl 5 <e and x + v E S2. Then IIF(x + h) - F(x + k)II, 5 IIF(x + h) - F(x + v)II,. + IIF(x + v) - F(x + k)II,,
< CIh - vI + CIv - kI < CIh - klN + 2Ce. Since e > 0 was arbitrary, we arrive at (4.5.2). Moreover, we get automatically that
x + H C Xo. It remains to note that in order to get a version of F that satisfies
0
(4.5.2) for every x, it suffices to redefine F by zero outside Xv.
4.5.3. Remark. (i) It follows from the proof that for X0 one can take a countable union of metrizable compact sets. Indeed, it suffices to choose in the set X0 above a full measure subset X1 which is a countable union of metrizable compact sets and take X, + H. (ii) The same reasoning gives the following more general statement: if E is a linear subspace of H equipped with some separable norm II E such that 'y-a.e. one has IIF(x + v) - F(x)II, < CIIvIIe, Ye E E, (4.5.3) II
then there is a full measure set Xv with X0 = X0 + E such that (4.5.3) holds true F(x + v) is Lipschitzian along E with constant C). for every x E Xo (i.e., v
4.5.4. Corollary. Suppose that F: X -+ Y is a -y-measurable mapping with values in a separable Banach space Y such that, for every h E H, one has II F(x + h) - F(x)II , 5 CIhI
4.5.
H-Lipschitzian functions
175
on a full measure set dependent on h. Then F has a Boret modification that is H-Lipschitzian and. moreover, satisfies (4.5.2) for every x E X. PROOF. Indeed, let C H be a countable dense set. We may assume that F is Borel. Indeed, since H is a separable metric space, there exists a Borel version of F, which satisfies the same condition by the equivalence of y and yh. By using Remark 4.5.3(ii), for every space IR' h,,, we find the corresponding full measure
set X = X,, + 1R'h,,. Put z = n,,=,
Suppose first that F is bounded.
The mapping TTF (which is defined, since Y is a separable Banach space) has the following property: y-a.e. one has
VnEll'I. This follows from the fact that a-'x + 1 - e- ty E Z for y®y-a.e. (x, y), which gives the estimates
II F(e-tx + e-th +
1 - e-2ty) - F(e-'x +
1 - e-2ty) Ily <_ Clhnl,,
Since the mapping h * -. TTF(x+h) is continuous, we conclude that TTF satisfies the condition in Lemma 4.5.2. Then, for every n, there is a full measure Borel set Sl,,, which is a countable union of metrizable compacts sets, such that Sln and h h) is Lipschitzian with constant C for every x E fln. It is readily verified that is a Borel mapping. By letting T11,F = 0 outside Q,,, we get a Borel mapping G. such that h 'H Y, is Lipschitzian with constant C for every x. Since T1/ F F in L' (y, Y), we can choose a subsequence C,,, that converges to F a.e. Denote by Sl the set of all points where {G,,, } converges
and define 4 as the corresponding limit. By Proposition 2.4.10, there exists a full measure set Slo C f2 such that llo is a countable union of metrizable compact
sets and, for every x E Slo, the set (Sl - x) fl H is dense in H. Then, for every x E 11o, the sequence {G,,,(x + h)} converges for every h E H, due to the easily verified fact that any uniformly Lipschitzian sequence convergent on a dense set is convergent everywhere. In particular, Slo + H = ilo. Therefore, for every x E flu, the mapping h i-, '(x + h) is Lipschitzian on H with constant C. It remains to put 'p = 0 outside SZo. Clearly, we get a Borel mapping. If the case where F is not bounded, we consider the sequence of bounded mappings tJ,, o F, where V,m(y) = y if Ilyllr < m and +y(y) = my/Ilyll, if IIyll, ? m. The mappings >y,,, o F have H-Lipschitzian versions F,,, and converge pointwise to F, whence by the same reasoning as above we get a version of F with the desired properties. D
4.5.5. Corollary. Let Y be a separable Banach space and let F: X -» Y be an H-Lipschitzian mapping with constant C. Then there is a full measure set X0, which is a countable union of metrizable compact sets, such that X0 + H = Xo, F is Borel on X0, and (4.5.2) is true for every x E Xo.
PROOF. By the previous corollary, F has a Borel version F0 such that h Fo(x+h) is Lipschitzian with constant C for all x E X. As shown above, there exists a full measure set it which is a countable union of metrizable compact sets and has the following properties: F = Fo on Sl, and for every x E fl, (4.5.2) holds true and
the set (U - x) fl H is dense in H. For every such x, one has Fo(x + h) = F(x + h) for all h E H. Therefore, we can put X0 = 0 + H. 0 In the next chapter (see Section 5.11), we discuss some differentiability properties of H-Lipschitzian mappings.
Chapter 4. Convexity
176
We pass now to the infinite dimensional analogues of the inequalities proved in Chapter 1. Let -y be a centered Radon Gaussian measure on a locally convex space
X and let H = H(7) be its Cameron-Martin space. Sometimes it becomes more useful to estimate the concentration of a function around its median rather than around its mean. For a measurable function f on a probability space (X, p), we denote by M(f) a median of f, i.e., M(f) is a number such that I, (x:
f(x) > ,1(f)) > 1/2, p(x: f(x) 5 M(f)) > 1/2.
A median exists (but may be not unique), since one can take
M(f) = sup{t: µ(x: f (x) < t) < 1/2}. Zero is a median of the function f - M(f). It is suggested to prove in Problem 4.10.20 that if p is a Radon Gaussian measure on a locally convex space X and f is a p-measurable function such that for p-a.e. x the function t ' f (x + th) is continuous for every h from H(p), then a median M(f) is unique. 4.5.6. Theorem. Suppose that a -y-measurable function f satisfies the condition
f (x + h) - f (x)I 5 CIhj , V h E H -y-a.e. Then, for any r > 0, the following estimate holds true:
7(x: If(x) - h'f(f)I > r) < 2[l
-
4/
2C2).
hh1J < expl
(4.5.4)
(4.5.5)
PROOF. We may deal with a version of f which satisfies (4.5.4) for all x E X and h E H. Note that
({f > M(f)}
n ({f 5 M(f)} +rU,r) C {If - M(f)I <- Cr}.
Indeed, if x = y1 + rh1 = 112 + rh2, where f(y1) > M(f ), f (y2) < M(f), h1, h2 E U , then by (4.5.4) one has f (x) > f (y) - Cr > M(f) - Cr. In a similar manner one gets f (x) < M(f) + Cr. Therefore, the measure of the set (If - M(f )I > r} is estimated by the sum of the measures of the sets X \ ({ f > M(f) } + rC' 1 U ) and X \ ((f < 16f (f) } +
Applying estimate (4.3.2), we get the desired
rC-1
inequality.
This theorem implies the integrability of f. There is also a similar inequality for the mean. 4.5.7. Theorem. Suppose that a -y-measurable function f satisfies the condition (4.5.4). Then f is integrable and for all r > 0 one has 2
r
(x: If (x)
- J f d-r> r) < 2exp2C2 ). (_
(4.5.6)
In particular, the function exp(a f 2) is integrable, provided a < YC2
PROOF. Let us take an orthonormal basis {1;,,} in X4 and consider the conditional expectations fn from Corollary 3.5.2. It is easily seen that the functions f also satisfy condition (4.5.4). It remains to apply the finite dimensional estimate.
4.6.
Correlation inequalities
177
4.5.8. Example. Let f be a -y-measurable seminorm on X. Then it satisfies condition (4.5.4). Put
lEf:= ffd7.
X(f):=sup{f(h):
y{x: If(x) - iEf l > t} < 2exp1 -7r2X(f)2t2l PROOF. According to Theorem 3.2.10(i), the function f is continuous on H(-Y).
Therefore, If (x + h) - f(x)l < f(h) <_ X(f)IhIH(.,) a.e. A number of additional results about the exponential integrability will be discussed in the next chapter in connection with the logarithmic Sobolev inequality.
4.6. Correlation inequalities This section is devoted to the so called correlation inequalities for a centered Gaussian measure -y, i.e., inequalities of the form
y(An B) > y(A)y(B).
(4.6.1)
The functional form of this inequality is
Jig dy >
Ji dry
f
g dy.
(4.6.2)
Note that (4.6.1) implies (4.6.2) for all nonnegative functions f and g from L2(7)
such that the sets A = if > t} and B = {g > s} satisfy (4.6.1) for all t, s > 0. Indeed, we may assume that If IIt,(-,) = 1. Let us consider the probability measure
I A:= f y. We have to show that J g dy <
1 g dµ. It suffices to prove that
p({g > s}) > y({g > s}),
V s > 0.
Put B = {g > s}. If y(B) > 0, then v:= y(B)-'yIB is a probability measure and estimate (4.6.1) reads as v({ f > t}) > y({ f > t}), whence
u(B)= y(B)
ffdv> ffdy=1.
Inequality (4.6.1) is known to be true for certain pairs of absolutely convex sets A, B in IR" with a centered Gaussian measure -y. Some of these cases are discussed in this section and in the complements to this chapter. It is an important open problem whether (4.6.1) remains true for arbitrary absolutely convex sets. For n = 2 a positive answer is given in [614].
4.6.1. Theorem. Let -y be a centered Gaussian measure on lR". Then for every absolutely convex set A and every strip n of the form II = {x: If (x) I < c}, where f is a linear function and c E IR', one has -f (A n II) > y(A)y(II).
(4.6.3)
Chapter 4. Convexity
178
PROOF. It suffices to prove the claim for the standard Gaussian measure yn
on R". Moreover, we may assume that II = (x = (x1,... ,x"): 1xt(< c}. Put
f(xl) = f I,,(xl,y)y.,-1(dy),
x1 E IR.,
y= (yt,-.. ,y"-I)
The claim is obviously true if n = 1. Hence, for any nonnegative log-concave function f on R', we have
f
+x
c
(4.6.4)
f(t)y1(dt) ? yi([-c, c]) f f(t)-Y1(dt).
-x
-c
Indeed, since the measure yl((-c, c]) is probability, (4.6.4) follows by Problem 1.10.11. Applying (4.6.4) to our function f, we get )'" (AnfI)
=
f
(xl)f(xl)yl(dx1)
_x
tx
>'yl ((-c, c]) f f(x1)7(dxl) = y1((-c,c])-y (A) = yn(n)yn(A). -x
where we used Fubini's theorem. The following useful inequality (called $idak's inequality) is easily obtained by induction.
4.6.2. Corollary. Let t;1, ... , tk be random variables whose joint distribution is centered Gaussian. Then, for any reals c1, ... , ck, one has
t
P(E111
k
t 5 C,) Ck) >_ flP(Ijl \1
Cl,...
(4.6.5)
i=1
4.6.3. Theorem. Let - be a centered Radon Gaussian measure on a locally convex space X. Then for every absolutely convex -y-measurable set A and every strip H of the form. II = {x: If (x) 1 < c}, where f E X. and c E R', one has
y(Anf) > y(A)y(II).
(4.6.6)
PROOF. The claim reduces to the finite dimensional case considered above, since, for every e > 0, we can approximate A by an absolutely convex cylindrical set C such that -y (A 11 C) < e.
4.6.4. Corollary. In the situation of the previous theorem. one has (4.6.7) ,r
x
X
protided f, g E L2(1) are such that the sets { f > t}, t E R', are absolutely convex up to measure zero sets, and the sets {g > s}, s E Rl, with 0 < y({g > s}) < 1 are, up to measure zero sets, symmetric strips of the form indicated in Theorem 4.6.3.
4.6.
Correlation inequalities
179
PROOF. If f and g are nonnegative, then (4.6.7) follows from (4.6.6) by Problem 1.10.11. Suppose that f > c and g > c for some c E IRI. Then (4.6.7) is true for f - c and g - c in place of f and g, which gives the same estimate for f and g. Now the general case follows by Lebesgue's dominated convergence theorem, for the functions f" = max(f, -n) and g" = max(g, -n) satisfy the condition in this corollary.
The following result with its proof is borrowed from [6711.
4.6.5. Proposition. Let -y,, be the standard Gaussian measure on IR", let A and B be two absolutely convex sets in 1R", and let U" be the group of all orthogonal
operators on IR" with its Haar measure m. Then
f 1 y"(An U(B)) m(dU) > y"(An B).
(4.6.8)
U In particular, if B is spherically symmetric, then y"(An B) >'y"(A)y"(B). PROOF. We shall see that in fact (4.6.8) is true for all spherically symmetric measures and all measurable sets A and B such that the sets A,, = A n 1Rlcp and B,D = B n 1R1 cp are symmetric intervals for every cp from the unit sphere S. To this
end, let us write y" as crop, where a is the normalized Lebesgue surface measure on S and p is some probability measure on (0, oo). We observe that a coincides with the image of m under the mapping c' '- Ucp, U" S, for any fixed cp E S. Therefore,
f(A n U(B)) m(dU) = J JP(A,, n B,,) a(dcp) a(d>,). SS U. Now we observe that p(A,, n B,) > p(A;,) since A,, n B0 is either A., or Be,. Noting that ,y. (A) =
f
p(Ap) a(d4p)
S
0
and similarly for B, we get (4.6.8).
It is interesting to note that inequality (4.6.2) that we would like to have for log-concave functions is valid for convex functions. Let us give a precise formulation following [3501.
4.6.6. Proposition. Let y be a centered Radon Gaussian measure on a locally convex space X and let f, g E L2(y) be convex functions such that I,(f) = 0. Then
f
fgdy >
ffdy J9d.
(4.6.9)
x x x PROOF. Let us consider first the case, where y = y,, is the standard Gaussian measure on IR". Suppose that f, g E W2,2(-Y,). Let (W1),>0, Wt = (W=1,... ,Wt ), be a standard Wiener process in IR". According to Proposition 2.11.1, we have
f(WT)=IEf(WT)+ JVP1._tf(wt)dwt a.e. 0
Chapter 4. Convexity
180
Using an analogous representation for OP1 _t f , the commutativity between Pt and
a., and the fact that Pt_5P1_1 = P1_ we arrive at the equality n
a.,P1_tf(ut) =E[a=,P,-tf( t)] +E f ax. 8x,Pi_sf(W.)dwa. j=1 0
Note that EE[a=,P1_tf(Hi)] = Pt(a=.P1-tf)(0) _ (az.Plf)(0)
v.f(b)7(dy) =0, R"
since
a.. P1f(x) = 8
,
f
W
f(x+y),y(dy) = f (z, -x,)f(z)'y(dz). R"
Clearly, we may assume that I, (g) = 0, passing to the function g - I, (g). In addition, we may assume that f and g have zero integrals with respect to -y, passing to the functions f - c1, g - c2. Therefore, we arrive at the following representation: t
1
JJax.o11Pi_8f(W.)dWdWt'.
f(W1) _ 0
0
An analogous representation holds for g. It remains to note that Jig d-, = lEf(K'1)g(K'1) > 0. Indeed, the matrices (A'j) = (8z, a=, Pi _d f (x)) and (B')) = (a=, a=, P1 _,g(x)) are nonnegative definite, which follows from the convexity of the functions Ptf and Ptg (implied by the convexity of f and g). Therefore, n
A'-7 B'-' = trace(AB) > 0, ij=1 since, letting c, be eigenvectors of A corresponding to eigenvalues ai > 0, we have
trace (AB) _ E a, (Bei, e,) > 0. If f and g are not supposed to be in W2.2(ryn), we =1
consider the approximations Ttf and Ttg, t > 0, where (Tt)t>o is the OrnsteinUhlenbeck semigroup. It is readily verified that Tt f and Ttg are convex and 11(Tt f) = e-tI1(f) = 0. Since Tt f, Ttg E W2.2(-yn), we arrive at (4.6.9) in the limit as t - 0. In order to complete the proof in the infinite dimensional case, we apply our standard finite dimensional approximations from Corollary 3.5.2, which give convex cylindrical functions fn and gn according to Remark 3.5.3. 0 There is a version of the correlation inequality for positively correlated random variables.
4.6.7. Proposition. Let y be a Gaussian measure on R" such that all elements of its covariance matrix K = (K;,.,) are nonnegative and let f and g be two functions in L2(,y) that are nondecreasing in every variable. Then
f
R"
fg d^y > f f d-y f g dy. R"
R"
(4.6.10)
Onsager-Machlup functions
4.7.
181
PROOF. Suppose first that f and g are bounded and nonnegative. We prove the claim by induction in n. Let n = 1. In this case, we do not need that y be Gaussian. We may assume that 11911L'(,) = 1. Then p = g y is a probability measure. According to Problem 1.10.11, it suffices to show that
p({f > t}) > y({f > t}), bt > 0. (4.6.11) If the set J = (f > t} is nonempty, then it is a ray (r, +oo) or Jr, +oo). Let us consider the probability measure v = y(J)-1yIj. For every s > 0, one has v({g > s}) _> y({g > s}), since the set {g > s} is also a ray (or empty) and its intersection with J is either J or {g > s}. Hence f g dv > f g dy, whence (4.6.11) follows by Problem 1.10.11. Suppose that (4.6.11) is proven for some n > 1. We may assume that y is centered and is not concentrated at the origin. According a well-known theorem of Perron, the maximal eigenvalue k of K corresponds to an eigenvector e = (a1,... ,a,) with nonnegative coordinates a, (see, e.g., [670,
Appendix]). We may assume that a > 0 and that k = 1. Denote by -yo the projection of y to IRn-1 regarded as a hyperplane L in 1R". Then the elements of the covariance matrix of yo are nonnegative. The conditional measures yy on the lines y + IR' en , y = (y1, ... , yn _1,0) E L, are Gaussian with means m(y) = a l y1 + +
an-lyn-1 and variance 1 (see Theorem 3.10.2). Note that if y = (y1,... yn_1,O), z=(z1,...,zn_1,0)ELandy,>z,,i=1,...,n-1,then m(y) _ m(z). By the one dimensional case, we have
f fgd-y =
R"
>
f f f(yl,... ,yn-1,t)9(Y1....
4n-l,t)yy(dt)yo(dy)
R"-1 y+Rle
f 9(y1,... f ( f f(y1,... ,yn-.,t)7'y(dt)y+Rle
1 Yn_1,t) Yh1(dt)) -to (dy)
Rn-1 y+R'e The expression on the right is the integral against yo of the product of the functions
ff(i.....y_i,t+m(y))yj(dt). f 9(y1, ... , y.- 1, t + m(y)) yl (dt) R1
R1
of y, which are nondecreasing in every variable yi. Hence we can apply the inductive assumption, whence the claim. If f and g are bounded, but not necessarily nonnegative, we apply (4.6.11) to the functions f + c and g + c with c sufficiently large and observe that this yields the claim for f and g. For unbounded f and g, we use (4.6.11) for the nondecreasing functions fk = max(min(f, k), -k) and 9k = max(min(g, k), -k) and pass to the limit as k -' oc.
Other proofs of this proposition can be found in (36), (769] (see also Problem 4.10.23).
4.7. The Onsager-Machlup functions Let p be a measure on a metric space X and let K(x, e) be the closed ball of radius a centered at x. In diverse applications one has to study the existence of the following limit: lim
p (K(a,E)
e-o A(K(b,e))
= I(a,b).
Chapter 4. Convexity
182
If this limit exists, then the function F(a, b) = log I (a, b) is called the OnsagerMachlup function (where we put log(oc) = oc, log(O) = -oo). We only discuss here the Gaussian measures, since many results on the Onsager-Machlup functions are based on the Gaussian case. Let - be a centered Radon Gaussian measure on a locally convex space X, let H = H(y) be its Cameron-Martin space with the Hilbert norm I I,,, and let q be a -y-measurable seminorm on X. We shall assume that a version of q is chosen that is a seminorm in the usual sense. For vectors h E H we shall investigate the existence of the limit lim
-y(17, + h)
y(V)
C_o
V. _ {x: q(x) -< r}.
(4.7.1)
We shall assume that
'Y(VE)>0, Ve>0 (otherwise one can assign 1 to the corresponding limit). As we already know, this condition is indeed a certain restriction on q, since it can happen in infinite dimensional spaces that -((VI) = 0 whereas -y(V2) > 0 (see Example 3.6.3).
It will be shown below that the limit in (4.7.1) exists. Note that by virtue of the Cameron-Martin formula (2.4.2), the following equality holds true: 7(
, + h) = expC -
1 Ih1 2
exp(h(x))ry(dx).
(4.7.2)
ti(ti:) J
Y
where h is the measurable linear functional, generated by h. Let us introduce some additional notation:
.4(f) =
exP(f(x)) -y (d,),
1
V. = {q < s},
7(VE)
Eq = { j E X,y: lim J,(f) exists}, C-O
Fq={f EX,;: ii mJ,(f)=1}. By virtue of (4.7.2), the existence of the limit in (4.7.1) is equivalent to the inclusion h E Eq.
A trivial example of an element from Fq is any function f that satisfies the condition If I < Cq ry-a.e. for some number C; in this case, one has exp(-Ce) < J. (f) < exp(Ce). Clearly, F. may not coincide with Eq. For example, this is the case if y is the standard Gaussian measure on 1R2 and q(x) = 1xI 1, x = (x1, X2)corresponding limit for f (x) = x2 equals I +'c e=, p(x2, 0. 1) dx2. As we shall see below, the reason why Eq and Fq may not coincide is similar in the general case.
4.7.1. Lemma. For all f E X., the following inequality holds true: 1<
ti(VE) J V
eXP(f(x))7(dx) <eXP1 2Ilfll22(.)).
\
(4.7.3)
4.7.
Onsager-Machlup functions
183
PROOF. Indeed, since our measure is symmetric, we have
f f d7 = 0. By
virtue of Jensen's inequality for convex functions, we get
I = exp(-
-
fl(s) '(dx)) <
y(Ve)
V,
f exp(f(x)) y(dx)-
v
On the other hand, according to Anderson's inequality, y(Ve + h.) < y(V, ), Vh E X.
For h E H this yields
fex(_i
1H +
V.
h(x)) y(dx) < y(Ve).
/
Since every functional f E Xf, has the form f = h for some h = R,,(f) E H with IIfIILz(,) = IhIM, we arrive at (4.7.3).
The following simple lemma is crucial below.
4.7.2. Lemma. The sets Fq and Eq are closed in X.. In addition. Fq is a closed linear subspace in X;. PROOF. Let us denote the norm in L'(-y) by II 112. According to (4.7.3), the IR1 are locally Lipschitzian. Indeed, applying the Cauchy functions Je: X,; inequality to the integral
f exp(f(x)) [exp(g(x)) t; and then estimate (4.7.3), we get 1J,(f + 9) - J,(f)1 <
e(2f)
- 1] y(dx),
2JE(9) + 1)
exP(IIIII2)
eXP(2II9II2) - 1 <- V'exp(IIfIh)eKP(p9I12)IIgII2,
since JE(g) > 1 and
0 <_ Je(2g) - 1 < exp(2II9II2) - 1 < 2II9II2eXP(2II9II2),
which follows from estimate (4.7.3) and the inequality et - 1 < te'. Therefore, on every centered ball of radius R in X;, the functions J. are Lipschitzian with the constant vre2R2 . This uniform in e > 0 local Lipschitz estimate implies that Eq and F. are closed in X. Indeed, let fn -+ f in X,;, f,, E Eq and b > 0. It follows from the estimate proven above that, for n sufficiently large, one has I Je (f n) - J e (f) I < b for all c > 0. By condition, there is co > 0 such that I Je (f ,) - Jo (f,,) I <- b
provided e, 8 E (0,eo]. Therefore.
lit(f) - J4(f)1 5 I e(f) - JE(fn)I + I MM - J8(fn)I + IJe(f.) - e(f)I
X.
which implies existence of the limit Um Jf(f), i.e., the inclusion f E E. In a similar o manner one verifies that Fq is closed as well.
Let us now prove that Fq is linear. Let f E Fq. For all a E (0, 1), according to (4.7.3) and Holder's inequality applied to the probability measure 7(Ve)-17I
15 Je(Af) <- e(f)''.
Chapter 4. Convexity
184
Therefore, 1f E Fq. Put TE(z) = JE(zf). The functions TE are holomorphic and satisfy the inequality
I T (Z) I! JE(Re(z)f) <exp('Re(z)2IIf112). Since Um Te(z) = 1 for z E (0,1), Vitali's theorem implies that these functions converge to 1 uniformly on bounded subsets of the complex plane (see [767, Ch. V, § 5.21). Hence of E Fq for all A E 1R'. Let g E Fq. By virtue of estimate (4.7.3) and the Cauchy inequality, we have
a-.0.
1<JE(f+9)
J.(2 f ))Ja(2g)-'1, Therefore, f + g E Fq which brings the proof to the end.
It follows from the estimate proven above that for any sequence e 0, there is a subsequence b, = e,,, such that lim JJ, (f) exists for all f E Xy (it suffices to tx find such a subsequence for the elements of any countable dense set in X;). We prove below more precise statements. Let us denote by Pq the orthogonal projection onto the closed linear subspace Fq in X,* and put
Oq=I - Pq. 4.7.3. Theorem. For all f E XX one has (4.7.4)
JL JE(f)=exp(2IlAgfII.2(.)).
PROOF. According to Proposition 4.4.3, the seminorm q coincides almost everywhere with the function sup,, [f,, (a) + an] , where the f,,'s are some elements in
Xy' and a >_ 0. In particular, f,,(x) + a,, <_ q(x) a.e., whence f,,(-x) + a,, < q(x) a.e., hence a < q(x) a.e. Due to our assumption about q, we get a = 0 for all n. As it has been noted above, fn E Fq. Letting f = Pq f + Aq f and denoting by Iv the indicator function of VE, we have
JE(f) =
7(V') J x
Iv, exp(Pgf +Agf)d7 = JE(Pgf) f exp(Agf)d7, x
since Aq f and exp(Pq f )Is; are independent random variables. Their independence
follows from the fact that the seminorm q is equivalent to a Borel function of { f,a } C Fq, and Pq f is in Fq, while the function ©q f is independent of any sequence
of elements of F9, since A. f is orthogonal to all such elements. Straightforward evaluations for the Gaussian random variable Aq f yield
f exp(Agf)d'.=exp{2 J(Af)2 d7] =exp[ IlAgf112 l
x By virtue of the inclusion Pq f E Fq we arrive at (4.7.4). X
(
)].
l
Let us now find the conditions under which Fq = X; (in this case Aq = 0). Let us introduce several new objects related to (4.7.1). Recall that the restriction qIN is continuous with respect to the natural norm of H. Put
Z = {a E H: q(a) = 0},
Z° = R,-'Z',
4.7.
Onsager-Machlup functions
185
where Z1- is the orthogonal complement of the closed linear subspace Z in H. Note that Z° is a closed linear subspace in the Hilbert space X;. In addition, by Proposition 2.8.4, the restriction of q on H is the same for all seminorms that coincide with q almost everywhere. In particular, the set Z is independent of such versions of q.
4.7.4. Lemma. The equality Fq = Z° holds true. PROOF. Suppose that f E F,,. We shall deal with a proper linear version of f. Suppose that a E H and q(a) = 0. Then VE + a = V. Therefore,
JE(f) =
7(VE) J exp(f(x)) y(dx) V. +a
7{VE) J
For any r > I and s by the following quantity:
exp(f(x + a)) exp(-2IaiH) exp(a(x)) 7(dx). the right-hand side of this inequality is majorized
exp(f(a))exp( rlIat(JE(9f))11,(JE(ra))1Jr 1/a
< exp(f (a)) exp -1 Ja1
(JE(sf))
exp 2 rIaIH )1
H
which tends to exp(f (a)) exp(I(r - 1)IaIH) as e - 0. This implies the estimate
1 = li pJE(f) <exp(f(a)) The same is true for -f. Hence f (a) = 0 (otherwise one of the two numbers ef or e'f(°) would be less than 1). Therefore,
U,a)c=(,)=f(a)=0. that is, f E Z°. Conversely, let f E Z°. As it was noted in the proof of Theorem 4.7.3, q has the form q = sup,, f,, a.e., where f E X;. Then obviously ff E F, and
Z={hEH:
VnEN}.
Since f. (h) = (R,f,,,h)H, the set Z1 coincides with the closure of the linear span {R1 in H. Therefore, Z° = R;' (Z1) coincides with the closure of the linear in X,Y, hence is contained in F,,. span of We observe that Aq f equals the conditional expectation of f with respect to the o-field generated by the elements h, h E Z.
4.7.5. Corollary. The equality Fq = X; holds true if and only if q J H is a norm.
PROOF. Clearly, q1 H is a norm if and only if Z = 0. The latter is equivalent to
the equality Zl = H, hence to the equality Z° = X;. It remains to apply Lemma 4.7.4.
186
Chapter 4. Convexity
4.7.6. Example. Suppose that. {q < 1} is a bounded set. Then Fq = X,*. Indeed, the boundedness of the set {q < 1 } implies that q on H vanishes only at the origin, hence is a norm.
4.7.7. Remark. According to Lemma 4.7.2, a sufficient condition for the equality Fq = X; is the existence of a set of continuous linear functionals from Fq that is dense in Xy with respect to the norm from L2(y). As noted above, the set Z = {a E H: q(a) = 0) is a closed linear subspace in H. Let lrq be the orthogonal projection onto its orthogonal complement Z'. It follows from the relations above that zr, = R, PR; i and Z° = R-' Z'. Thus, we get the following expression for the Onsager-Machlup function. 4.7.8. Corollary. Let h, k E H(-y). Then li
VV
o 7(
In particular, X(h, k) = 1agk12 PROOF. We have lim
exp(2largkIH
+ k) =
- 2lxghlH
1TghJN.
2
y(h) y(V = exp(
Ih12) lim Jc (h)
r)
= exp(_. 2IihiI2) exp(2lloghliz) exp(- 2IIPghllz) = exp(- 2I7rghI , ).
0 4.7.9. Example.
(i) Let y be the Wiener measure on CIO, 11 and let q be a -y-measurable seminorm. If
li mE(expw(tr) I q < s) = 1, where IE(F[A) := y(A)-'
J
dt E [0,11,
(4.7.5)
Fdy, then
it
lim E(exp J w(t) dwt 19 < ) = 1.
V
E L210,11.
(4.7.6)
0
(ii) If IIXIILI < Cq(x) for some C, then (4.7.6) and (4.7.5) hold true.
(iii) If 7 is an arbitrary centered Gaussian measure on C[0,11 satisfying (4.7.5), then
lim1E(expf Iq
PROOF. Note that the estimate 1I4L' < Cq(x) shows that q is a norm. Condition (4.7.5) means that the functionals bt, t E [0,11, belong to F. Therefore, Fq contains the closure of their linear span in L2(y). Every continuous linear functional on C[O,11 is the limit of a pointwise convergent sequence of linear combinations of the functionals bt. Hence claims (i) and (iii) follow from the fact that Fq is a closed linear space.
0
4.8.
Small ball probabilities
187
4.8. Small ball probabilities This section presents some basic facts about the behavior of Gaussian measures on small balls, that is, we consider the quantity -y(q < c), where -y is a Gaussian measure and q is a seminorm. The most important for applications special case is the behavior for small positive s of the probability P(sup ICt I < c), where is a
Gaussian process. We shall discuss first the classical Wiener space. Let U be the closed unit ball in Rd and let u: [0, oc) xU -. AR' be the solution
of the initial value problem au/at = 2"'. u, u(0, x) = f (x), with the boundary condition ulat,^ = 0. Then u(t. .r)
_
Y'n (x) J f(y),pn(y) dy, n=1
where 0 < Al < A2 < ... are the eigenvalues of the boundary value problem 0, plat.- = 0, corresponding to the normalized eigenfunctions {Wn } C
L2(U). It is known that A, > 0 and that p1 > 0 in the interior of U. For example, if d = 1, then A, = ir2/8 and Vp, (x) = cos(irx/2). Let (wt)t>O be a standard Wiener process in Ed. Put r(x) = inf{s: x + u admits the following representation (see, e.g., [639]): u(t,x) = 1F(f(wt +x)I,,=,>t). Taking f = 1, we get
P(r(0) > t) _ E
I Y9n(y) dy. t.
n=1
Observe that ewt/t: is a standard Wiener process for every e > 0. Therefore, denoting by P"w the Wiener measure on C(]0, TI, ad) and letting ' IIc be the II
sup-norm on C((0, TI, Rd), we get
PI'(IIzIIt,
< t-) =
P( max lu',I < e) = P( max Ieu,;_2I < e) O
= P(0< max
8ST/e'
1)
e-A"T/`'
_
Wn(0) f i0n(y)dy
(4.8.1)
n=1
which gives
P1i (IIxii'. < e) - Cl exp(-
Al
for small positive e. In particular, if d = 1, we get
l
P1t (I1xIIc < f J
4
/
expl
z
\
-W2There are several other examples /where the Gaussian measures of balls are found explicitly (e.g., the balls in a Hilbert space). In typical applications, however, only asymptotics or estimates are available. Moreover, very often such asymptotics
or estimates in terms of elementary functions are more convenient than explicit expressions. The reader will find a number of results of this kind in Section 4.10. Now we shall investigate the problem of existence of the limit
v(x: q(x) < F) lim t 0 µ(x: q(x) < e)
(4.8.2)
Chapter 4. Convexity
188
where p is a centered Radon Gaussian measure on a locally convex space X, v is a Gaussian measure equivalent to p, and q is an arbitrary p-measurable seminorm on X. The case where v is a translation of µ has been studied in the previous section.
We shall see in Chapter 6 that the Radon-Nikodym density of v with respect to p has the form exp Q, where Q is a p-measurable second order polynomial (i.e., an element of Xo » X, t$ X2, see Section 2.9). In Chapter 5 we shall formulate the results obtained in this section in a compact form in terms of the second derivative D,?Q of Q. Here we make use of the following representation of F obtained in Section 5.10: there exist real numbers c, cn, and an, n E N. such that x
rn < x, F, an < x, n=1
n=1
x
X
Q(x) = c+
an(en(E}2 - 1}
cnen(x) + n=1
(4.8.3)
n=1
where both series converge almost everywhere and in all LP(-y), p E [1, oo). Therefore, our problem can be formulated as follows. Let {yn } and {qn } be jointly Gaussian centered random variables such that the Cn's are standard Gaussian. Suppose that q = sup I?In I < x almost everywhere and that {q < r} has positive probability
for every 6 > 0. Let x
x
n=1
n=1
(4.8.4)
Note that if Q is given by (4.8.4), then expQ is integrable if and only if an < 1/2 for all n. In addition,
x E an ({n - 1)) }
lE [exp (c + E n=1
n=1
x =
exp(c+ 2 E n=1
1
x 2 c"anl TI exp(-an) / n,1
1 - an
(4.8.5)
This is readily verified by the direct evaluation taking into account that the product
x
on the right in (4.8.5) converges if E an < oc and an < 1/2. n=1 Put JJ = IE(expQJq < e). It will be shown below that the finite and positive limit lim Jf exists for every choice C-O
of q precisely when E lanl < oc. If an and an are, respectively, the positive and n=1
negative elements of the sequence {an}, then the convergence of one of the two x x
series E a,-,, E an together with the divergence of the other one implies that n=1
n=1
li m Jr = 0, respectively, limo Jf = +oo. Finally, if both series diverge, then one can choose qn = tnS, in such a way that there will be no definite limit of Je as e -+ 0.
Small ball probabilities
4.8.
189
It follows from what has been said that the problem of investigating the limit in (4.8.2) is equivalent to the problem of the study of aim 6--0
(4.8.6)
1 ({q < e} I expQdy
q.
for a centered Radon Gaussian measure -y and a --y-measurable second order polynomial Q (or, equivalently, a random variable (4.8.4)).
4.8.1. Lemma. Let y be a centered Radon Gaussian measure on a locally convex space X, let V be a -r-measurable absolutely convex set of positive measure, and let Qi, Q2 E Xo ®X2 be nonnegative. Then
Jex(-Q2)di <
fexp(Qi - Q2) dy 5 fexPQi dy. _-
X
(4.8.7)
X
V
PROOF. Since exp(-Q2) < exp(Q1 - Q2) < expQ1, it suffices to consider separately the cases where Q1 = 0 and Q2 = 0. Let Q2 = 0. It will be shown in x x is Section 5.10 that Q1 = c+ E anon, where c >- 0, an > 0, an < oe and n=1
n
an orthonormal basis in X;. Clearly, we may assume that c = 0 and that expQ] is integrable (otherwise the claim is trivial). The measure
v:= IIeXPQIIIexpQ]
y
is centered Gaussian (see Example 5.10.17) and its covariance majorizes the covari-
ance of -y. Indeed, let f E X
x Then f = 1 c, ,, where
E 12. The integrals
n=1
of f 2 with respect to v and y equal the sums of the corresponding integrals of the cnE 's by the independence of the random variables cn. Therefore, the claim reduces to the one dimensional case in which it is obvious. Hence v(V) < y(V), 0 which is our claim. The case Ql = 0 is analogous. 4.8.2. Corollary. Let ? be a centered Radon Gaussian measure on a locally convex space X, let V be a -y-measurable absolutely convex set of positive measure,
x
x
n=1
n=1
and let Q = E an (t n -1), where an > 0,
sequence in X. Then
exp Q dy <
I V
an < oo, and {t;n } is an orthonormal
fexPQd.
(4.8.8)
X
PROOF. We have to consider the case where exp Q is integrable, which correk
sponds to the condition an < 1/2 for all n. If Q is replaced by Qk = E an(tn -1), n=1
then the claim is true by the previous lemma. Now it remains to note that the functions exp Qk converge to expQ in L' (y). Indeed, the sequence exp Qk converges a.e. to expQ and is uniformly integrable, since by (4.8.5) there exists p > 1 such that sup II expQkIlLP(-,) < 00.
0
k
4.8.3. Theorem. Let -y be a centered Gaussian measure on a locally convex space X, let q be a -f-measurable seminorm which is a norm on H(-y), and let Q
Chapter 4. Convexity
190
x
be
given by (4.8.3) with E IanI < oo. Suppose expQ is integrable. Let us put n=1
VV = {x: q(x) < c}. Then Ei o
-Y(K)
JexPQ(x)l.(dx) = exp(c - E an
(4.8.9)
PROOF. It follows from our assumptions that Q can be written in the form
x
x
Q=C+f+Ql-Q2, f=EC, n,
Q,
x a.C2,
Q2=
n=1
n=1
I3
,
n=1
q
where c E IR1, (cn) E 12, an > 0, On > 0, an/jn = 0 for every n,
an < 00, n=1
x E 3n < oo, and {ln} is an orthonormal basis in X. Clearly, we may assume that
n=1
c = 0. We shall assume first that f = 0 and that exp(8Q1) E L1(-y). It follows from Proposition 4.4.3 and our condition that q(x) = sup fn (x) a.e., where {f,,} C X.; is a sequence with the dense linear span (if the linear span of { fn } is not dense in
X., then there is a nontrivial vector h E H(y) with fn(h) = 0 for all n, whence q(h) = 0). It is easily seen that our claim is true if Ql and Q2 are finite sums of the squares of finite linear combinations of the fn's, since in this case Q1 < cq2 and
Q2 < cq2 for some c > 0, which yields exp(-2ce2) < expQ < exp(-2cc2) on VE. Moreover, the general case reduces to the case where Q2 = 0. Indeed, letting JE (7, Q) =
exP Q dy,
1
7(K)
V,
and taking the centered Gaussian measure v := IIeXP(-Q2)IIL.(,) exp(-Q2)
'Y,
we have
Je(7,Q1 -Q2) =
JE(V,Q1) JE (V, Q2 )'
since
7(VV) = f eXP(Q2) dv V.
J
exP(-Q2) d7-
X
In addition, the fn's a r e orthogonal in L2(v) and II
nhI
2ivl = (1 + 28n)-1. Hence
Q2 = E 0. (1 + 2)3.) -1(.2, where {(,) is an orthonormal basis in X,,, i.e., Q2 n=1
satisfies the initial condition with respect to v. n
Let Sn = E a,t:?. Let us fix q > 1. By Lebesgue's theorem, there exists n such that
=1
JexP(2Q1 - 2Sn) dry < X
q2.
4.8.
Small ball probabilities
191
Using Lemma 4.8.1 and the fact that Q1 - S, > 0, we get the estimate
/eSn(
I <Je(7,Q1)=Jr('Y,QI-Sn+ S)= J(7,2Sn) <
Je(
2S)
-S)d7
Je(7,2Q1 -2Sn)
JX
exp(2Q1 - 2Sn) d7
J(7, 2Sn).
4
Since q > I was arbitrary, this estimate shows that it suffices to get the equality lim Je(7,2Sn) = 1. By induction, this reduces to the case a1 = ... = an-1 = 0, e-o since the measure v := 11 exp(2Sn _ 1) EI
V(.) exp(2Sn_ 1) - 7 is centered Gaussian and
Je(7,2Sn) = Je(v,2an£n)Je(7,2Sn-1). an`n- Then a(e)2 < 1/8 by the integrability of exp(8Q1). Let us fix again q > 1. Let us take a functional rl from the linear span of {fn } such that Set
x(17)2 < 1/8 and
2-1
Then we find e1 > 0 such that Je(7,4i12)
Ve E (0,e1)
q,
Since 2 < 2( - 17)2 + 2172, estimate (4.8.7) implies that, for any e E (0, e1), we have Je(7,4(C-17)2) <-
q
f exp(4((- n)2) d7 5 (1 X
- &i( -
7)2
Je(7,4172) 1/4
)
f
< q.
Since q > 1 was arbitrary, the claim is proved under the additional assumptions made above. The case where f 96 0 reduces to the case considered above. Indeed, by the Cauchy inequality, we have
IJe(7,Q+1)-Je('Y,Q)I=7(Ve)If expQ(expI-1)d'y v.
<
J(-y,2Q)
J(7,21) - 2Je(7,1)+ 1,
which converges to zero, since the first factor on the right is uniformly bounded in e E (0, 1] by (4.8.7) and the second factor tends to zero as e goes to zero by Theorem 4.7.3 and Corollary 4.7.5. Finally, let us drop the assumption of the integrability of exp(8Q1). Since a, < 1/2, there exists A > 1 such that the function exp(AQ1) is still 7-integrable. The functions G(z, e) = Je (y, zQ) are holomorphic in z if 0 < Re z < A and uniformly bounded in e > 0 and z with 0 < Re z < A. Since they converge to 1 as e - 0 for all sufficiently small real z, the convergence takes place for all z in [0, 1] by Vitali's theorem (see [767, Ch. V, § 5.21). 0 m
4.8.4. Remark. It is seen from the proof given above that if Q = E anCn, n=1
where E Ian < oo. and i,, E Xy are such that o(( )2 <-1, then lim JJ(-y,Q) = 1. 6-0 n=1
Chapter 4. Convexity
192
x
This follows also from (4.8.9). Indeed, let us write Q = c+ E An(17n-1), where {,1n) n=1
is some orthonormal basis in X,*,. We shall see in Proposition 5.12.16 that E an = n=1
2-'trace D,2Q = (Q,1)Loi,rl. Hence it remains to note that c = (Q,1)L2iyi. Let us give another example where the existence of the limit in (4.8.2) follows without knowledge of the eigenvalues of the corresponding quadratic form.
4.8.5. Example. Let Q be a sequentially continuous quadratic form on X and let q be the same as in Theorem 4.8.3. Then
limy(45e} ,[ expQd-y
t -0
1.
q<e
PROOF. According to Proposition 5.10.16 in the next chapter, Q has the form indicated in the previous remark. [J Combining Theorem 4.8.3 and Corollary 4.8.2 ae see that
lim J,(y,Q) = 0 if Q = Ql + Q2, where Q1 and Q2 are second order polynomials such that Q1 x satisfies the conditions in Theorem 4.8.3 and Q2 = E an(Cn - 1), where an > 0, n=] ao x an < oo and £ an = oo. The proof of this fact is readily reduced to the case n=1
n=1
Q1 = 0. Letting Sk =
k
an({n - 1) and assuming that exp(2Q2) is integrable, we
n=1
get by Corollary 4.8.2 and the Cauchy inequality that JE(1', Q2)2 < J.(y, 2Sk)Jr(y, 2Q2 - 2Sk) < Je(y, 2Sk)
The first factor on the right tends to exp(- E an) as a n=e
r exP(2Q2) dy Jx
0, whence the claim. Fi-
nally, note that there is no loss of generality to assume the integrability of exp(2Q2), since an -+ 0, hence, for sufficiently large k, this assumption is satisfied for the function Q2 - Sk in place of Q2 and the measure Q exp Sk jj L 1(7) exp Sk y. Considering -Q2 in place of Q2 above, we get infinity as the limit of Jf (y, Q). We shall now see that the conditional convergence of the series of the an's does not yield the existence of the limit JJ (ry, Q).
x
4.8.6. Example. If Q = E an(Cn - 1) + E an (tn - 1), where (an) n=1
T2,
n=1
(on)E12,an>0,an <0,anan =0forall nand
t
an = n=1
jan j = 00,
(4.8.10)
n=1
then with a suitable choice of a norm q, there is no definite (finite or infinite) limit of the quantities J1(y, Q). Moreover, one can choose q in such a way that JJ (y, Q) will oscillate between 0 and oo.
Small ball probabilities
4.8.
193
PROOF. For notational simplicity, we only consider an example of {an} and q where there exist two different cluster points; in the general case, one uses a modification of this construction. The sequence (an) will be constructed as a sequence of finite blocks
an.l, ... , an.2m(n), m(n) = n22n+2 an.j = n 2 ' 2 , an.m(n)+j = -an,j if j = 1,... ,m(n). m(n)
Then
j=1
2m(n)
an,j = n and
a2.j = 2-"-1. We shall find two sequences of positive
j=1
numbers en < 1 and cn,j, j = 1,... , 2m(n), such that en > fn+I, Cn,j > nm(n) (this is needed in order to get a finite norm), and letting 2m(n) . , Jn(E,Cn,1,..Cn.2m(n))
1
_
exp(an.jt2 -anj)P(t)
J
,
where p is the standard Gaussian density on IR1 and
Anj(E) = J
p(t)dt,
-ccn.,
one has
m(n) 1
j=1 J. (e2n, Cn,1 ,
1,
i<2n,
Me-n,
, C,,.2.(.)) <
IJn(e,Cn.l,... ,Cn.2m(n)) - 11 < 2-"-1, where
<2 "
1 - 4anj
VC E (0, e2n+1),
x m(n)
M = fl f VF + 4an,j. n=1=1 Such sequences are constructed inductively. Suppose that we have already constructed e,, i < 2n, and ck.,, k < n. Let us take R > nm(n) so large that, for every i < 2n, one has m(n)
Jn(Entl,... ,tn(m),81,... ,Sm(n))
II
j=1
< 2-"-1, 1 - 4On j
dti, S,
R.
2m(n)
This is possible by (4.8.5) and the equality E an,j = 0. Further, there exist j=1
b E (0, R-1) and r > 6 such that, letting b, = 6 and r,
r, we get
m(n)
Jn(e,61,...,bm(n),rl,...,rm(n))<2e-" rj
1+4an,j<2Me-".
(4.8.11)
j=1
Indeed, as 6 - 0, the product of the first m(n) factors in the expression for Jn m(n)
tends to the value exp(- E an,j) equal exp(-n), and, according to (4.8.5), the j=1
Chapter 4. Convexity
194
m(n)
product of the remaining m(n) factors tends, as r - +oo, to fl
Let
j=1
us put £2n = Cn.j =
R E2n_16
,
62£2n -1
R
rR
j = 1, ... , m(n), c,,,j =
m(n) + 1,... , 2m(n).
£2n -1
Then we have E2nCn.j = 6
if
j < m(n),
E2ncn,j = r if m(n) < j < 2m(n).
Finally, let us take c E (0, E2n] such that IJn(E,C.,,1,... ,Cn.2m(n)) - 11 <2 n-1,
dE E (0.c].
(4.8.12)
2m(n)
This is possible by the equality E an,j = 0. It remains to put E2n+1 = c/2. This j=1
construction yields the desired example if we take for y the countable product of the standard Gaussian measures on 1R1 and set
x Q(x) = Fan (xn - 1),
q(x) = sup jx, /A, 1,
n=1
n
where {A,} is the sequence formed by the finite blocks cn,j, j < 2m(n). Note that q < oo almost everywhere and y(q < e) > 0, Ye > 0, due to our choice of a;. The sequence J,,,, (y, Q) tends to zero, whereas the sequence JE,,,., (y, Q) tends to a positive number. In the general case, if condition (4.8.10) is satisfied, we can assume that the numbers a,+, and ia; I are less than 1/2. Choosing two subsequences nk and mk such that n4
-2-k < J=1
o+
m4
aj < 2-k, j=1
one can consider a construction that is completely analogous to the one described above and differs only in a more complicated notation. In particular, instead of e-" on the right in (4.8.11) we get exp(- E a ),and in (4.8.12) in place of 1 we j=1
get exp (- F, a' - E a ). Clearly, one can get an example where J, (y, Q) has j=1
j=1
as cluster points 0 and +oo; then .J (y, Q) oscillates in the interval (0, 00).
0
Note that for a suitable sequence of positive numbers q,, the measure y can be regarded also on a separable Banach space Z of the sequences x = (x,,) such that IIxII = sup lgnxn I < oo and Um lq,,xn I = 0. If {qn} decreases sufficiently fast, then
n-x
the sets {q < e} are compact in Z. Therefore, the limit in (4.8.2) may not exist for equivalent Gaussian measures even in the case of compact sets {q < e}. The reason is that the corresponding Radon-Nikodym density does not always satisfy the condition in Theorem 4.8.3. The situation may be different if the norm q is prescribed. It follows from [498, Proposition 8 and Remark, p. 288] (see also [4841) that there exist a centered Gaussian measure y on X = 12 and a sequence (an) E l2
4.9.
Large deviations
with n=1
195
Ianl = oo such that, letting q(x) = IIxII, one gets a positive finite limit
in (4.8.6). Finally, note that in Section 5.12 below, we shall give representation-free formulations of the results in this section; in addition, we shall consider the case where q may be a seminorm on H.
4.9. Large deviations Let -y be a centered Radon Gaussian measure on a locally convex space X, let U be the closed unit ball in the Hilbert space H = H(-y), and let
-y'(A)=y(e-1A), a>0. For any two sets A, BCX,weput
AeB=(xEAIx+BCA). Denote by BAL the Ben Arous-Ledoux class of all Borel subsets V of the space X such that limi0nf 7e(V) > 0.
Note that if V E BAL, then Al' E BAG for every A > 0. Any Borel set V such that AV C V for all \ > 0 and 'y(V) > 0, belongs to BAG. Put
I(A) =
2
zinf I(x)2,
1(x) = lxl,,, where I(x) = oc if x ¢ H.
Let us introduce the following functionals on B(X) :
r(A) = sup{r > 0: 3 V E BAG, (rU + V) n A = O}, where r(A) = 0 if for no r such a set V exists, and
s(A) = inf {s >- 0: 3 V E BAG, (A e V) n sU is nonempty}, where s(A) = +oo if for no s such a set V exists.
4.9.1. Theorem. For every Borel set A in X, one has limsup e2log 7e(A) <
e-0
-lr(A)2,
(4.9.1)
2
limiofe2log-ye(A) > -2s(A)2.
(4.9.2)
PRooF. Let r > 0 be such that (rU + V) n A is empty for some V E BAG. Then
rye(A) < 1 --y(e-'rU,, +e-1V). Since V E BAG, there is a > 0 such that -y(e-'V) > a for all sufficiently small positive e. Therefore, by virtue of estimate (4.3.3), we get the existence of a number c(a), which depends only on a, such that for all sufficient small positive e one has -y' (A) <
exp(
2e2
+c(a)E
whence estimate (4.9.1) follows at once, since one can /choose for r any number less
than r(A).
Chapter 4. Convexity
196
In order to prove (4.9.2), let us pick a positive number s, for which (AeV )nsU,, is nonempty for some V from ,BAG. Then there is h E sU with V + h C A. Hence, for every e > 0, one has
-((A) > 7`(V + h). According to the Cameron-Martin formula, we get y E(V + h) = exp( Ih12
exp(r sh(x)) ly(dx).
J
c-1V
Since V E BAG, there is a such that y(e-1V) _> a > 0 for any sufficiently small e. By Jensen's inequality, we obtain
f exp(- h(x)) y(dx) > y(e-'V) exp[-`y(e 11V) l
E-It,
E-I4.
Applying the Cauchy inequality, we arrive at
f h(x) y(dx) < Ihl E-1V
Therefore, for all sufficiently small e > 0, there holds the inequality z
y`(A) > aexp(-I2Cz
- IQc )
This yields the estimate
liminfe2logye(A) > -IhIH/2> -s2/2.
E-u Since one can choose for a any number bigger than s(A), estimate (4.9.2) follows.
0
4.9.2. Lemma. The following inequalities hold true: I r(A)2 > I (A)
for every closed set A C X.
1 s(A)2 0. Let 0 < r < I(A). It follows from the definition of I(A) that 2rU fl A = 0. Since /U is compact in X and A is closed, there is an absolutely convex neighborhood of zero V such that ( 2rU + V) n A is still empty. It is clear that V E BAG. Therefore, r(A) > v. This implies the first inequality. Let A be open and I(A) < oc. Then one can pick a vector h E A n H. There exists an absolutely convex open set V such that V + h c A, which means that the set (A e V) n IhIHU is nonempty. Therefore, s(A) < IhIH, whence the second inequality. 0 This lemma implies the following classical large deviations principle.
4.9.3. Corollary. For every Borel set A in X, the following estimates hold true:
-I(A°) < liminfe2logyc(A) < limsupe2logy`(A) < -1(A). E-0 a-0 where A° is the interior of A C X and A is its closure.
4.10.
Complements and problems
197
4.9.4. Example. Let X be a separable Banach space with the norm II ' [[ and let or be the minimal positive number with U,, C {x E X : jjx[O < o} (which is the
norm of the natural embedding H -+ X). Then, for both sets A = {x: jjxjj > 1) and A = {x: lixi[ > 1}, we have I(A) = o-2/2. Therefore, li me21o87
(x:
11xIl ? £ f
2a
Let us mention a result from [4511 related to large deviations. Suppose that -y is a centered Radon Gaussian measure on a locally convex space X with the Cameron-Martin space H and let A and B be two Borel subsets of X. Put
d(A.B)=inf{la-bl,,: aEA,bEB,a-bEH}. if (A - B) n H is nonempty; otherwise d(A, B) := oc. Note that d(A, B) < oo if A and B have positive measures (since then y(A+ H) = y(B + H) = 1).
4.9.5. Theorem. Suppose that y(A) > 0 and y(B) > 0. Then lim sup 4t log f T11H dry < -d(A, B)2. e-.o
See (2341 for related results. The constructions described above can be applied to Laplace approximations. Let us mention a result from [54].
4.9.6. Theorem. Let F E L' (-y). Then, for every Borel subset A in X, one has 1
-cj < lims'upe2log J exp(-E2F(ex)) y(dx)) < -c2, A
where
C2=inf 9ER(
t+1$(An{F
4.10. Complements and problems Gaussian symmetrization on functions In [224] A. Ehrhard introduced also Gaussian symmetrizations for functions. Using the same notation as in Section 4.1, for any Gaussian k-symmetrization S(L,e) in IR" and any Borel function g on IR", we put
S(f)(x) := S(L,e)(f)(x) = inf{a E IR`: 4'((x,e)) < yk((L` +x) n{j < a})}. The next result (see [224] for the proof) describes basic properties of symmetrization on functions.
4.10.1. Theorem. Let f and g be continuous functions on 1R". Then: (i) For all a E IR' one has
{S(f)>a}=S({f >a}), 'yn({S(f)
Chapter 4. Convexity
198
(ii) If f and g are nonnegative and square-integnable with respect to the standard Gaussian measure y" on 1R", then
J (iii)
fgdyn
fS(f)S(g)d)n.
If f on the set {a < f < b} has the modulus of continuity w(f), then S(f) on the set {a < S(f) < b} has the modules of continuity w(S(f)) < w(f). In particular, if f is Lipschitzian with constant C, then S(f) is Lipschitzian
with the same constant. (iv) If f is convex, then S(f) is convex as well. (v) If f is Lipschitzian, then
f IVS(f)fzd?" < f IOfIZdy" The convexity of Gaussian measures is related to the properties of equimeasurable rearrangements of functions. Let y a centered Radon Gaussian measure on a locally convex space X and let f be a y-measurable function. We define a function on IR1 with the standard Gaussian measure It by the relation
f'(t) = inf{.9 E IR': 4+(t) < y({ f < s})}.
4.10.2. Theorem. If the function f is convex, then f' is convex as well. If f satisfies the condition If (x + h) - f (x)l < L{hl,,,,,, dh E H(y) for y-a.e. x, then the function f' is Lipschilzian with constant L. Proofs can be found in [172], [223].
On the correlation inequality For the proofs of certain special cases of the correlation inequality and an additional discussion, see (101], [186], (416], [471], [472], [615], [675], [699], (7001, (701], [7691, [770]. See [6711 for other interesting results relating to the correlation inequality. For example, it is proved there that (4.6.1) is true for centrally symmet-
ric ellipsoids. Also, in the case of the standard Gaussian measure on R", (4.6.1)
is true if A and B are contained in the ball of radius Rn = I'(1 + n/2)'1"/f 2-le-s%:/i, whereas the validity of this inequality for all A and B from the ball of radius yr would imply its validity in general. The following problem remains open as well. Let A C 1R" be an absolutely convex closed set and let S be a symmetric strip with yn(S) = yn(A). Is it true that y, (M) _> y"(AS) for all A > 1? See [458] for some positive results. In relation to the correlation inequality, let us mention the following result.
4.10.3. Theorem. Let -y be a centered Gaussian measure on 1R" with covari-
ance K = (K,,,);'j., and let y be the centered Gaussian measure with covariance K(A) _ (Kij(A))" where K;.,(A) = K,., if i. j > 2 or i = j = 1 and i1_' K1, (A) = K,,I(A) = AK1,3 if j > 2. Then
Gn(ai.....a". K(A)) := yA(x: [x,I < a,, i = 1,... n) is monotonically nondecreasing in A E [0, 1] for any fixed at, ... , a,,.
4.10.
Complements and problems
199
PROOF. Suppose first that K is nondegenerate. Using that K is positive definite, it is readily verified that K(A) is nondegenerate for all A E [0, 1]. Denote by n
n
fn (K(A), - ) the density of ya. Put A = 11 [-a,, ail, B = ji [-a,, ail. Interchangi=2
i=1
ing differentiation in A and integration, using the chain rule and Lemma 1.2.4, we get d
A dAGn(a1,...
,a,,K(A)) a n E Kj,, 8K1.7 fn (K(A), xi, ... , xn) dxl ... dxn
A j=i
n' n 2
02
>2Ki.j axlax; fn (K(A), xl,... , xn) dx1... dxn JJ B 0 = j
n
2 E Klj-fn(K(A),al,x2,... xn)dy2...dyn, xj B j°2
since 2
I
B j=2
xn)
Ki .i axj
fn (K(A), 0, xz,
--
,
dx2 ... dyn = p
by the symmetry of B and fn. The function fn (K(A), a1, x2, ... , xn) can be written as
,(al )fn-1 (Ko(A), x2 - a1 K1.2(A), ... , xn - a1Kl.n (A))
with some positive op(al) and positive definite (n - 1) x (n - 1)-matrix Ko(A). Therefore, it suffices to show that
,x,, -aiKl.n(A))dx 2...dXn
f >2Kl.j'G(al)aa B j=2
is nonnegative. This follows by Theorem 1.8.5, since the foregoing expression coincides with the derivative in t of the function t '-+ µ(B - tk), where µ is the centered Gaussian measure on IRn-1 with density fn-l (Ko(A), )) and k =
(aiKl,2(A),... ,a,K,.n(A)). If K is degenerate, the claim follows by approximat0 ing K by the sequence K + m-1 1. 4.10.4. Corollary. Suppose that K(A) depends on A = (Al,
An) in the
following way: K;,j(A) = Kj,j and K,,j(A) = A;AjKj.j if i # j. Then
Gn(a,,... ,an,K(A)) := yy(x: Ix,I < ai, i = 1,... ,n) is monotonically nondecreasing in every A, E [0,1] for any fixed a1, ... , an. In particular, n
y(x:
(x,I f y(x: lx,I
PROOF. The first claim follows by renaming the coordinates. Now we obtain Sidak's inequality (already proved in Corollary 4.6.2) by letting A, - 0.
0
Chapter 4. Convexity
200
Some small ball estimates and asymptotics Let y be a centered Gaussian measure on a separable Hilbert space X and let K be its covariance operator. Denote by of = = ok > ok+1 > ... the eigenvalues
of K arranged in the nondecreasing order. Let us put Mk = fl (1 - a2
/ai)-t/z
j=k+1
Suppose that o1 > 0. Obviously, the distribution density pa of ra(x) = IIx - a112, a E X, admits an explicit representation. However, in many applications it is difficult to work with this explicit expression, and estimates in terms of elementary functions are preferable. We shall mention here a result from [783] which extends earlier results from [842] and [343]. For additional information, see (166].
4.10.5. Proposition. If k = 1, then g. (t) < M(2)(2otoz)-1 exp(-(t1/2
- Ilall)/(2oi)).
Ifk> 1, then (k/2))-1(t/(2a2))_ 1+k/2exp(-(t1/2
Pa(t) < ?if (k)(2a1i
- IIaH)/(2a2)).
It should be noted that if p is a Borel probability measure on an infinite dimensional separable Hilbert space X with the closed unit ball U such that z has no atom at zero, then µ(U + a) has no nontrivial lower bound independent of a E U, since, according to Problem 4.10.24, one has inf µ(U + a) = 0. aE(
For certain norms q on C(0,1], lower or upper bounds for log P" (q < c) with small s > 0 are known. Typically, log Pu (q < c) is compared with E or . ° (log E) and the corresponding conditions are expressed in terms of the Schauder basis or wavelets decompositions. Let us formulate several results in this direction. Let Id = (0, 1]d (considered with the Euclidean norm) and let l; = (te)1E1d be a centered Gaussian process. We shall assume that f has continuous paths and to = 0. The measure induced by C on the space Co(Id) of continuous paths vanishing at zero (equipped with the sup-norm) is denoted by pf . Let f : (0, oo) (0, oc) be a nondecreasing continuous function. Put
Af(x) =
sup
Ix(t) - X(s)I'
tC1d.e#t
f(It - si)
x E Co(Id).
For example, if f (u) = u°, where a E [0,1], then A f = fl II° is the standard Holder
norm of order a. In the case a = 0, the corresponding norm is equivalent to the sup-norm on Co(ld).
4.10.6. Theorem. Suppose that EI& - EaI2 < r(It -
9()2,
Vt, s E Id,
where r is a nondecreasing function such that r(u)/f (u) < Mu;3 for all u E (0, 11 and some positive constants M and, 3. Then there exists a constant C > 0 such that µf (x E Co(Id): '\1 (X) < 25) > exp(-Ce-d/^3), Ve > 0.
4.10.7. Corollary.
Suppose that
6,I2 < It - sla,
Vt, s E Id,
4.10.
Complements and problems
201
where 0 < 6 < 2. Assume that 0 < a < 6/2. Then there exists a constant C > 0 such that A, (x E Co(Id): IIxIIa < E) >-
exp(-CE-2d/(6-2a)),
Ve > 0.
Let {hk,} be the Haar system, i.e., h00 = 1 and, for k > 0, 1 < 1 < 2k,
if (I - 1)2-k < t < (I - 1/2)2-k, -2k/2 if (I - 1/2)2-k < t < 12-k, 2k/2
hki(t) _
0
otherwise.
It is known that the Haar system is an orthonormal basis in L2[0,1]. Letting e
Pk1(t) =
fhici(u)du, 0
we get the Schauder basis in Co(0,1], i.e., every function x E Co[0,1] is given by the uniformly convergent series x 2i
X(t) = xW'Poo(t) + E L. xkl`Pk1(t) k=o1=1
4.10.8. Theorem. Suppose that d = 1, M > 0. 0 < 6 < 2, 0:5 a < 6/2 and 1E If, - f,I2 < It - sI6.
Vt, s E [0,1].
(4.10.1)
Let q be a ^y-measurable seminorm on C(0.11 such that
x q(x)
2-(1/2-a)k
< 1b1 k=o
sup
Ixk1I,
11x E C[0,1].
1<1<2k
Then there exists a constant C > 0 such that µt (q(x) < E) >
exp(-CE-2/(6-2a)),
de > 0.
Proofs are given in [725]. For related results, see [T24), [691]. In some cases, the estimates given above are sharp. Namely, as shown in (725], if d = 1 and
Elf, - {.I2 = r(It - 81)2. then under certain additional technical conditions on r and q, one has
A' (q(x) < e) <
exp(-CE-2/(6-20)),
VO < E < 1.
For example, the results above yield the following estimates.
4.10.9. Example. Let d = 1. 0 < 6 < 2, a E (0,1/2), and let t be either the fractional Brownian motion of order 6 or fe = n1 - rp, where (>h),F)0,1) is the fractional Ornstein-Uhlenbeck process, i.e. the centered Gaussian process on [0, 1] with the covariance function exp(-It - sI6). Then there exist positive numbers c and c' such that -ce-2/(6-2a) < log 14 (IIxII, < E) < -C E-2/(6-2a)
Chapter 4.
202
4.10.10. Example. Let p > 1, a E (0, 1/2), and let
Convexity
II. be the Holder
II
norm of order a on C[0,1] with the Wiener measure P'r'. Then there exist positive
numbers c i. = 1,... , 4, such that -cle-2 < log Pn
e) < < log P" (IIxII. < e) <
-C3F-2/(1_2c,)
-C2e-2,
-C4e-2/(1-2"
Miscellaneous remarks One can derive from Corollary 2.4.9 that every absolutely convex set of positive ,y-measure contains a ball from H(y). The next. proposition from [7351 enables one to estimate the radius of this ball.
4.10.11. Proposition. Let -y be a centered Radon Gaussian measure on a locally convex space X and Let A be an absolutely convex set of positive y-measure.
Then A contains the open ball from H(y) of radius r. where r is determined from the equation 4'(r) = z(y(A) + 1).
PROOF. Let h E H(y) be a unit vector. Let us take a closed hyperplane Y in X complementary to R' h and put A = A n (y + 1111h), y E Y. The lines y + R' h are equipped with the metrics d(y + th, y + sh) = It - sI. The length IA,I of the convex set. A is not larger than the length of A° for all y E Y (this is easily seen from the convexity of A combined with the fact that the centrally-symmetric sets A, and A_1, have equal lengths). Let -yy be the conditional Gaussian measures on the lines y + R'h. Since A0 is a symmetric interval, by virtue of Anderson's inequality, we get y°(Ao) > yy(Ay) for all y r= Y. Therefore,
24,(')&j) - I = 7"(A.) ? 7(A), which shows that A contains the interval {th: ItI < r}, whence our claim.
0
4.10.12. Remark. Let -1 be a nondegenerate centered Gaussian measure on a separable Banach space X with the unit ball U. As shown in 14831, for every y(U + tx) is strictly decreasing on (0, +oo). In nonzero x E X, the function t addition, it has been proved in [4831 (answering a question raised by W. Linde)
that the equality f+4 x y(dx) = 0 implies that a = 0. U+a
4.10.13. Remark. It is worth mentioning that in [6221 a construction is given of a centered Gaussian measure y on a separable Hilbert space X and a set A with y(A) > 0 such that
y(A fB(x,r)) Urn
y(B(x.r))
= 0 y-a.e.,
where B(x, r) is the closed ball of radius r centered at x. In [6241 there is an example of a function f E L1(y)rsuch that liminf1--y (B(x,s)) -0
'
J
f(y)y(dy): xEX,
0<s
JJ
8(z.0)
On the other hand, the following result is obtained in [626). Let y be the countable product of the standard Gaussian measures on R1 regarded as a measure on the
4.10.
Complements and problems
203
c, where
weighted Hilbert space X of the sequences
with 0 < ck +I /ck < q, q < 1. Then, for every f E L' (ry ), one has ry
(B(x, s)) '
r f (y) ry(dy) -+ f in measure ry, B(z.S)
as r 0. In addition, according to [626], the a.e. convergence takes place as well, provided c++1 < c,i-°, where a > 5/2. Related problems are discussed in [623), [626], [766].
Problems 4.10.14. Show that the function f (x, y) =
-'(4'(y)
- 4'(x)) is concave on the
half-space {x < y} in IR2.
4.10.15. Derive Anderson's inequality from Ehrhard's inequality. 4.10.16. ([486]). Let -y be a Radon Gaussian measure on a locally convex space X, let A be a 'y-measurable convex set of positive measure, and let B be an open convex set
such that -y(A n B) = 0. Show that there exists a continuous linear functional f on X such that inff f (x) > ess sup f (x).
.c- B
zEA
i.e., inf,EB f(x) > f(y) for a.e. y E A. Hint: take the topological support S of the measure ryIA, which is convex by Corollary 4.2.4, and apply the Hahn-Banach separation theorem (see [670. Theorem II.9.1]) to the convex sets B and S.
4.10.17. Derive Corollary 4.4.2 from Theorem 4.4.1, using the properties of concave functions and the function 4'.
4.10.18. Let t. t E T, be a centered Gaussian process such that sup Itt I < oo a.e. tET
Prove that the set ft r, t E T} is relatively compact in L2(P). Hint: otherwise there exist d > 0 and a sequence {,,, such that El{,,, -,112 > d2 for all q from the linear span of tt,. i < n; use Lemma 1.8.8 to show that P(sup I{t I < M) = 0 for every M > 0.
4.10.19. Let 0 < r < p < oc. Show that there is a number K(r.p) with the following property: for any Banach space X and every Radon Gaussian measure ry on X, one has (1IIx1I°7(dx))'/P < K(r,p)(IIIxlL''y(dx))
Hint: use Example 4.5.8 in the case 1 < r < p < oc; in the case 0 < r < p = 1 apply the Cauchy inequality to the function IIxII = iixII 2lixii'-'`2 and use the result for p = 2 - r. 4.10.20. Show that if p is a Radon Gaussian measure on a locally convex space X and f is a µ-measurable function such that, for µ-a.e. x, the function t - f (x + th) is continuous for every h from H(p), then a median M(f) is unique. Hint: see Proposition 6.7.7 below.
4.10.21. Let p be the same measure as in the previous problem and let F be a Borel mapping from X into a normed space Y such that IIF(x + h) - F(x)II,. S IhI,rt,.>,
Vh E H(p), for p-a.e. x. Prove that
E V(p) for a < 1/2. Hint: show
that the function f (x) = IIF(x)II, satisfies the condition of Theorem 4.5.7.
4.10.22. Let p and v be two Gaussian measures on a separable Banach space X such that p(V) = v(V) for every closed ball V in X. Show that p = v. Hint: note that µI = p s (p s v) and vt = v s (p s v) also coincide on all balls; use Problem 3.11.47 to show
Chapter 4. Convexity
204
that H(µ1) = H(vi) as Hilbert spaces and then apply Corollary 4.7.5 and Corollary 4.7.8 to show that µl = vi. (Note: the result holds true for non-Gaussian measures, but the proof is much more difficult, see [6261).
4.10.23. An alternate way of proving Proposition 4.6.7 is this: letting (Se)e>o be the semigroup with generator Lip(x) = Ocp(x)/2 - (Kx,VV(x)), show that the function St f is nondecreasing in every variable; then use the identity /'
f fgdry -
J fd7 f fd'Y=f f (VStf,Vg)d-ydt. 0
Hint: see [36[.
4.10.24. Let µ be a Borel probability measure on an infinite dimensional separable Hilbert space X with the closed unit ball U. Show that inf µ(U + a) = 0, provided µ has no atom at zero. Hint: fix e > 0, take an absolutely convex compact set K and 6 > 0 such that µ(K) > 1 - E and µ(6U) < e, then take v V K with norm less than 62/2; note that there is a closed hyperplane E such that v + E does not intersect K and prove the existence of a unit vector a such that (U + a) n K C 6U.
4.10.25. Let -y be a Radon Gaussian measure and let f be a )-integrable convex function. Show that f f d-y > M(f), where M(f) is the median of f. Hint: consider the one dimensional case and use Theorem 4.10.2; another proof is given in [456].
4.10.26. Let -y be a centered Radon Gaussian measure on a locally convex space X and let q be a -r-measurable seminorm such that -1(q < E) > 0 for any E > 0. Show that there exist a full measure Borel linear subspace Xo C X and Borel linear functionals gn on X0 such that q = sup Ig I a.e. and note that sup Ign I is a Borel seminorm on Xo. Hint: n
n
use Proposition 4.4.3 and Proposition 3.11.1; observe that an = 0 in (4.4.1).
CHAPTER 5
Sobolev Classes over Gaussian Measures It is my conviction that it will be possible to prove these
existence theorems by means of a general principle... provided
also if need be that the notion of a solution shall be suitably extended.
D. Hilbert. Mathematical problems. The general problem of boundary values 5.1.
Integration by parts
In this section, we discuss integration by parts formulae for Gaussian measures. Let us start with the differentiation of functions. The various types of differentiability can be described by the following scheme of the differentiation with respect
to a class of sets M. Let X and Y be two locally convex spaces and let M be a certain class of nonempty subsets of X.
5.1.1. Definition. A mapping F: X -. Y is said to be differentiable with respect to M at the point x if there exists a continuous linear mapping from X to Y. denoted by DF(x). such that, for every fixed set M E M, one has uniformly in h from M:
F(x + th) - F(x)
= DF(x)h.
li o t Taking for M the collection of all finite sets, we get the Gateaux differentiability. If M is the class of all compact subsets, we arrive at the compact differentiability (which, for normed spaces, is called the Hadamard differentiability). Finally, if X, Y are normed spaces and M consists of all bounded sets, then we get the definition of the Frechet differentiability. In the finite dimensional spaces the Hadamard
differentiability is equivalent to that of Frechet and is stronger than the Gateaux differentiability. For locally Lipschitzian mappings between normed spaces, the Gateaux and Hadamard differentiabilities coincide (Problem 5.12.22). In infinite dimensional Banach spaces, the Frechet differentiability is strictly stronger than the Hadamard differentiability. For example, the function 1
f: V[0,11-IR', f(x)= fsiux(t)dt,
(5.1.1)
0
is everywhere Hadamard differentiable, but nowhere Frechet differentiable. The same is true for the mapping F: L2[0,1] -+ L2[0,1], F(x)(t) = sinx(t). (5.1.2) If E is a linear subspace in X equipped with some stronger locally convex topology, then one defines the differentiability along E (in the corresponding sense) at the point x as the differentiability at h = 0 of the mapping h - F(x + h) from 205
Chapter 5. Sobolev Classes
206
E to Y in the corresponding sense. The derivative along E is denoted by D£ F. If E is one dimensional, this gives the usual partial derivative OAF, defined by the formula
8hF(x) = lim
F(x + t h) - F(x)
t-0
t The derivative DE F is a mapping from X to the space C(E, Y) of all continuous
linear mappings from E to Y. Therefore, if we equip C(E, Y) with some locally convex topology, we can consider the second derivative DE2F and so on. Suppose that E and Y are normed spaces and that C(E, Y) is equipped with the operator norm. If the Frechet derivative DEF(x) exists everywhere, then DEF is again a mapping with values in a normed space. Thus, the n-fold Frechet derivative DI F can be defined inductively as DE (DE -1F) (or as D£ -1(D F) ). The mapping D, "F can be regarded as taking values in the normed space C.(E,Y) of all continuous n-linear Y-valued mappings with the norm sup{Il'(hi,... ,hn)Ilr, Ilh;IlE <_ 1}. For example, the space C(E,C(E,Y)) can be identified with the space of all continuous bilinear mappings from E x E to Y. If a mapping F is n-fold Ftechet differentiable along E for all n, then we say that it is infinitely Flechet differentiable along E. The n-th Frechet derivative of F along the whole space is denoted also by FW. In most of the results below connected with differentiability, we shall be concerned with mappings F taking values in a Hilbert space Y and differentiable along a Hilbert space E such that DEF(x) is a Hilbert-Schmidt operator between E
and Y, i.e., DEF(x) E N(E,Y). Since 41l(E,Y) is again a Hilbert space, the derivative turns out to be a Hilbert space valued mapping; consequently, higher derivatives take values in Hilbert spaces. This is in contrast with the general case, where the space C(E,Y) of all bounded operators is not Hilbert, so that one leaves the framework of Hilbert spaces. We shall be especially interested in the case where E is a separable Hilbert space
continuously embedded into X. For example, let X = IR" and let E =12. Denote by 'y the countable product of the standard Gaussian measures on the real line. The x functions F(x) _ E n-2xn and G(x) _ E n-2x2 are defined almost everywhere n=1
n=1
with respect to the measure -y. Let us put F = 0 and G = 0 outside the natural domains of definition of these functions. Then F and C are Ftechet differentiable along E at every point x; for the vectors x from the corresponding natural domains
of definition, the functional DF(x) is represented by the vector v = (n-2) E E, and the functional DEG(x) is represented by the vector u(x) = E E (at other points both derivatives are zero). In either case, y-a.e. there is no Gateaux derivative along the whole space X. In a similar manner, both mappings f and F given b y (5.1.1), (5.1.2) are infinitely Ftechet differentiable along E = W '(0,1) (and along E = C[0,1]). For any h E E, one has DEF(x)(h)(t) = cosx(t)h(t). Indeed,
h(.)) - sin(x( )) -
2 sup Ih(t)I2,
whence the Frechet differentiability along C(0,1]. The reason that there are more functions differentiable along E than along X is that E is smaller and its topology is stronger.
5.1.
Integration by parts
207
Notation. Let X be a locally convex space. Everywhere below the symbol .FC" stands for the collection of all functions f on X of the form
f(x) = ' P (I1(x),... ,l (x)),
V E Cb (IR"),
1, E X', n E N.
Such functions will be called smooth cylindrical. We shall need the following classical integration by parts formula.
5.1.2. Theorem. Let p be an integrable locally absolutely continuous function on the real line and let a function f be either locally absolutely continuous or every-
where differentiable. Suppose that the functions f'p and f#' are integrable. Then the follounng equality is valid:
+x
tx
f fi(t) p(t) dt = - f f(t) d (t) dt.
(5.1.3)
-x
PROOF. It suffices to prove our claim for bounded functions f. In the general case one can take a sequence of continuously differentiable functions On such that supn 1O n1 < oo, Bn(t) = ton [-n, nj, 19n1 < n + 1 and jOn(t)j = n + 1 if jtj > n + 1. Then formula (5.1.3) for fn = On(f) and the Lebesgue theorem imply the validity of this formula for f . In the case of a locally absolutely continuous bounded function f the claim follows from the classical integration by parts formula for closed intervals, since by the integrability of p there exist two sequences of numbers aj -oo and
+oo such that jp(a,)j + jp(b1)j -. 0. For functions f that are everywhere differentiable, the proof is somewhat more tedious, since such functions need not be locally absolutely continuous. However, this proof is brought to the end using the integration by parts formula for absolutely continuous functions and the obvious observation that f is absolutely continuous on every closed interval where p has no zeros, since on such intervals the function f' is integrable. 0 b;
5.1.3. Definition. A Radon (possibly, signed) measure p on a locally convex space X is called differentiable along a vector h E X in the sense of Fomin if there exists a function Oh' E L1(µ) such that, for all smooth cylindrical functions f, the following integration by parts formula holds true:
f Ohf (x) p(dx) x
f
f (x)fh(x) p(dx).
(5.1.4)
x
The function Oh" is called the logarithmic derivative of the measure p along h.
The measure Qh p is denoted by dh p and is called the derivative of p along h. By induction one defines higher derivatives dhp and mixed derivatives dh,
5.1.4. Example. A measure p on the real line is differentiable along 1 precisely when it has a locally absolutely continuous density p with respect to Lebesgue
measure and p' E L'(IR'). In this case ,31 = p'/p.
Chapter 5. Sobolev Classes
208
PROOF. If a density a with the aforementioned properties exists, then the Fomin differentiability follows from the classical integration by parts formula. Con-
versely, if there exists 8, then it is easily verified that the function B(t) =
f3r(s)ii(ds)
serves as a density for µ, whence the existence of an absolutely continuous density follows.
5.1.5. Remark. Originally, Fomin defined the differentiability of µ along h as follows: for every Borel set A, there exists the limit `im µ(A + t h) - µ(A)
(5.1.5)
This definition is equivalent to the one given above and is a special case of the definition introduced earlier by Pitcher [6081. Definition 5.1.3 admits a straightforward generalization to nonconstant vector fields (in particular, for measures on manifolds).
In infinite dimensions, a nonzero measure cannot be differentiable along all vectors. However, Gaussian measures possess large collections of vectors of differentiability.
5.1.6. Proposition. Let y be a Radon Gaussian measure on a locally convex space X. Then H(y) coincides with the collection of all vectors of differentiability. In addition, the measure y is infinitely differentiable along H(-y). If h E H(y), then
,3Z (x) = -h(x).
PROOF. Let f E.FC,c. By the Cameron-Martin formula, one has f (x + t t) - f (x) y(dx) = f (x) r(t, t) - I y(dx),
f
f
where r(t,x) = exp(th(x) -
f
IhEH(,)). The left-hand side of this expression tends
f
z
to ah f dy as t 0, whereas the right-hand side tends to h f dry by the Lebesgue theorem. The infinite differentiability along h follows from the formula above. If h V H(y), then the measures yth and y are mutually singular for all t > 0, whence one readily deduces that y is not differentiable along h (see, e.g., (5.1.5) or Problem 5.12.24).
5.1.7. Corollary. The mapping S: h - yh from the Cameron-Martin space H(y) to the Banach space of all Radon measures on X equipped with the variation norm is real-analytic. In addition, IISIk>(a)II < k'12 whenever IaIH(.) < 1. In particular, all functions A -. y(A + Ah), h E H(y), extend to entire functions on the complex plane.
PROOF. The analyticity follows from the estimate Ild"yll 5 kk12.
IhIHe,, < 1,
which is valid for the total variation of the derivative of any order k along h. * yk, where the Indeed, we may assume that y has zero mean. Then y = yl * identical measures y, coincide with the image of the measure y under the mapping
5.1.
Integration by parts
209
x - k-1'2x. It is easily verified that one has do y = dh y, * - - - * dhyk. It remains to note that Ildh-y,II = /Ildhyll 5 %/k_A11L2(,), whence Ildh vII <- kk/'2llhII 3(,)
Note that Ski h = -8kh = -(k, h) H(.) for any vector k E H(y) if we take a proper linear version of Xh = -h. Hence the measures dh, ... dh,,-y are given by densities with respect to y, which can be easily expressed by means of the Hermite polynomials in k .
5.1.8. Theorem. Let y be a Radon Gaussian measure on a locally convex space X and h E H(-y). Suppose that F: X - Rl is a -y-measurable function such that, for -y-a. e. x, the function t i-- F(x + th) is locally absolutely continuous. If the functions 8hF and Fh are integrable with respect to y, then the following formula holds true: JOhF(x) -y(dx) = X
Jx
F(x)h(x) y(dx).
(5.1.6)
PROOF. Let Y be a closed hyperplane complementing the line R' h (if h = 0, then the claim is trivial). Formula (5.1.6) follows from (5.1.3) and the theorem on the Gaussian conditional measures yv on the real lines y + IRih, y E Y, since the restriction of h = -ah to the line y + IR' h coincides with -Oh'.
5.1.9. Corollary. Let y be a Radon Gaussian measure on a locally convex E is a measurable mapping with space X and h E H(-y). Suppose that F: X values in a separable Banach space E such that, for y-a.e. x, the mapping t ,-+ F(x + th) is absolutely continuous and almost everywhere differentiable. Assume, in addition, that the mappings 8hF and Fh are integrable with respect to y. Then one has
f 8 h F(x) y(dx) = I F(x)h(x) y(dx). X
(5.1.7)
X
PROOF. Since both sides make sense, it suffices to prove that we get the equality
when applying to them any continuous linear functional. The latter follows from the theorem above. 5.1.10. Corollary. Suppose that the conditions in Theorem 5.1.8 are satisfied and that F E LI (y). Then the measure µ := F y is differentiable along h and one has
-h + 8hF/F. PROOF. It suffices to note that 8h(VF) = 8hW - F+p8hF for any V E.FC.
5.1.11. Remark. A Banach space E is said to be a space with the RadonE is differenNikodym property if every absolutely continuous function f : [0, 1] tiable almost everywhere. This property is known to be equivalent to many other properties such as the almost everywhere differentiability of all Lipschitzian functions with values in E (see [195], [598]). All reflexive Banach spaces (e.g., all
Hilbert spaces) and all separable dual spaces possess the Radon-Nikodym property. If E is a space with the Radon-Nikodym property, then the existence of 8hF almost everywhere follows from the absolute continuity on almost all lines x + IR' h. In particular, in this case, if F is Lipschitzian along h with a constant independent of x, one has equality (5.1.7).
Chapter 5. Sobolev Classes
210
Let us now prove a special integration by parts formula which will be useful later. In the proof we use the following lemma which is of independent interest.
5.1.12. Lemma. Let y be a centered Radon Gaussian measure on a locally convex space X and h E H(y). Suppose that a function f E L2(y) is _absolutely continuous on the lines x + 1Rlh for almost all x. If 8h f E L2(y), then f h E L2(y) and one has Jf(x)2h(x)2.y(da)
5 8 f
[4ghf(x)2 + f(x)2lhl2(,)] y(dx).
x
X
PROOF. It suffices to prove the claim in the case, where IhIH(,) = 1. Put el = h and complement el to an orthonormal basis {en} in H(y). The mapping J: x -. X -- R°`, takes y to the product µ of the standard Gaussian n=1
measures on the real line. Since x = En°_1 F;(x)e1 a.e. according to Theorem 3.5.1, the claim reduces to the case where y = p and h = et = (1, 0.... ). Now, by Fubini's
theorem, the whole thing reduces to the one dimensional case, where h(t) = t. Since both sides of the inequality to be proved are homogeneous, it suffices to prove it assuming that. 11 f h-(,, ) = I. One has t2 f (t)2 < 8f (t)2 log If(t) I whenever t2 < Slog I f (t)I and f(t)2t2 < t2et2/4 whenever t2 > Slog If(t) 1. By virtue of the logarithmic Sobolev inequality, we get
+x Jt2f(t)2p(t,O, 1)dt< f[8fP(t)2 + t2e2/4] p(t,O, 1)dt, +oo
-x
-x
whence our claim, since / t2et /4p(t, 0,1) dt = 2f < 8.
0
5.1.13. Theorem. Let y be a centered Radon Gaussian measure on a locally convex space X, let h, k E H(y), and let f and g be two functions from absolutely continuous on the lines x + Rlh and x + R1k for a.e. X. Suppose that ahf, ahg, akf, 8k9 E L2(y). Then hf, kg E L2(y) and the following equality is L2(-y),
valid:
f [f(x)h(x)
- ahf(x)} [9(x)k(x) - 0k9(x)] n(dx)
X
_ (h, k),H,,> f f (x) g(x) y(dx) + f akf(x) ahg(x) y(dx). X
(5.1.8)
x
PROOF. By virtue of condition and Lemma 5.1.12, both parts of equality (5.1.8) make sense. Similarly to the previous proof, the claim reduces to the two dimensio-
nal case (for el we take the vector ch if h 96 0, then find a vector e2 1 el in the plane containing h and k, and complement these two vectors to a basis in H(y)). Thus, we may assume that y is the standard Gaussian measure 12 on 1112- If h and k are independent, then it follows by the condition that f and g are elements of the Sobolev space W2.1(12), and 8hf, 8kf, 8r,9, 8kg coincide with the Sobolev derivatives. Then (5.1.8) is valid for all h, k E R2, provided that the corresponding partial derivatives are understood in the Sobolev sense. Therefore, it suffices to
5.2.
The Sobolev classes 1V P-' and DP-"
211
prove (5.1.8) for the basis vectors h = el and k = e2. If f, g E Co (1R2), then (5.1.8) follows from the usual integration by parts formula for the standard Gaussian density p on 1R2 taking into account the equality 81, 8129 = 8=20"19. In the general case we find two sequences {f) } and {g.,} of compactly supported smooth functions convergent in %,2.1 (12) to f and g, respectively. Using Lemma 5.1.12, we get (5.1.8)
by passing to the limit in the equalities for f, and gj. If the vectors h and k are linearly dependent, then we arrive at the one dimensional case, which is considered in a similar manner. Note that if 8kg is absolutely continuous along h. 8hg is absolutely continuous along k and OhOkg = Sk0hg E L2(-Y), then (5.1.8) follows by the integration by parts formula. Indeed, the left-hand side can be written as
I [8hf9k+ f8hgk+ fgOhk-8hfO9 k'- fMhOk9-8hfgk+8hfOk9]d7 = f [fehgk+fgahk-fahak9] dy = J [akfah9+fak.9h9+f9ahk-f,9hOk9J d7, which is (5.1.8), since 8hk = (h, k),, for a proper linear version of k.
5.2. The Sobolev classes IVP,' and DP,' Let y be a centered Radon Gaussian measure on a locally convex space X and let H = H(y). There are three different ways of defining Sobolev spaces over the Gaussian space (X, -y): as certain completions, by means of generalized derivatives,
and with the aid of integral representations. In this and the next sections, the necessary definitions are introduced: the relations between them are discussed in the subsequent sections. Recall that the space of all Hilbert-Schmidt operators (see Appendix) between Hilbert spaces H and E equipped with the Hilbert-Schmidt norm is denoted by ?{(H, E). The symbol 9ik stands for the Hilbert space of all k-linear Hilbert-Schmidt forms on a Hilbert space H (see Appendix). This space is naturally isomorphic to the space 7 {(H.fk_1). Classes H
For all p > 1 and r(J[ N. the Sobolev norm II E
lltt a.- is defined by the formula
r
Ilflltt o =
p/2
F,
(0, ...a;kf(x))2]
k=0 X i,.... dk>1
Y(dx))
,
(5.2.1)
where 8, stands for the partial derivative along the vector e, from an arbitrary orthonormal basis (e,) in H. Denote by the completion of the linear space FC" with respect to the norm II Ilµo.r. Note that the same norm can be written as IIDHfllLP( ,1ik)k=O
In a similar way one defines the Sobolev spaces WP"r(y, E) of mappings with values in a Hilbert space E. The corresponding norms are denoted by the same symbol 11
Note that if two sequences (f,) and {g.} from FC" are fundamental in the norm 11
Ills-p.,, where r E IN, p > 1, and converge in LP(y) to f, then the sequences and have equal limits (denoted by D f) in LP(y, fk), k < r.
Chapter 5.
212
Sobolev Classes
Indeed, suppose first that p > 1. For any fixed h E H and any I E X', we get by the integration by parts formula
f euahf,d,=-il(h) f&ui, dy + f e1if,hdy, which tends to
-il(h) f eilfdy+ f e"fhd7. The same is true for g,. Since I was arbitrary, this means that 8h f, and 8hg, have equal limits in LP(-r), whence the equality of the limits of DJ, and DHg,. For higher derivatives and vector-valued mappings the reasoning is similar. In the case p = 1 derivatives are also well-defined, but the reasoning above should be modified. Namely, by the same argument for every p E C6 (IR1), we get a common limit of D. (w o f)) and D ('p o gj), whence the claim. This observation enables one to define the derivatives D f (called the Sobolev derivatives) for all f E WP"(-y) (and similarly for f E WP''(-y, E), where E is a separable Hilbert space). Finally, let us put
H`(7) =
n
Wp'r(- ),
W"(7, E) = n
p>1 rEIN
E)
p>1, rEIN
5.2.1. Remark. (i) The classes WP-"(-y) are stable under the compositions with C6-functions; if n = 1, they are stable under the compositions with Lipschitzian functions. Indeed, if a sequence of smooth cylindrical functions fj converges to f in the space WP-' (-r) and P E Cb (1R1), then the sequence p o fj converges to o f in LP (-y) and is fundamental in WP-' (y), which is easily seen by the chain rule.
(ii) Note that, by the density of X' in X;, the classes WPr(y) do not alter if we replace X' by X. in the definition of YC".
5.2.2. Remark. Suppose that f E W'-'(-y). Let {e"} be an orthonormal basis in H and 8, := 8e,. Then by the same reasoning as in Remark 1.3.5 one has k
Jk(f) _
1)
-- f 8i ...8,kkf(x)7(dx)
Characterization via partial derivatives Let E be a Hilbert space.
5.2.3. Definition. Let F: X
E be a -y-measurable mapping.
(i) F is said to be nay absolutely continuous if for every h E H, there exists a mapping Fh : X -+ E such that F = Fh -y-a. e. and, for every x E X, the mapping t - Fh(x + eh) is absolutely continuous. (ii) F is called stochastically Gateaux differentiable if there exists a measurable mapping DH F from X to 7{(H, E) such that for any h E H, the expression
F(x + th) - F(x) - D. F(x)(h) t tends to zero in measure y as t 0. The mapping DHF is called the stochastic derivative of F.
5.2.
The Sobolev classes ll'r and Dp"
213
The n-fold derivative D, "F is defined inductively.
5.2.4. Definition. Let 1 < p < x. Denote by DP-' (-y, E) the space of all mappings f E L"(?, E) such that f is ray absolutely continuous, stochastically with Gdteaux differentiable and DHf E LP (y,N(H.E)). Let us equip the norm IIf
IIfIILP(-..E) +
Then for n = 2, 3.... we define the spaces DP-" (,y. E) inductively by the following formula: Dp."(y E)
=ffE
DP."-'(y. E): D. f E
Dp."-1 (y,n(H, E))
The corresponding norms are given by the equalities IIJ
DP " = If1 Lr[,.E + ,IIDkHfIILP(,.xk(H.E)) k=l
Let us put DP."(-y): = D' ' (y, IR') and
D"(y) = n
n
Dp.r(y)
p>l.
Dp.r(y,E)
p>1. rElhl
5.2.5. Remark. Let f E Dp"(ry) and 0 E C, (IR') (or, more generally, cp E Cb (IR' )). Then it is readily seen from the definition that ;p of E DP-"(y). If n = 1, for any Lipschitzian function ip. In addition, DH(,po f) = then ;po f E
If f E D" ;(I) and E Cx (lR') is such that the derivatives of , have at most polynomial growth at infinity, then cpof E D' (-y). yp'(f)DH f .
5.2.6. Lemma. The spaces
(ry. E). II '
11D'') are complete.
PROOF. To simplify notation we only consider the space D-'(-r). Suppose that t f j) is a Cauchy sequence with respect to the norm 11 11D) , . Passing to a subsequence we may assume that it converges almost everywhere to some function f E L' (y) and that the sequence DH f j converges to some mapping G E L' (y. H). Moreover, we may assume that I1f,-1 - f,IID= <_ 2-'. Letting
S,(x) = If,.l(x) - fl(x)I + IDHfi+1(x) -
DHf,(x)IH,
we get the integrable series S(x) _ E Sj(x). Let h E H be a unit vector and let J=1
Y be a closed hyperplane such that X = Y j) lR'h. We shall deal with versions of f j that are locally absolutely continuous on the lines x + 1R' h. By Theorem 3.10.2, there exist Gaussian conditional measures y", y E Y, on the lines y+$3'h such that they have densities with respect to the natural Lebesgue measures. Denote by v the image of y under the projection to Y. For v-a.e. y E Y, the series S is y`-integrable and fj(y + th) - f (y + th) for a.e. t. For every such y and any interval [a, b], the sequence of functions t - f, (y + th) is fundamental in the Sobolev class 11"1 ' (a. bJ, hence it converges pointwise to a locally absolutely continuous limit, which yields the desired version of f . Clearly, ah f = (G, h),, a.e.
Chapter 5.
214
Sobolev Classes
Now we shall discuss the generalized derivatives defined by means of the integration by parts. This will enable us to introduce more general Sobolev classes E), where E is a separable Banacb space.
5.2.7. Definition. Let p > 1. (i) We shall say that a function f E LP(y) has the generalized partial derivative
g E L'(-y) along a vector h E H if, for every
f
from .FC', one has the
equality
8hr(x) f (x) ti(dx)
f vp(x) 9(x) y(dx) + J ip(x) f(x) h(x) y(dx). X
(5.2.2)
X
Put 8hf := g. (ii) Let Z be a normed space such that there exists a countable set L of continuous
linear functionals separating the points in Z. We shall say that a mapping F: X -. Z has the generalized partial derivative 8hF along h if for each I E L. the function (1, F) is in LP(y) and has the generalized partial derivative equal (1, (9h F).
Note that the definition above can be extended to functions f such that fh E L' (y) for all h E H (which is the case, e.g., if f log If I is integrable). The following condition is sufficient for the existence of the generalized partial
derivative8hf of a function f: fhEL'(y)and the sequence converges in L 1(-) .
Let now E be a separable Banach space. A mapping ' : X -- C(H, E) will be called measurable if, for every h E H, the mapping x -+ 'l'(x)(h), X - E, is measurable in the usual sense (i.e., when E is equipped with the Borel or-field). Similarly, a mapping 4 0 f r o m X to the space Ck(H, E) of k-linear continuous map-
pings from H to E is said to be measurable if, f o r every collection h;, i = 1, .... k, E, is measurable. Note that, for such a the mapping x " 'I'(x)(h1, ... , hk), X mapping, we get the measurability of the function II*PIIc., where
IIAIIc,, := sup{IIA(ai,... ,ae)IIE, a, E H, Ia,l < I} Denote by LP(-. Ck) the class of all mappings 41: X -. Ck(H. E) measurable in the sense explained above and satisfying the condition II'I'Ilc,(H.E) E LP(-y).
5.2.8. Remark. (i) The separability of E in the previous discussion can be replaced by the weaker condition (mentioned in Definition 5.2.7(ii)) of the existence of a countable family L of continuous linear funetionals separating the points in E.
Then the measurability of a mapping F: X -- E should be understood as the measurability of all functions 1 o F, I E L, and the concept of measurability of Ck(H, E)-valued mappings is changed accordingly. (ii) The reason why we employ such a form of measurability is that the space C(H) of bounded operators on an infinite dimensional separable Hilbert space H is nonseparable with the operator norm, and the usual concept of measurability for C(H)-valued mappings is not convenient. On the other hand, the space C(H) has a countable collection of continuous linear functionals separating the points (e.g.,
A --. (e Ae, ), where {ej} is an orthonormal basis in H). By the aid of the concept of generalized derivatives one defines the Sobolev classes GP," (1, E).
5.3.
The Sobolev classes HP-'
215
5.2.9. Definition. Let GP,1(y, E) be the class of all mappings F E LP(-y, E), for which there exists a mapping W E LP(y, C(H, E)) such that, for every h E H, the mapping %P(-)(h) is the generalized derivative of F along h. Put T. We shall say that D, ,F is the generalized derivative of F. By analogy, one defines the classes GP-"(y, E) of n-fold differentiable mappings for n > 1; in this case the generalized derivatives DNF, k = 1, ... , n, take values in the spaces LP(-y, CO- If we are in the situation described in item (i) of the previous remark, then the classes GP-" (-y, E) are defined inductively. The natural norm in the class GP."(-y, E) is defined by n
IIfIlca." =E k=0
5.2.10. Remark. We observe that by the symmetry of the usual derivatives of smooth functions, the mappings D, f , k _> 2, take values in the space of symmetric k-linear forms (or symmetric operators). In other words, Oh, ah,, f does not
depend (as a measurable mapping) on the permutations of hl,... , hk, h; E H. p > 1.
It will be shown below that
5.3. The Sobolev classes HP,' In this section, y is a centered Radon Gaussian measure on a locally convex space X and H = H(y) is its Cameron-Martin space. Recall (see Chapter 2) that the Ornstein-Uhlenbeck semigroup (Tt)t>o is defined by the formula
Ttf (x) = f f
(e-tx + V'117- -e-2t y) y(dy).
x Let L be the generator of the Ornstein-Uhlenbeck semigroup (called also the Ornstein- Uhlenbeck operator) on L2(y) (see Appendix and Chapter 2). The proof of Proposition 1.4.5 did not use the finite dimensionality of the space, hence it remains valid in the general case.
5.3.1. Proposition. The operator L has domain of definition
D(L) _
{i:
rk2111k(f)Ili2(,) < 00 k=1
on which it is given by Lf = -
kIk(f) k=1
Let r > 0. Put
JtnI2letTtf Vrf r(r/2)dt, f E LP(-y), 0
where
r(a)
Jt1et dt. 0
216
Chapter 5.
Sobolev Classes
By the same formula we define V, on LP(-y, E), where E is any separable Hilbert (or Banach) space.
5.3.2. Lemma. For any p > 1, the mapping V, is a bounded linear operator on LP(-y) with norm 1. The same is true for LP(y, E), where E is a separable Banach space.
0
PROOF. It suffices to apply the estimate IITtfIILP(,) 5 (If11Lah)-
Let f = E 1n (f) be the Wiener chaos decomposition. Then n=0
0C
3C
Vr(f) = I-(r/2)-1 7 Jt2
etet dtIn(f) = F(n+ 1)_r/2In(f), 'c
n=00
n=0
whence
Vr = (I - L)-r/2
(5.3.1)
Note that (5.3.1) can be shown in a different way. Namely, for every nonpositive
self-adjoint operator L on a separable Hilbert space E generating the semigroup (S,)t>o one has xftQ_e_tS
(I - L)-° = f(Q)tdt, a > 0.
(5.3.2)
0
Indeed, by the spectral theorem, it suffices to prove (5.3.2) for the operator L of multiplication by a nonpositive measurable function ' on the space L2(m), where m is a probability measure on some measurable space. Then St is the multiplication operator by ett' and (I - L)-° is the multiplication by (1- >')-°. Hence the claim follows from the equality
(1 - y)-Q =
T(a)
f to-le-let" dt,
valid for every y < 0. Applying this representation to the Ornstein-Uhlenbeck semigroup on L2(y), we arrive at (5.3.1). 5.3.3. Corollary. For every p > 1, (V,)r>o is a strongly continuous semigroup of contractions on LP(y).
PROOF. If f is a bounded measurable function, then V, f = (I - L)-r12 f is (I-L)-r/2-'/2 again a bounded function. Since (I -L)-r/2(I -L)-'/2 = on L2(y), the semigroup property follows on every LP(-y). It suffices to verify the continuity of the mapping r i--. Vr f for every bounded measurable f . Since I Vrf 15 sup If I, the LP-continuity follows from the L2-continuity. The strong continuity of (Vr),>0
on L2(y) is clear, since V, = (I - L) )-rJ2 and the operator (I - L)-1 is unitary equivalent to the multiplication b y ( 1 + z )- 1 for some nonnegative function ty on a suitable space L2(m). 0
The operators V,. can be included in a two-parametric family as follows. For any a > 0, put V(a) f =
1
(r/2)
fr/2e_tTtfdt. 0
5.3.
The Sobolev classes HP`
217
The same arguments as above show that ar/2V (°) is a contraction on L"(-y) and lira ar/2V(°) f = f in LP(y) for every f E L"(y). In addition, a-+x V r(") = (aI - L) -r/2 on L2(y) Note the following identity:
V(°) _ (I + (3 - a)V(°))
where (1+(3-a)V2°)
r/2
r/2
13 - a1 < a,
V(3),
(5.3.3)
is defined by means of the series > cn((3-a)V°)n n=o
corresponding to the expansion (1 + x)r12 = X£ cnxn (the corresponding operator n=0
series converges in norm by the estimate 113 - al < a). It suffices to verify identity (5.3.3) on bounded measurable functions, hence on L2(y), where it is obvious by the spectral theorem, since -L is unitary equivalent to the multiplication in some L2(m) by a nonnegative function 1G and V(°) is the multiplication by Note that the mappings Vr are injective on L"(y), p > 1. In the case p > 1 this follows from the symmetry of TT on L2(y), which implies that the dual operator to Vr on L1(y) is Vr on L9(-y), where 1/q + i/p = 1. In addition, the image of Vr is dense in LQ(y), since it contains the spaces Xk. This implies that Ker Vr = 0. (a+V,)-r/2.
For any p > 1, the same follows from (5.3.3) with 3 = 1 and a - oo taking into account that ar/2V() f -+ f in L'(-y) as a - oo. Therefore, the space Hp.r(y)
Vr(L'(y)),
IIfIIHP.- = IIV,'fIILD(,,),
is complete. Put
H" (y) = n Hl`(y), H(y) = n H°'r(- ). p>l.r>l
The norm II 11 HP` is often denoted by II In a similar manner one defines the classes
r>1
Ilp,r
E), H` (,y, E),
(y, E),
where E is any separable Hilbert space. It follows from (5.3.1) that II(1- L)r/2fIIL2(. ),
5.3.4. Example. Let f E Xn. Then f E
fE
H2.r(y)
for all p > 1 and r > 0 and
IIfIIHv. = (n + 1)r/2IIfjIL,,(,)
PROOF. Note that Vr(f) = (n + 1)-r/2 f.
5.3.5. Lemma. One has HP2.r(y) C HPI.r(y) if p2 > pi and Hp-r2(y) C Hp-r, (y) if r2 > r1. PROOF. The first claim is obvious. The second one follows from the equality Vr, = Vr,Vr,_r, and the continuity of the operator Vr,_r, on Lp(y).
It will be shown below that for any p > 1 and n E IN, hence the classes HP,n(y) are stable under the compositions with Cb -functions (if n = 1, they are stable under the compositions with Lipschitzian functions).
Chapter 5. Sobolev Classes
218
5.4. Properties of Sobolev classes and examples
An important feature of the Sobolev classes over Gaussian measures is their invariance with respect to the measurable linear isomorphisms. For this reason, completely different by their geometric properties infinite dimensional spaces possess identical collections of functions smooth in the Sobolev sense. Recall that the situation is different for functions differentiable in the FYrcchet sense. On many Banach spaces there are no nontrivial Frechet differentiable functions with bounded
supports (hence there are no nonzero FY6chet differentiable functions tending to zero at infinity). To this class belong such spaces as C[0,11 and 1'. It is known that, for a separable Banach space X, the existence of a nontrivial Frechet differentiable function f with bounded support is equivalent to the separability of X' (see [194, Ch. II, Theorem 5.3]). If such a function with the locally Lipschitzian derivative exists on both X and X', then X is linearly homeomorphic to a Hilbert space (see [194, Ch. V, Corollary 3.6)). Finally, the situation is absolutely hopeless with the continuous functions having compact supports: such functions may be nontrivial only on finite dimensional locally convex spaces. In contrast to what has been said, on general spaces there is a lot of functions from Sobolev classes, in particular, there exist nontrivial Sobolev class functions with compact supports. Throughout this section -y is a centered Radon Gaussian measure on a locally convex space X.
5.4.1. Proposition.
(i) Any function f E has a version fo with the following property: given linearly independent vectors hl,... , hn E H, for -y-a.e. x, the function (t1, ... , t) -+ fo(x + t1h1 + is in + Bi« (R") and is locally absolutely continuous in eachtj fora.e. (tl,... ,tn). The same is true if f E WP'r(y), where E is a separable Hilbert space. (ii) Let f E WJO (y) and let (e., } be an orthonormal basis in H. Then f has a modification fo such that (t1, ... , fo(x + t1e1 + is in + W'X and is infinitely differentiable for all x E X and n E I. PROOF. Let us take a sequence of smooth cylindrical functions f, such that
fi -. f a.e. and If - fi IIty, < 2-?. Letting r
,
SJ(x)=F
k=0
one gets the integrable series E S; (x). Let us put fo(x) _ slim f, (x) for all x J=1
where this limit exists and zero at all other points. We shall show that fo is a version of f with the desired properties. Let X = E ® Y, where E is the linear span of hl.... , h and Y is a closed linear subspace. By Corollary 3.10.3, there exist Gaussian conditional measures yy, y E Y, on the affine subspaces y + E such that yy has a Gaussian density with respect to the natural Lebesgue measure
on E + y. Let v be the image of y under the projection to Y. Then, for v-a.e. x y E Y, the series E SJ(x) is integrable with respect to y9. For every such y, we 3=1
have 11im IISJIIV(,,,) = 0, and the function (t1,...
fo(y+t1h, +-
with respect to the nondegenerate Gaussian belongs to the Sobolev class measure yy on lR" identified with y + E, whence the first claim in (i). The second claim follows by a similar reasoning applied to the one dimensional conditional
5.4.
Properties of Sobolev classes and examples
219
measures, taking into account that if a sequence of smooth functions on the line converges almost everywhere and is fundamental in 14'l J(yi), then it converges pointwise to a function that is locally absolutely continuous. The vector case is analogous. In (ii) we may assume that -y is the countable product of the standard Gaussian
measures on 1W and that {e"} is the standard basis in 12. Denote by H. the linear span of e1,... ,e,,. Let ff be a sequence of smooth cylindrical functions convergent to f in every space 14"(-1). Denote by 11 the set of all points x such that f (x) = lim f j (x) and .r-ac
liminf J IID fj(x+u)IIh y"(du) < oo,
b'n, r E N.
R^
By condition and Fatou's theorem, we have y(1l) = 1. There is a full measure subset Do c S such that for every x E Sto. the set (Cl(, - x) n H. is dense in H. for every n. By the Sobolev embedding theorem, for every x E 11o, the functions h .- h (x + h) on H" converge to f uniformly on every ball, and the same is true for all their derivatives along H", whence our claim.
E). where E is a separable Banach 5.4.2. Proposition. Let F E space. and let h E H be a nonzero vector. Then F has a modification which is locally absolutely continuous on the lines x + IR' h and the partial derivative of this modification along h exists -y-a. e. and coincides with Bi,F (the generalized partial derivative) a.e.
PROOF. Let Y be a closed hyperplane complementing IR' h. By the aid of the
conditional measures on the lines y + R' h, y E Y, it is easily verified that the mapping
Fo(x) = F(y) + J dhF(y + sh) ds,
x = y + th, y E Y,
(5.4.1)
0
is a modification of F with the desired properties. In particular. the existence of ahFo a.e. and the equality ahF() = 8hF a.e. follow by [214, Theorem 111.12.81.
5.4.3. Corollary. Let f E GP,"(y) and :p E Cb (1Rl) (or, more generally, If n. = 1. then ,oo f E GP-'(y) for any Lipschitzian function sp. In addition, D (y,o f) = p'(f)D f . So E Cs (IR')). Then cp of E
PROOF. We shall consider only the case n = 1. because the general claim is deduced from this case by induction using the reasoning below. According to Proposition 5.4.2, for every h E H, the function f has a locally absolutely continuous modification. For this modification, the function y,o f is also locally absolutely continuous along h and has almost everywhere the partial derivative p'(f )ah f , where ah f is the generalized partial derivative of f. The integration by parts formula implies that p'(f )ah f is the generalized partial derivative for o(f). Since V is Lipschitzian, the mapping y'(f)D f can be taken for D., (<po f). 5.4.4. Lemma. Suppose that f,, E WP`(y, E), where p > 1 and E is a separable Hilbert space. If f" -» f a.e. and supllf"IIW","(i,E) < oc, then f E The same is true for the class HP-'(-y, IE).
Chapter 5. Sobolev Classes
220
PROOF. Using that LP, p > 1, is uniformly convex (see (195. Ch. 3[)1 it is readily verified that the space LP(7, E) is uniformly convex as well. Hence LP(y, E) has the Banach-Saks property (see [195, p. 78, Theorem 11). i.e., every bounded sequence {gyp,} in this space has a subsequence {y;,^} such that the sequence n `(vi, + + p;") converges in Lt'(y, E). This property applied to the sequences { D,k fn }. k = 0,1,... , r, yields the claim. In the case of (ry. E) we write f" = Vg, and apply the Banach-Saks property to the sequence {gn} that is bounded in LP (1, E).
El
5.4.5. Proposition. Let {en } be an orthonormal basis in H and let EA" be the conditional expectation with respect to the 'y-field A. generated by el,... ,e". If f belongs to one of the classes WP-'(,y). p _> 1. or GP`(- r), p > 1, then EA" f does also. and (EA" f } converges to f with respect to the corresponding Sobolev norm. If r > 2, then LEA" f = E'4" L f . Finally. one has (5.4.2)
The same is true for mappings with values in a separable Hilbert space.
PROOF. We shall deal with the version of EA" f given by (3.5.5). Clearly, (5.4.2) is true for smooth cylindrical f. Hence it extends to f E GP-'(-t), by using
that for any function r, that depends on fl,... J,, E X.. one has BhlEA"cp = BphEA"y7, where Ph is the projection of h to the linear span of R, f,, i < n. Then Corollary 3.5.2 yields the claim for GP-'(-y), p > 1. In a similar manner it is proved for IVP-'(1), p > 1. It remains to note that T1EA^ f = EA^Tt f,
b' f E LP('.).
(5.4.3)
which is equivalent to the equality
JJf(e'Pnx +
1- a-2tPnz + S"y) 7(dy) y(dz)
XX
_ f f f (e-'P"x + e`'Snz +
1 - e-2ty) ry(dy) y(dz).
XX
where Px = el(x)el +
+ e;,(x)e." and Sn = I - P". The foregoing equality holds true, since the images of ' g y under the mappings 1 - e-t Pnz + Sny and 1 - e- ty + e-'S,,z coincide. Indeed, both images are centered Gaussian x measures, and an easy calculation shows that. the variance of any f = E CA, is j=1
x
n
j=1
J=1
E c - e-2' E cl with respect to both measures. By (5.4.3) we get LEA^ f =
EA^Lf (if f E
and V,.EA" f = EA"Vf, V f E LP(y), which yields the claim for HP-'(-r). The vector case is analogous.
5.4.6. Proposition. Let E be a separable Hilbert space. Then: C GP-'(ry E) and 4'"(,E) C for any p E (1.0o) and n E N: (ii) The derivatives in the sense of DP-"(-f, E) (respectively. IS E)) serve (i)
as the generalized derivatives in the sense of GP-" (-f. E); (iii) DP-" (y. E) = 6VP-"(y, E) for all p E 11, oo) and n E IN.
5.4.
Properties of Sobolev classes and examples
221
PROOF. The inclusion DP-"(-r, E) C GP-"(-y, E), which makes sense if E is Hilbert. follows from the integration by parts formula (5.1.6). In addition, for any f E DP."(-y, E) and h E H, the derivative Oh f in the sense of DP,' (-y, E) serves as the generalized derivative in the sense of GP."(-y, E). If f E WP"' (y, E), then there is a sequence of smooth cylindrical mappings fn convergent to f in WP-' (-y, E). This enables one to pass to the limit in the equality
f
ahV fn d, = - J Vahf" dy + f _hf,, dy
for smooth cylindrical functions V. Hence f E GP,' (y, E) and ah f is its generalized derivative. By induction, we get the claim for all n E IN. Note we can since use an analogue of (5.4.1) to define versions of D -'F for F E D,,"-'F takes values in the separable space of Hilbert-Schmidt mappings, hence the corresponding integral exists in the usual sense (as a Bochner integral). The (y, E) follows by Proposition 5.4.1. By induction, one inclusion (y. E) C gets E) C DP-"(-y, E) for any n E IN. Let us show the inverse inclusion. To this end, note that every F E D"' (y, E) is approximated by {P"F}, where P. is the orthogonal projection to the linear span of the first n elements of any fixed orthonormal basis in E. Hence the verification reduces to the case where E = 1R1. If p > 1, then F E GP-'(-y). By Proposition 5.4.5. the functions EA,F converge to F in GP-'(-y). Therefore, our claim reduces to the finite dimensional case, in which it follows by Proposition 1.5.2. If p = 1, then we may assume that F is bounded, for it is approximated by gyp, o F, where {ip, } is a sequence of smooth functions with uniformly bounded derivatives of any order such that Vj(t) = t if t E [-j, j[ and cps (t) = j sgn t if t % [-j - 1, j + 11. By the same reasoning as in Proposition 5.4.5, it is proved that the functions E'1^F have generalized derivatives D, ,4'E ' F, k = 0, ... , r, that converge to the respective derivatives of F in LP(y, l'lk). Therefore, our claim reduces again to the finite dimensional case.
5.4.7. Corollary. Let E be a separable Hilbert space. Then
E) _ {F E GP-"(y, E): In particular,
E LP(y,Nk), k S n
(y) = GP'(y) for any p > 1.
The same reasoning as in Example 2.9.3 proves the following useful property of (T,),>o.
5.4.8. Proposition. Let E be a separable Hilbert space and let f E LP(-y, E), where p > 1. Then, for any t > 0, one has: (i) For a.e. .r, the mapping h T,f(x + h). H -. E. is infinitely Frechet differentiable. In additions. for every h E H and y-a.e. x, one has ahTtf (x) =
e
e- 2t
J
f (e-'x +
1 --e-21 y)h(y) y(dy).
(5.4.4)
X
(ii) If p > 2, the derivatives f) are Hilbert-Schmidt mappings, and, moreover, the mappings h '-» f)(x + h). H -- ?ir(H, E), are continuous with respect to the Hilbert-Schmidt norms on ?{,.(H. E).
(iii) If f En, LP(-y, E), then Ttf belongs to the classes D'(-y, E). H' (-y, E), and It" (1, E).
Chapter 5.
222
Sobolev Classes
PROOF. (i) Let h E H. By Fubini's theorem, for ry-a.e. x. the mapping
g.: y'-+ f (e-tx +
1 - e-2ey)
is in LP(-y, E). Denote the set of all such points by Q. For any x E Q, the CameronMartin formula yields
Ttf(x+Ah) = =
JX
f
f (e-tx+ 1- e-2t(y+Ae-t(1 - e-2t)-112h]).y(dx) f (e-tx +
1 - e-1y)A(t, Ah, y) 'y(dy),
x
where
Ate-2t
Ae-c
A(t, Ah, y) = expt
h(y) - 2(1 - e-2t) 1 --e - a
IhItt(,l).
Differentiating in A at zero (which is possible, since g; E LP(-) and A'-+ A(t, Ah, ) is differentiable as a mapping with values in L9 (-y), q = p(p - 1)-1), we arrive at (5.4.4). The infinite Frechet differentiability of Tt f along H follows from the infinite Frtchet differentiability of the mapping h e-+ Lo(t,h, ), H -. L9(-y). The latter is readily verified by use of the equality e -t
=Q( 1-e- a hl(x),...
ah,
e -t ,
i - e- t
where Q is a polynomial on IR" whose coefficients are polynomials in the quantities e-2t(1 -e-2t)-1(h,h))H and a-21(1 -e-2t)-l(h;,h,)x. Therefore, the L9-norm of the derivative 8,, ah,. A(t, h, x) is uniformly bounded in hl,... , hn from the unit ball of H. (ii) Now let p > 2. By equation (5.4.4), we have for a.e. x e-2e
0C
s = 1 - e-2e
n=1
I1 n=1 X
2
f (e-tz+
1 - e-2ty)F"(y) Y(dy)
,
where {en} is an orthonormal basis in H. Therefore, by Bessel's inequality, we have
EI
(x) I E-
e 1
e t 2r
n=1
Jf(_t+
1- e-try) I2 ti(dy).
X
Hence DHTt f (x) E 7{(H, E) for a.e. x and e--2t e21
II DHT:f IlL (-,.x(H.E))
<_
e
gi
t
2e
1
fJi(etx+ XX
f
I f f (e-tx +
P/2
2
1 - e-try) I
1- e-try) j" y(dy) ti(dx) =
l
e gi
Y(dU) J
Y(dx)
tzr Jf(z))(dz). X
Therefore, Tt f E Dr" (ry, E). Writing Tt = T,,r Tt/,,, we get by induction that Ttf E DP-"(7,E), n E N. It is readily seen that DHTtI = e-a/2Te12DHTe/2f. It follows from what has been proved and Example 2.9.3 that the mapping h -.DHTt f (x + h) with values in 1i(H, E) is continuous for a.e. x. By induction we get the Hilbert-Schmidt continuity of all higher derivatives.
5.4.
Properties of Sobolev classes and examples
223
(iii) We obtain from (ii) that Tt f E D' (-y, E) if f E f `P>1 LP(-y, E). We shall see below that D°°(y,E) = W°°(y,E) = H°°(y,E), whence the last claim. Let us give a short independent proof of the inclusion T1 f E WOO (-r, E). To this
end, let us take smooth cylindrical mappings fn such that f,, - f in LP(-y, E). Clearly, Tt fn E W°°(-r, E). The previous estimate shows that the sequence {Tt f } is bounded in WP"n (y, E) for every p > 1 and n E IN. By Lemma 5.4.4, we get the claim.
5.4.9. Example. Let f be a bounded Borel mapping on X with values in a separable Hilbert space E. Put
F(x) = Jf(x+y)v(dv). x
Then F E D' (y, E). PROOF. As shown above, the function F is infinitely differentiable along H and
(9hF(x) =
JX
f (x + y)h(y) y(dy),
h E H.
Let {h;} be an orthonormal basis in H. Since {h,} is an orthonormal sequence in L2(y), by virtue of Bessel's inequality, we get from the equality above
I8h,F(x)IE--JIf(x+y)IEy(dy) _ suplii . n=1
X
Hence F E DA1(y, E) for all p > 1 and II DH F(x)II x(H.E) 5 sup If IE, whence the claim follows by induction.
5.4.10. Example. (i) Let f be a -y-measurable function. Suppose that there exists a number C such that one has for a.e. x
If(x+h) - f(x)I 5 CIhIH, Vh E H. Then f E DP-'(,y) for any p > 1. Recall that functions of this type are called H-Lipschitzian.
(ii) Let B E B(X). Put
d8(x):=inf{IhIH: x+hEB, hEH}, and d8(x) = 0 if (x + H) f1 B = 0. Then dB is an H-Lipschitzian function with C = 1 and dB E DP"1(y) for any p > 1.
PROOF. (i) We shall deal with a version off that is Lipschitzian along H with constant C (see Lemma 4.5.2). As we already know, f E LP(y) for all p > 1 (see Theorem 4.5.7). It remains to note that 8hf exists a.e. and IBhfI <_ C, since the function t f (x + th) is Lipschitzian, hence differentiable a.e. (ii) The validity of the Lipschitz condition along H is obvious (since the distance to a set in a metric space is a Lipschitzian function). Hence we only need to verify the measurability
of dB. For any fixed t > 0, the set {dB < t} can be written as (B + tV) U A, where V is the open centered unit ball in H and A = X\(B+H). It was proved in Chapter 3 that there exists a linear space E of full measure obtained as the union of a sequence of metrizable compact sets K. Since H C E, then (B + tV) U A
Chapter 5.
224
Sobolev Classes
up to a set of measure zero coincides with (BE + tV) U AE, where BE = B fl E, AE = An E = E\(BE+H). It remains to use the fact that the sets BE, IV, and H are Souslin, hence so are the sets BE + H and BE + tV, whence the measurability of AE and (BE + tV) U AE.
5.4.11. Example. Let E be a separable Hilbert space, G E
E) and
let ep be a bounded H-Lipschitzian function. Then pG E D2.(7, E) and D. (pG) = DH v $ G + oDH G,
where, given u E H, v E E. the symbol u®v stands for the operator h In particular, IDH`p(x)IHIG(x)IE +Iw(x)IIIDHG(x)IIx(H.E)
(u. h)Hv.
a.e.
PROOF. Let h E H. Then the equality ah(VG) = BhWG+Vi9hG follows from the previous example. It remains to note that IIu®rII?uH.E) = IuIHIrIE and that L2(7,r1(H,E)). DHv
5.4.12. Proposition. Let K C X be a compact set and let U D K be an open set. Then there exists a function f E H' (7) such that f is infinitely Frechet differentiable along H, 0 < f < 1, f = 1 on K. f = 0 outside some closed totally bounded set S C U (S is compact if X is complete), and sup I DH f I < x. PROOF. We construct first a function g E H" (7) such that 0 < g < 1, g > 2/3 on K and g < 3/5 outside some closed totally bounded set S C U. To this end, let us denote by V the closed absolutely convex hull of an arbitrary compact set Q D K with measure greater than 1/2. We shall deal further with the linear span E of the set V, setting all the functions zero outside E (note that 7(E) = I by the zero-one law). Clearly, K C E. Since K is compact, there exists an absolutely convex neighborhood of zero IV in X such that K + IV C U. In addition, there exists r E (0. 1) such that 2rV C IV. Put 10(x)
= inf{p, (x - y), y E K}.
x E E.
The function r/' is the distance to the set K with respect to the norm p, Since the sets K + cV are closed by the compactness of K, we get that w is a Borel function. Let
p(x) = 1 - r-10(v(x)), where 0(t) = t if t E [0, r] and 0(t) = r if t > r. Then p(x) = 1 whenever x E K, and yp(x) = 0 for all x outside K + rV. In addition, 0 < ;p < 1. Let us pick m E V, for which 7(mV) > 8/9. Finally, put g(x) = where (Tt)t>o is the Ornstein-
Uhlenbeck semigroup and t > 0 is chosen in such a way that 1 - e-t < r/8 and 1 - e- tm < r/8. With this choice, for any x E K, one has a-tx E K +rV/8 and. for any y E mV, one has 1 - e- ty E rV/8, whence a-tx+ 1 - e- ty E K+rV/4, which yields
p(e-tx +
1 - e-21y) > 3/4.
2/3. In a similar manner, for any x E E\(K+rV) and y E mV. we get a-tx V K+5rV/7, since et - 1< r/7, whence a-tx + 1 --e-2t y V K + (5/7 - 1/8)rV due to our choice of t. Then Therefore, the estimate -y(mV) > 8/9 implies
o(e-tx +
1 - e-21y) 5
2 7
+
1
8
<
3 7?
5.4.
Properties of Sobolev classes and examples
225
which leads to the estimate 3/7 + 1/9 < 3/5. For completing the proof it remains to put f (x) = 01(g(x)), where 0 is a smooth function on the real line
such that 0 < 01 < 1, 01(s) = 0 if a < 3/5, and 01(s) = 1 if s > 2/3. The function thus constructed is Lipschitzian along H, since so is the function p,, which is a measurable seminorm, and all the subsequent functions arise as the results of compositions with Lipschitzian functions and Tt. By Proposition 5.4.8,
0
f is infinitely Frechet differentiable along H.
5.4.13. Remark. As it follows from the proof, if the compact set K is metrizable, then the support of f can be made metrizable as well; for any sequentially complete space, the support off can be made compact metrizable. If K is an absolutely convex compact set of positive measure, then our construction enables one to find a function f E HOC (y) such that 0 < f < 1, f = 1 on K and f = 0 outside 2K. Note also that this construction yields the existence of Hl(-y)-partitions of unity on X (cf. [300]).
5.4.14. Corollary. Let Z C X be a closed set, W a neighborhood of zero in X and U = Z + W. Then there exists a function f E Hl(y) such that 0 < f < 1, f = 1 on Z and f = 0 outside U. PROOF. We may assume that X is complete, passing to its completion and then restricting the function we constructed to the initial space. Let us take an absolutely
convex compact set V of positive measure. Let Z = Zfl{n -1 < p, < n}. The sets Z are compact, and one can take for them the functions of the form f,, =
constructed in the proof of Theorem 5.4.12. Put f = E f,,. Problem 5.12.37 n=1
suggests to verify that this series defines a function with the desired properties. 0 Another interesting example arises in the theory of stochastic differential equa-
tions. Let A: Rd -+ £(Rd) and B: Rd - Rd be mappings of the class Cb and let a;t be the solution of the stochastic differential equation dCt = A(41) d147t + B(it) dt,
where {W,} is a standard Wiener process in Rd. Denote by p( the measure generated by l;t on the path space x = C((0.1),Rd). It is known (see [361, Ch. IV] or (504, Ch. 4]), that there exists a Borel mapping F: X -+ X such that pE = yoF-1, where -1 is the Wiener measure on X (the distribution of the process {W1}).
5.4.15. Example. Under the foregoing assumptions, the mappings bt o F, t E 10, 11, turn out to be elements of the class W' (y, Rd). In particular, choosing (X, y) for a probability space for (11i)t>o, we get that, for any fixed to, the mapping w
ttfl (w) belongs to W'° (y, Rd).
The proof of this claim can be found in (361].
5.4.16. Example. Let F E G" 1(y) be such that D F = 0. Then F coincides with some constant a.e. The same is true for the class WP- (-y).
PROOF. It suffices to note that, for a.e. x and any h E H, the function t -. F(x + th) coincides a.e. with some constant.
O
5.4.17. Example. Let A be a set such that IA E GP-'(-y). Then we have either y(A) = 0 or y(A) = 1.
Chapter 5.
226
Sobolev Classes
PROOF. Put 2. Then cy is a Lipschitzian function. According to Corollary 5.4.3, we get the equality 21AD,,IA = D IA = 0.
Finally, note that the classes GP-1 (-y, H) are larger than DP-" (-y, H) (see also Problem 5.12.28).
5.4.18. Example. Let y be the product of the standard Gaussian measures , on the real line and let F: R7O - H = 12 be given by F(x) = (2-n f (2nxn)) 0= where the function f on R1 is defined as follows: f(t) = 1 - Itl if Itl < 1 and f is extended periodically to all of RI. Then IF(x + h) - F(z)I,, < IhIN, and F E G2 I (y, H), but F 0 D2"(y, H).
dz E ROO,b'h E H,
PROOF. We have
x
2
x
-nIf(2nxn+2nhn)-f(2nxn)l2<[,2 2'22nhn=IhiH.
[
n =1
nn=1
Clearly, IF(x) 1, < 1. It is easily verified that the mapping
: R°O -+ G(H) defined
by '(x)(h) = (f'(2nxn)hn)x is defined y-a.e. and serves as the generalized 1
derivative of F. Hence F E GP-1 (y, H) for all p > 1. Since I f'(t)I = 1 a.e., we have
x
IIW(x)IIx =
If'(2nxn)12 = OC
y-a.e.,
n=1
whence the last claim.
5.5. The logarithmic Sobolev inequality This section is devoted to the extension to the infinite dimensional case of the logarithmic Sobolev inequality, the Poincar6 inequality, and certain other estimates
of the integrals of Sobolev functions via the integrals of their derivatives. The next three results follow immediately from the finite dimensional case considered in Chapter 1. 5.5.1. Theorem. Suppose that y is a centered Radon Gaussian measure on a locally convex space X. Then, for any f E W2.1(y), one has
ff2logIfJd.)<JjDfI2d..Y+!(Jf2dY)log(ff2dY).
(5.5.1)
In addition,
1@-!)2<_JX
IDfI d.
(5.5.2)
5.5.2. Theorem. If exp(oJD f 1y) E Ll (y) for some o > 1/2, then one has exp(If1) E LI(y) and
Jexp[/ X
Jfdv} dY
(fexP(oIDufI) dy X
(5.5.3)
5.5.
227
The logarithmic Sobolev inequality
In particular, if expJexp[i (ID f IH) E L'(y), then
-
ffd] d-y < f exp(IDHfl2)d?'.
(5.5.4)
x If, finally, one has I D. f I H < C, then exp(o f 2) E L' (y) for every a < X
(2C2)-l.
5.5.3. Theorem. The Ornstein-Uhlenbeck semigroup (Te)e>o is hypercontractive: whenever p > 1, q > 1, one has IlTtf1IQ 5 IIflip
forallt>0such that e2t>(q-1)/(p-1). 5.5.4. Corollary. Let p > 2. Then the operator 1,,: f '-.
from L2(7)
to LP(y) is continuous and (5.5.5)
IIIn(f)IIp 5 (p- 1)n12IIf112.
In addition, for every p E (1, oo), the operators In are continuous on L1(y) and (5.5.6)
1IL.IIc(Lp(,)) 5 (M - 1)n/2,
M = max(P, P(P - 0-1). PROOF. The same reasoning as in the finite dimensional case applies.
where
0
5.5.5. Corollary. Let f E X and p > 2. Then, for any r E IN, one has (p 1)n/2(n+ 1)J/2I1fliL2(,) IIfllH7,- 5 IIfilyp.r < (p-1)"/21IfIIH2,r = are equivalent to the L2-norm on Xk. In particular, all norms II
-
(5.5.7)
PROOF. Inequality (5.5.7) follows from Example 5.3.4. This inequality implies 0 the equivalence of the aforementioned norms.
5.5.6. Remark. Note that the Ornstein-Uhlenbeck semigroup (T1)t>o is hypercontractive on the spaces LP(y, E) of mappings with values in a separable Hilbert
(or Banach) space E. Indeed, if q and p are related as above, one has for any F E LP (-y, E):
IITtFIIL9(,,E) 5
IIFIIE fl Lp(1) =
IIFIIL,(,.E)
Therefore, the corollaries of the hypercontractivity established above are valid for vector-valued mappings. If E is a separable Hilbert space, then (5.5.2) yields
Ji-
f
fdy12dy-
f IDHflh(H.E)d7,
fEW2.1(y,E)
X x E x Below we return to vector Poincare inequalities.
5.5.7. Corollary. Let f E Xd. For any a E (0, 2e ), there holds the inequality
y(x: If(x)I > tIIf112) 5 c(a,d)exp(-at2/d), where
c(o,d) =expa+
d d-2ea'
Chapter 5. Sobolev Classes
228
PROOF. We may assume that Ilf 112 = 1. By the estimate {I f IIp < (p - 1)d/2, we get d-3
J
eXp(alf I21d) d / <_ E n=0
a,,
x
an
ni + td n Ilf II2nn/d < expa + F
-d
nci
n -
2n
(d
-' 1) n.
According to the Stirling formula, if a < 2 , then the sum of the series on the right-hand side is dominated by d(d -
5.5.8. Corollary. The spaces Xd are closed with respect to convergence in d
measure. Moreover, any sequence from ® Xk that converges in measure, is conk=0
vergent in LP(y) for every p E (1, oc). The same is true for the spaces Xd(E) of mappings with values in any separable Hilbert space E. d
PROOF. Let { fn } C ® Xk converge in measure to a function f. If we prove k=0
that supn IIfnIILz(,) < o, then Corollary 5.5.4 will yield the uniform boundedness of {fn } in all LP(-y), which, by the Lebesgue-Vitali theorem, leads to the desired conclusion. Suppose that supn IIfn Il L2 h) = cc. Passing to a subsequence, we may
assume that cn := IlfnlIL2(,)
x Then gn := f,,/c
0 in measure and
II9nllL2(,j = 1. By Corollary 5.5.4, one has supn Ilg IIL,(-,) < cc, whence, by the Lebesgue-Vitali theorem, we get that 119nhIL2(,) -- 0, which is a contradiction. The same reasoning shows that every Xd is closed with respect to convergence in measure (cf. [6731). For vector-valued mappings, the proof is analogous (it is also possible to use the scalar case, since (fn, f,,) E: E X2d).
5.5.9. Corollary. The norms from LP(-y), p E [l, oc), are equivalent on every X,,. In addition, for every p > 0, the topology on X,, induced by the metric from L°(y) coincides with the topology of convergence in measure. Finally, if q > p > 1, one has n/2 1
IIflla <_ Iif1IQ <- (-)
lull,,,
Vf E X,,.
5.5.10. Corollary. Let E be a separable Hilbert space. Then a sequence { Fl } d
from ® Xk (E) converges in measure precisely when for every k = 0,... , d, the k=0
sequence fX D,, F, dy converges in fk(H.E). PROOF. Convergence in measure of our sequence is equivalent to its convergence in L2(-f, E) and yields the convergence of the integrals of all derivatives. Suppose the latter holds true. Since DHd FJ = f y DH FJ dy a.e., the vector version of the Poincare inequality mentioned above yields the convergence of the sequence D, -1FJ in L2(y,fd-I(H.E)). By the same argument, we get the convergence of
D, FJ in L2(y,'hk(H,E)) for any k = 0,... d - 2. The next result is the infinite dimensional extension of the Poincare inequality for vector mappings and any p > 1.
5.6.
Multipliers and Meyer's inequalities
229
5.5.11. Theorem. Suppose that -y is the same measure as in Theorem 5.5.1,
P? 1 and E is a separable Hilbert space. Let f E IW'p"' (y, E) be such that
fx
Np < oo.
II
Then
f I f (x) - f f dyI E y(dx) 5 (n/2)pA-1pNp,
(5.5.8)
x where Mp =
J
I tI pp(0,1, t) dt.
PROOF. The desired estimates follow from the finite dimensional Corollary 1.7.3 (see the proof of Theorem 4.5.7).
Multipliers and Meyer's inequalities
5.6.
As above, -y stands for a centered Radon Gaussian measure on a locally convex
space X and H = H(y) is its Cameron-Martin space. Note that the operators Tf and L are special cases of operators of the form
'paf = E p(n)In(f )
(5.6.1)
n=o
We shall find conditions under which ID, is bounded on LP(-y).
5.6.1. Lemma. For every p > 1 and N E IN, the following inequality holds true for every f E L"(y):
IIT`(f - Io(f) - ... - IN-I(f))IILP(,) <
K(N,P)e
where K(N,p) = (N+ 1)(Af - 1)'x'/2 and M = max(p, p/(p- 1)). The same is true for mappings with values in any separable Hilbert space E.
PROOF. Assume first that p > 2. Let s be such that p = e2a + 1. If t > s, we have by the hypercontractivity inequality
T,Tt-s(f-Io(f)
IN-i(f))II 2LP(s)
Te-3(f - Io(f) - ... -I:-V- i (f)) II Ls( ) =
I
k=N
e-k(1-')1k(f)IIL, 2 <e-2N(t-°)IIlIIi=(y)
(ti)
=
E
e-24-°)IIIk(f)IIi
k=N
<-e-2N(t-a)IIf1ILP(ti)
Chapter 5. Sobolev Classes
230
whence the desired inequality, since eNs = (p - 1) N12. If t < s, then, by Corollary 5.5.4, we have N-1
IN-.(f)) I
IITt(f - Io(f)
LP(y)
e-ktIIIk(f)IIL"(,y)
< >2 k=0
+ IIfIILP(y)
N-1
< > eks-kt1IfIIL9(y)
+ 11f 11 Lo(,)
k=0
<
(NeNs-Nt +
1)eNse-N1
1)11EII Lo(,) <- (N +
II f II
L"(,)
The case 1 < p < 2 follows by the duality.
The next result is called the multipliers theorem.
x
5.6.2. Theorem. Let ak be numbers such that E IakIN-k < oo for some k=0 x N E IN. Suppose that p(0) = 0 and that V(n) _ E akn-k if n > N. Then the k=0
operator Q. defined by equality (5.6.1) is bounded on LP('y) for all p E (1,00). PROOF. Put
x
N-1
>2 ca(k)Ik + > cp(k)Ik = S + R. k=0
k=N
We know that S is bounded on LP(-y) by the continuity of every Ik. Set
x AN(f) =J Ttlf
00
-Io(f)-...-IN-1(f)Jdt=
k-'Ik(f) k=N
0
This operator is well-defined at least for smooth cylindrical f. By Lemma 5.6.1, we have IIAN(f)IILP(y) S
K(N,p)N-'IIfIILP(,).
Note that for every n E IN, one has
Ant(f) _ 00>2 k-nIk(f) = f k=N
Io(f)
IN_1(f)Jdt1 ...fin
(0,oo)n
Therefore,
e-N(tl+...+tn)dt1...dtn
IIAN(f)IILP(y) s K(N,p)IIIIILP(,) 1 (O.x)n
=
K(N,p)N-nhIfIILP(,)
Using the identity 00
x
0o
R(f) = E (`ank-n)Ik(f) _
k=N n=0 we arrive at the inequality
anAN(f)+ n=0
x IIR(f)IIL"(,) sK(N,p)>2IafIN-nIIfIILP(.), n=0
5.6.
Multipliers and Meyer's inequalities
231
whence the desired conclusion.
Note the following commutation relationship between D and *oDHF,
(5.6.2)
where 0(n) = p(n + 1). This identity is readily verified for the elements of Xk and then extends to all elements of W2,1 (-y). The same is true for the mappings with values in any separable Hilbert space E. In particular,
DH (I - L)-1/2 F = (2I - L)-112DHF. More generally, for any integer k and any F E W2.,(-y) with r > IkI, one has
DH (I - L)k'2F = (21- L)k/2DH F.
(5.6.3)
This identity holds true also for vector-valued mappings F E W2.,(1, E), where E is a separable Hilbert space. For the proof it suffices to verify (5.6.3) for Hermite polynomials. Our next aim is to establish the so called Meyer's equivalence of different norms on Gaussian Sobolev classes. Recall that the Hilbert transform of a function f E Co (Wt1) is defined by
4f (x) =
Jf(x + t) - f (x - t) dt. _x
t
It is known (see [844, Ch. XVI, p. 256, Theorem (3.8)]) that 9 is a continuous linear operator on every LP(1R1), p > 1. By analogy, one defines the Hilbert transform for functions on R1 with values in a separable Hilbert space E. Then it is continuous on LP(IR1, E) (see [121], where this result is proved even for a wider class of Banach spaces; the Hilbert space case makes no essential difference with the scalar case).
5.6.3. Lemma. For every p E (1, oc), there exists a constant NP such that 11DH (I -
5 NP11911 LP(,)
(5.6.4)
for every smooth cylindrical function g. Moreover, for any smooth cylindrical mapping G with values in a separable Hilbert space E one has IID,, (I - L) - 112GII LP(,.N(H E)) 5 NPIIGIILP(,,E).
(5.6.5)
PROOF. We shall start with the scalar case. For any (x, y) E XxX and 0 E IR1, we write
z s := xcos6+ysin6, ye := -xsin0+ycos0.
RB(x,y) = (xe,ye),
By virtue of formula (5.4.4). for any h E H and any bounded g, one has
ahT,9(x) =
e
1 - e-21
J 9(e`z + 1 - e X
y)h(y)'7(dy)
Chapter 5. Sobolev Classes
232
Therefore, using (5.3.1) and making the substitute cose = e', we get
x 8h(1 - L)-1/29(x) =
1
f
Jt_h/'2e_tah(Teg)(x)dt 0
/2
JJ cos9llogcos01-1/2h(y)g(xe)'Y(dy)do.
ox Since y is symmetric, we have
f h(y)9(xe) y(dy)
f h(y)9(x-e) y(dy)
X
X
Thus, we arrive at the following representation: ,r/2 eh(1 - L)-1/29(x) _
f f K(e)h(y)[9(xo) - 9(x-e)] y(dy)do,
o x
where K(O) = z cos0llogcosOl-'/2. Note that K0(0) := K(0) - (fB)-' is a bounded function on (0, it/2). This is readily verified taking into account that the function I logzI-'/2 - (1 - z)-1/2 is bounded on (0,1). Therefore, the operator ,r/2
Tf(s) = f K(0)[f(s - 0) - f(s+0)] de, 0
defined on smooth functions f extends to a continuous operator from LP[-ir, 7r] to L9[0, a/2] with norm cp. Indeed, if we put f = 0 outside [- 7r, a], then T - 9/v/2- is bounded, since 3x/2
x/2
(fT-4)f(s)= f vl'2-Ko(0)[f(a-0)-f(s+0)]de+ f f(s-0) a f(s+0) de s/2
0
which gives a bounded operator on LP(-a, a]. Let us apply this result to the function f (s) = g(x,), where x and y are regarded as fixed parameters and g is a smooth cylindrical function. This yields the following estimate: /2
a
JITf(s)IPds < pf If(s)Ipds. -,r
0
Put w/2
S(x, y) := f K(e) [9(xo) - 9(x-B)] de. 0
(5.6.6)
Multipliers and Meyer's inequalities
5.6.
233
Since -y®7 is invariant under the transformations R we have n/2
f
r J
I S(x, y)I°7(dx) ti(dy) = f f f l S(x y)I °'r(dx) yr(dy) ds
0Xx
Xx
/2
2
XIf x 0f
IS(Rs(x,y))Ipdsy(dx)ry(dy).
(5.6.7)
Using the relation n/2
S(R,(x,y)) = f K(O)[g(x.+e) - 9(za-e)] d6, a
which follows from the identity R. o Re = R,+e, and estimate (5.6.6), we see that the right-hand side of (5.6.7) is estimated by
f
x
f f f l g(xe)I P d97(dx)'y(dy) = 4cy 19(x)1 p '(dx). XX X Using the notation introduced above, for any h E H, one has p
ah(I - L)-1129(x)
7r
fS(x,y)h(y)7(dy).
(5.6.8)
x
in H. For every fixed x, we denote by Let us take an orthonormal basis F(x, - ) the orthogonal projection of the function y '-- S(x, y) to the space X1, i.e., is an orthonormal basis in X1, we obtain from F(x, y) = Il (S(z, )) (y). Since equality (5.6.8):
ID.(1- L)-,,g(x)12 =
(f S(x.y)I (y)1'(dy) x
_
1
f F(x,y)hn(y)ti(dy)J2
=
f F(x,y)27(dy)
n=I X X Using that F(x, ) is a centered Gaussian random variable and applying estimate (1.6.5) to the function F(x, - ) E X1 for every fixed x, we get IIF(x,
KpII F(x, )IIt.9
<- Kpblpll S(x, .)Ilr.o(,)-
Therefore,
ftDHi -
L)- i/2g(x) N
7(dx) < n' KPMP f f I S(x, y)I p7(dy) ti(dy)
xx
X
< 4(cKpMp)p f 1 g(x)I p 7(dx). ap/z
X
It remains to be noted that the same reasoning applies to the Hilbert space valued cylindrical mappings, since the Hilbert transform is bounded also in the spaces of O vector-valued functions of real argument.
Chapter 5.
234
Sobolev Classes
5.6.4. Corollary. For every p E (1, oo), there exist constants \, and µp such that the following inequalities hold true for all smooth cylindrical functions V: II (I -
IILP(,)
5 I\p{IkPII LP(,) +
IIDHIPIILI(y.H))
<- Apll(I - L) 112WIILP(,)
(5.6.9)
PROOF. We have
II D II LP(,.H) = II D (I - L)-112(I - L)'1'AILP(,.H) < N,11(1IIi%IILP(, = II (I - L)-1/2(1- L)112 IILP(,) <- 11(1- L)'/2
since (I -
L)-1/2 is
a contraction on LP (-y). Further, II(1 - L)1/2OILP(,) = II (I - L)-1/2(1-
L)VII L.(,)
5 II(I -
L)-1/21PIILP(,)
+ 11(1 -
L)-1/2
LwIILp(,)
IIOILP(,) + II(I - L)-112LpIILp(,)-
Finally, note that II (I - L)-'12L'PII LP(,) 5 NpII DH;PII LP(,.H),
which is verified as follows. Let g be a smooth cylindrical function. Then
J(I_L)2Lcpgdy=JLo(I_L)_h12gdy=_f(DH'P,Dfl(I_L)_/2g)Hd.y, which is estimated by NpII Holder's inequality and (5.6.4).
Lp(,,H)IIgl Lo(µ), where q = p(p - 1)-', due to
0
5.7. Equivalence of different definitions One can extend Meyer's inequalities obtained in the previous section to higher orders of differentiability by using the multipliers theorem. As above, -y is a centered Radon Gaussian measure on a locally convex space X and H = H(y) is its Cameron-Martin space.
5.7.1. Theorem. Let p > 1 and r E N. Them exist positive constants mp,r and 1tfp,r such that, for any smooth cylindrical function f , one has mp,rllD,1 f IILP(,.,cr) 5 II (1 - L)r12fII LI(,) < A'fp.r [II D» f II
+ Ilf IILP(,)]
(5.7.1)
Analogous estimates hold true for E-valued mappings, where E is a separable Hilbert space.
PROOF. According to (5.6.2), for any F E W2'2(y, E), where E is a separable Hilbert space, one has
DH (I - L)-' F = D,, (21 - L)-'/2D,, (I - L) `1/2 F = DH (I - L)-1/2T,,DH (I - L)- '12F, where co(n) = '1+ n/ v/2 -+n. Therefore II DH (I - L)-1F IILr(ti.n,(H.E)) 5 G' aKIIp'IILP(,.lr),
5.7.
Equivalence of definitions
235
where C is the norm of D (I - L)-'12 and K is the norm of 1I', . Hence, IIDti GIILv(,.'H;(H.E)) < C2KII(I -
L)G1LP(,.E1
for all smooth cylindrical E-valued mappings G. Continuing in this way, we arrive at the estimates IID,kGII LP(,.'hk (H.E)) < C(p, k)fl (I - L)k 2GII LP(,,E).
k E N.
On the other hand, by (5.6.2) and Corollary 5.6.4, one has
II(I - L)GII LP(,.E) = II(I - L)'"2(I -
L)1'2GIILp(,.E)
< Apll(I - L)'12GII LP(,.E) + "pllDH (I -- L)'i2GII LP(,. ((H.E))
=1'p11(I - L) 1`2GIIL9(-.E) + tpII(2I ApII(I -
L)112GIILP(,.E) +),,Il(21-
L)'12(I -
LP(,.?t(H.E)) L)--1/2(I
- L)1 /2DHGII LP(,.?l(H.F.1)
< apIIGIILP(-.E) + (A2 + \Pbf)IIDHGIILP(,.N(H.E)) + ant11IIDH
where M is the norm of the operator IP.,, with ;Vi(n) = n + 2/ n -+I. Continuing -L)(k_ 1)/2. we estimate the norm in this way and writing (I - L)k/2 as (I -L)'/2(I of (1 - L)k'2G via the norms of D,, G, j < k. Finally, the norms of 1 < j k - 1, can be estimated via the norms of G and D G by virtue of the Poincare inequality. Indeed, let us suppose first that k = 2 and that G,, are scalar smooth cylindrical functions uniformly bounded in LP together with the second H-derivatives. Denote by v. the integrals of D,,G,,. By the Poincar6 inequality, the sequence DHC,, - t', is bounded in LP(-y, H). Put F., = C - n,,. Then DH F = DH C - r,,. In addition, the sequence
f
x
Fnd-
=JGdf
x
is bounded. Therefore, by the Poincare inequality. the sequence is bounded in LP(-y), whence the boundedness of {i } in LP(-f). Since the vn's are centered Gaussian random variables, they are uniformly bounded in L2(7), which implies the boundedness of {v,,} in H. This shows that is bounded in LP(7,H). The same reasoning applies to smooth cylindrical mappings G. with values in any separable Hilbert space E. In this case v E 7-!(H, E) and, according to Problem 4.10.19, the quantities lIV,,IILP(,.N(H,E)) are estimated via C V,,lIL2(,.1(H.E)), hence via GpIIVn Ilat(H.E). Moreover, it is easy to see from our reasoning that for every e > 0,
there exists C(e) such that IDHGIILP(,.H(H.E)) :5 EIIGIILP(,.E) + C(e)IID,; Gjll.P(,,h2(H.EI)
(5.7.2)
for every smooth cylindrical E-valued mapping C. Now, by induction, we arrive at the estimates FIIGIILP(-,,E) +C'k(F)II DH GIILP(,,,tk(H.E))+
where j = 1,... , k - 1, which bring the proof to the end.
(5.7.3)
0
5.7.2. Theorem. Let E be a separable Hilbert space and let p > 1. r E V. Then the classes E), E), and DP,'(ry,E) coincide and their norms are equivalent (in addition. the corresponding notions of derivative coincide).
Chapter 5. Sobolev Classes
236
PROOF. Meyer's equivalence together with the density of cylindrical mappings in LP (-y, E) yield the equality of the classes E) and HP"'(y, E) and the equivalence of their norms. It remains to apply Proposition 5.4.6. D
We observe that estimates (5.7.3) extend to the mappings G E 1'I'"(-Y. E).
5.7.3. Corollary. The classes Wx(y), H'°(y), D' (-y) coincide. If E is a separable Hilbert space, then the classes W' (y. E), H"(y, E), and D" (y, E) cox x incide. In addition, f = E precisely when E E oa
for allp> I. PROOF. By the previous theorem, W'(-y) = H"(y) = D'(-t). Let f E Since f E H2.,(-I) for every r E IN, we get En'Illn(f)II'z(.,) < oo. Conversely, suppose that this series converges for r = 4k, where k E IN. Then 5 c(k)n-2k. Since Iiln(f)tIH=.k < (n + 1)k'2IIIn(f)IIL2(,), the series
E
whence f E
converges in the space
n=o
5.7.4. Corollary. For any separable Hilbert space E and any k E IN, one has
XA. E 11'x(y.E) = D'°(y.E) = H"(1,E) The derivative in all Sobolev classes defined above is denoted by D or by Ot,. According to Theorem 5.7.2, the definitions of derivative in and DP-' lead to one and the same mapping up to a modification (in addition, the derivative along H turns out to be well-defined also for the classes for which it had not been introduced originally). Equality (1.5.2) extends easily to the infinite dimensional case.
5.7.5. Corollary. Let g E
f
(y). f E I{r2.2(y). Then
f gLfdy.
In a similar manner we have the infinite dimensional generalization of Proposition 1.5.6 (which was in fact justified above).
5.7.6. Corollary. For any f E
(y, E) = W2.1 (y, E) =
(y, E), where
E is a separable Hilbert space, one has
D,,Ttf
5.7.7. Lemma. Let f E WP-1(y), p > 1. Then DJ = 0 a.e. on the set (f = 0}. The same is true for the mappings from the class E) taking values in any separable Hilbert space E. PROOF. This property is obvious if we use the characterization of the Sobolev classes via the directional differentiability. Indeed, it suffices to show that, for any
h E H, having chosen a version of f that is absolutely continuous along h, one has Ohf = 0 a.e. on the set (f = 0}. By the aid of the conditional measures on the lines parallel to h, the claim reduces to the one dimensional case, in which it follows from the definition of the derivative and the fact that almost all points of any measurable set on the real line are its limit points (hence our derivative equals zero almost everywhere where it exists on the set { f = 0}). The vector case follows from the scalar one. 0
5.7.
237
Equivalence of definitions
In the same manner as in the finite dimensional case, one defines the local Sobolev classes on a locally convex space X with a centered Radon Gaussian measure y.
5.7.8. Definition. Let W ( ) , p > 1, r E IN, be the class of all functions f, for which there exist an increasing sequence of measurable sets X (called localizing) and a sequence of functions -On E W' (-y) with the following properties:
1) y(U Xn) = 1 and, for y-a.e. x, the union of the interiors of the sets n=1
(Xn - x) n H considered with the topology from H equals H; 2) ''nIX'., = 1; 3) WnJ E Wpx(y) In a similar manner one defines the classes of vector-valued mappings Woe (y, E), where E is a separable Hilbert space, and the classes H « (y). H, ('y, E). ,+I
By analogy we define the classes G o" (y, E) (replacing in the definition above the condition Wn E W'00 by Vn E G4-k(y) for all q > 1 and k E IN). Note that the classes Ha (y, E) can be defined for any r > 0. The following result is readily deduced from Lemma 5.7.7.
5.7.9. Lemma. Let f E D,± (t('n f) a. e.
If n > m, then one has
on Xn, for all k = 1.... , r and D,, f := lim n-xDH (?P f) does not
depend (up to a modification) on a choice of a localizing sequence {X.} and a sequence {ilrn } with the properties stated in the definition. The same is true for the classes GpaC ('y) and for vector-valued mappings.
5.7.10. Lemma. Let f E Wjo'(^y) be such that f and ID,, f I ,, belong to L7(y) for some p > 1. Then f E WP,'(-?). The same is true for the classes Gf '(y) and for vector-valued mappings.
PROOF. By Corollary 5.4.7, it suffices to show that f E GP-'(-r), that is, (DH f,h)H is its generalized derivative for every h E H. Using the functions ipn E W'° (y) from the definition of the local Sobolev classes and Proposition 5.4.2,
we get the versions of f,, = b,J that are locally absolutely continuous on the lines x + IR'h. Suppose that x belongs to the union of the interiors of the sets (X,, - x) n H. Then there exists an interval (-e, e) such that x + th E Xn for all t E (-e, e) and all sufficiently large n. Then f (x + th) coincides with fn (x + th) for all t E (-e, e), hence is locally absolutely continuous. In addition, the derivative of f (x + th) in t coincides for a.e. t with (DH f,, (x + th), h) = (DH f (x + th), h),,. H
Since IR'h is covered by the interiors of the sets (Xn - x) n H, we conclude that t f (x + th) is locally absolutely continuous on JR' and its derivative equals (DH f (x + th), h) for a.e. t. Therefore, the integration by parts formula applies H (y). The same reasoning proves the claim for the and yields the inclusion f E classes GP" (-y). The vector case follows from the scalar one.
O
5.7.11. Example. Let y be a centered Radon Gaussian measure on a locally convex space X, let E be a separable Hilbert space, and let f E W « (y, E) be such
that
I DH f (x)I°K(H E)y(dx) = N. Then f E WP-' (-y, E) and estimate (5.5.8) holds true.
J
Chapter 5.
238
Sobolev Classes
t if PROOF. Suppose first that E = Ift1. Let f, = g,,(f) where Itl < n and nsgnt if Itl > n. Then f E WP'1(y). It is readily verified
that ID H f I H < DH f (H . Estimate (5.5.8) for f follows from Theorem 5.5.11.
Since f - f a.e. and the integrals of IDHf,,Iv are uniformly bounded, we get by Fatou's theorem and (5.5.8) that the sequence
J
f dy is bounded. This yields
the boundedness of Moreover, we get in WP-'(-y), whence f E simultaneously the desired inequality for f. In the vector case, it suffices to note that fo = If I,,, belongs to Wl (y) and I DHfo(x)I H < II DHf(x)IIx(H.E) a.e. Hence fo E LP(-y), whence the claim.
5.7.12. Example. Let y and E be the same as in the previous example and let f E 4 ' i a (y, E), r E I N and p > 1 . be s u c h that. J ID, ,f (x)l
(H.E)y(dx) < oo.
Then f E WP''(7). PROOF. Follows by induction from the previous example.
5.8. Divergence of vector fields Let p be a Radon measure on a locally convex space X and let v: X - X be a measurable mapping, which we shall call a vector field. The differentiability along a field is defined by means of the integration by parts formula. For a function f on X, we put
&f (x)
(f'(x),v(x)) = i
fp x
tv (xt) - f(x)
whenever this limit exists.
5.8.1. Definition. The measure It is called differentiable along the field v if there exists a function 3,, E L1(µ) such that
f 8,; f (x) µ(dx) = - f f (x).3. 1(x) p(dx), X
d f E )rC" .
(5.8.1)
X
The function ,3' is called the logarithmic derivative of p along the field v or the divergence of v with respect to µ. In the latter case it is denoted also by by. 5.8.2. Example. Let X be an infinite dimensional locally convex space. Then no Gaussian measure on X is differentiable along the vector field v(x) = x. PROOF. Suppose that y is a Gaussian measure on X differentiable along v. To simplify notation, we shall assume that y is centered. Let be an orthonormal basis in H(-y). Put E 1 Fi(x)ei. Let us denote by an the a-field generated smooth function
Al (x) = E(1 - F,(x)2). i=1
5.8.
Divergence of vector fields
239
The random variables 1 - F,' are independent and have one and the same distribution. Hence the series F,(1 - F,2) does not converge. This contradiction proves
0
our claim.
5.8.3. Theorem. Suppose that 7 is a centered Radon Gaussian measure on a locally convex space X. Then every field v E H) has divergence by E L2(-y), and, for each f E U,,2.1 (_y), the following equalityf holds true:
f
(D,, f (x), v(x)) -y(dx) = H
X
f (x) bv(x),y(dx).
J
(5.8.2)
X
If {e,, } is an orthonormal basis in H and v = n 1 linen, then bv(x)
(o V. (x) - vn (x)en (x))1
(5.8.3)
n=1
where the series converges in
L2(-y). In addition, Il6vIIL:(7) 5 I10lW21.
PROOF. For any vector field v of the form v(x) = F,"=l v,(x)e,, where v,(x) E i-tr2.1(ry), we get by virtue of the integration by parts formula
f (D f(x), v(x)) H -t(dx) = - J f (x) (trace,,Du v(x) - E F, (x)vi (x)) y(dx).
(5.8.4)
i=1
Put
n
EF,(x)v,(x).
bv(x) =
i=1
Note that, for any two such vector fields v and u, according to (5.1.8), there
holds the equality
f
6v bu dry = f (v, u),, dry + f trace,, (D,, vDH u) dye
(5.8.5)
Indeed, it suffices to verify this equality for v = vie, and u = use,. Then one applies formula (5.1.8) and the easily verified equality trace,, (D,, (v,e,)D,, (u,ej)) = 0, via,,, -I
In the case u = v, the second integral on the right-hand side of (5.8.5) is dominated by JIlDHv(x)IIv(dx), since (trace,, (T2)I S IITII2, Indeed,
dT E x.
_
n-l
IITIIh IIT*IIx = IITII2
for any orthonormal basis {e, } in H. Therefore, we arrive at the following important estimate:
f 6v(x)2'Y(dx) 5
J[ivxi21
+ IIDHv(x)Il2t] (dx).
(5.8.6)
Chapter 5. Sobolev Classes
240
Formula (5.8.4) can be written as
f (Dxf(x),v(x))'t(dx) = - f f(x)6v(x) y(dx). Since, for any v E W2.1(1, H), the sequence vn :_ F-"=1 vie, converges to v in 10.1(y,H), we get, by virtue of (5.8.6), that the sequence {6v"} is Cauchy in L2(-) and, hence converges to some element, which we take for 6v. Clearly, one gets also equality (5.8.2).
Note that the two parts of the series in (5.8.3) may fail to converge separately.
5.8.4. Example. Let K E R (H). Then
6K = 6K' = 6K, a.e., where K, = (K + K')/2. In addition,
Dx6K(x) = -Kx - K'x = -2K,x.
(5.8.7)
PROOF. Indeed, let {en} be an orthonormal basis in H and let P" be the orthogonal projection to the linear span of e1, ... , en. It suffices to prove our claim
for the operators Q" = P"KP,,, since 6Qn -, 6K and 5Q;, - 6K' in L2(y) by virtue of the convergence KPn -i K in the Hilbert-Schmidt norm, which implies the convergence P,,KP" K and P"K'P, - K' in the Hilbert-Schmidt norm. Then
(Kei,ei)x
_
n
n
n
6Q_, (x)
i=1
- F(Ke, e,)ei(x)t,(x), a=1 j=1
since n
n
(x) _ j=1
n
Fej(x)(Ke.,e,)xe,
e', (x)PnKej =
1=1J=1
Clearly, the same expression is obtained for Q;,. Equality (5.8.7) is readily verified for finite dimensional operators and then follows for all Hilbert-Schmidt operators. Note that in the finite dimensional case, 6K(x) = trace K - (Kx, x). 0
5.8.5. Example. Let K be a symmetric Hilbert-Schmidt operator on H(Pw ). Then there exists a symmetric kernel Q E L2([O,1]2) such that t
-
/1
= f J Q(s, u) dw(u) ds = 0o
1
q, J vi(u) dw(u)
I
where {cp,} is the eigenbasis of Q corresponding to eigenvalues {qi} and the series converges in lVo' 1 [0, 1] for a.e. uJ with respect to the Wiener measure PRR*.
PROOF. Using the isometry between H(Pw) = IV02'1 [0. 1[ and L2[0,11 given by the integration operator, we get 11
Kh(t) =
JJQ(s*u)h'(u)duds* 0
`dh E H(P' ),
0
where Q E L2([O,1]2) is some symmetric kernel. Now the claim is obvious.
S.S.
Divergence of vector fields
241
5.8.6. Remark. Using the equality
f
v(x) [DH f (x)] y(dx)
x
f
f (x)bv(x) y(dx)
x
one defines the divergence for the operator-valued fields v E W2,1(y, NO, k E IN, too. In this case, clearly, 6v will take values in fk_1, No = IR1. The existence of divergence given by (5.8.3) in this case follows by the same reasoning as above.
5.8.7. Remark. Clearly, for every f E W2,2(-y), one has 5DH f = Lf.
5.8.8. Proposition. The operator b :
(5.8.8)
NO -
I (y, Rk_ 1) is con-
tinuous for any r E IN and k E IN. In addition, for every v E W°°(y, H), the series in (5.8.3) converges in all spaces WP-'(-y) and 6v E W'(-Y).
PROOF. To simplify notation we shall consider the case r = 1. Suppose that v is a smooth cylindrical vector field. For every smooth cylindrical function f, by the commutation relation (5.6.2), the multipliers theorem applied to V(n) _
vin/ 1 -+n and Meyer's inequality (5.6.9), one has 11(I - L)-1/2DHf II L"(., H) = 11(1- L)-1/2(2I - L)1/2(21- L)-1/2DHf II L. (,H)
f bvfdy=- f
(v,DHf)Hdy=-f ((I-L)1/2v,(I-L)-1/2DHf)H d7,
which, by virtue of Meyer's inequality and Holder's inequality, is estimated by
where p-'
+q-1 =1.
Hence, II6VIILP(,) <- Const IIVIIWP.I(,,H),
whence our first claim.
If r > 1, then for k = 2,...,r - 1, one proves that
D by E LP(y, Hk) by the k-fold integration by parts in f 8e,,
(bv) f dry
making use of the estimate on 11(1The second claim follows from the first one. Indeed, let {en} be an orthonormal
basis in H and let Pn be the projection in H onto the linear span of el,... en. Then
n
bvn,
where v" = P"v.
i=1
It is readily seen that v" - v in WP,r(y, H) for any p > 1 and r E IN. Hence
bvn
6v in
WP,r-I(-).
0
5.8.9. Proposition. For every p E (1, oo), there exists a constant NP such that one has 11(1- L)-I/2bvII LP(,) <- NPIIvIILP(,.H)
for every v E WP,
H).
Chapter 5.
242
Sobolev Classes
PROOF. It suffices to note that, for every smooth cylindrical function f, one has
r
1 f(I - L)-1/26vdy =
I(I - L)-`,2fbvdy = - J(D(I - L)-112f,v),. dy,
which by Holder's inequality and (5.6.5) is estimated by NpII fuIghIv((p, where q =
P(P - 1)-1. The divergence, similarly to the derivative, is local in the following sense.
5.8.10. Lemma. Let v E 1V2.1(y, H). Then 6v = 0 a.e. on the set {v = 0}. PROOF. Let be an orthonormal basis in H and let P be the orthogonal projection on the linear span of the vectors e1, ... , e,,. Then E W2.1 (-y' H), moreover, for the mappings our claim is valid by virtue of Lemma 5.7.7 and equality (5.8.3), in which the sum is finite. Note that Pv = 0 on the set {v = 0}. Hence 0 a.e. on the set {v = 0}. By Theorem 5.8.3, one has bv in L2(y). Choosing a subsequence convergent almost everywhere, we get the claim.
WI
The local property enables one to define the divergence for the fields v E (y, H). Choosing any localizing sequence of functions 0. E W-(-Y), we get
2,1
that the sequence 6(0,,v) is stationary a.e. Its limit is taken for 6v. By virtue of the local property, this limit up to a modification does not depend on our choice of 8,,. In addition, for any v E W2,1 (-y, H), we get the previously defined function 6v. The divergence is an abstract version of the so called extended stochastic integral (introduced by Hitsuda and Skorohod). The usual stochastic integral of an adapted square-integrable process u can be written in the form of the divergence of the field
v defined as follows. We shall assume that a probability space for the Wiener process wt is the space C(0,1) with the Wiener measure P. Put H = H(P). Let IIt
v(w)(t) =
J0
u(s,w)ds.
Then v is a vector field with values in H. This field may be nondifferentiable, yet, but the condition imposed, due to Girsanov's theorem (discussed in Section 6.7 below), implies the differentiability of P along v. Indeed, assuming first that u is bounded, we see, by Girsanov's theorem (see Section 6.7 or 1504, Ch. 7]), that the process wt + ev(t) induces the measure Pf, whose Radon-Nikodym density with respect to P is given by
>feu(t,w)2dt].
rr
A, = exp I J eu(t, w) dwt - 2 By using this formula, it is readily verified that, for any smooth cylindrical function f, one has the equality 1
f
J OvfdP = J f(w)(J u(t,w)dwt ) dP,
5.9.
Gaussian capacities
243
it
whence by = - J u(t, w) dwt. Then this relationship extends to all adapted square0 integrable processes u.
5.9. Gaussian capacities In this section, we define Gaussian capacities and discuss their properties. Using capacities, one gets some classical results (some limit theorems, the zero-one laws, etc.) in a sharper form, since capacities provide a finer characterization of the smallness of sets than measures. For instance, a point has positive C2,,-capacity in R, but zero C2,1-capacity in IR3.
Let p be a nonnegative Radon measure on a completely regular topological space X (certainly, we shall be particularly interested in the special case, where p is a Gaussian measure). Let T E G(LP(p)) be an injective linear operator taking nonnegative functions to nonnegative ones. There is a standard procedure associating with T a set function CT on X, called the capacity generated by T. Throughout this section we denote by II . IIp the norm in LP(p) if this does not lead to any confusion. Some additional information about capacities can be found in [107, Ch. I], [273], [297, Ch. 3], [404], (411], [4121.
5.9.1. Definition. For any open set U in the space X, we put CT(U) = inf{IIfIIp: f E L'(p), Tf > 1 p-a.e. on U}, and CT(U) = +oo if there is no such f. Then, for every subset A of X, we put CT(A) = inf{CT(U): A C U, U is open}. It is clear that in the definition of the capacity C7. on open sets one could consider only nonnegative functions, since T I f I> T f. We shall deal further with the operators Vr. The corresponding capacities are denoted by Cp.r and called Gaussian capacities. However, almost all results in this section are valid for general capacities CT.
5.9.2. Lemma. The capacity CT is subadditive, i.e., U B) G CT(A) + CT(B)
for any two sets A and B. In addition.
Cr(B) >
IITIIc(1Lp(a))p(B)'lp
dB E 8(X).
PttooF. It suffices to consider open sets A and B. For any e > 0, let f and g be functions from LP(p) such that IIfIIp 5 CT(A)+e, IIgIIp 5 CT(B)+e, TfIA > 1 and TgI B > 1 a.e. Put h = max(lf 1. Igi) Then IIhlIp 5 (Ilf IIp + IIgIIp)'1p < IIf IIp + Ilgllp <-
CT(B) + 2e.
On the other hand, ThI ,a > T f I A > 1 and ThI B > TgI B > 1 a.e., since Th > T f and Th > Tg a.e., due to our assumption. Therefore, CT(A U B) < IIhIIp, whence
the first claim. Let us prove the second claim. Let B be open and let f E LP(p) be such that T f mB > 1 a.e. Hence IITIIc(LP(H))IIIIIp > IITfIIp > p(B)'IP. Therefore, this estimate is valid for the infimum.
If p = 1 and T = I, then CT(B) = p(B), but typically CT is not countably additive even on the Borel o-field.
Chapter 5. Sobolev Classes
244
5.9.3. Proposition. The capacity CT has the following properties:
CT( n Kn).
(i) If compact sets Kn decrease, then lies n
n_1
Or
(ii) For arbitrary sets An, one has CT(U An) < n=1
C1-(A,). n=1
PROOF. (i) Indeed, for any open set U, containing the compact set nn 1 Kn,
one can find n such that Kn C U. In order to prove claim (ii), it suffices to consider open sets An. In addition, we may assume that the series of their capacities
converges. Put A = U' 1 A. Let e > 0. Let us choose fn E LP(p) such that T fn > 1 a.e. on An and (5.9.1) IIfnIIp I on A a.e.. since T f > T f,, by the estimate f > fn. Hence CT(A) <_ Ilfllp. Note that if Ip < r,', Ifnlp. Since. by (5.9.1), one Ilfnllp
has
x
x
Cr(A,)p + e < oc,
Ilfnlip <_
n=1
n=1
we get f E LP(s). In addition, applying (5.9.1) once again, we get a.
Ilf lip < [ Ilfnllp] n=1
x
x
1/p
Cj (An) +
lifnlip <
< n=1
n=1
whence CT(A) < n 1 CT(A,) +e. 5.9.4. Definition. A function f on X is called quasicontinuous with respect to the capacity CT if, for every e > 0, there exists an open set U with CT(U) < e such that f is continuous on X\U. We shall say that a certain property is fulfilled quasi-everywhere if it is fulfilled
outside a set of capacity zero. Note that any open set U of zero µ-measure has capacity zero (since the function f =_ 0 satisfies the condition Tf > I a.e. on U). However, for closed sets, this is not true (see Problem 5.12.42).
5.9.5. Lemma. If a function f is quasicontinuous and f > 0 p-a.e. on an open set U. then f > 0 quasi-everywhere on U.
PROOF. Denote by S the topological support of p (i.e., the intersection of all
closed sets of full measure). Since p is Radon, one has p(X\S) = 0. As noted 0. Let e > 0 and let Z be a closed set on which f is continuous e. By the subadditivity of Cr, we may assume that Z C S such that (replacing Z by Z fl S). Let us show that U n f f < 0} C X \Z, whence our claim follows. Indeed, suppose that z E U n Z and f (z) < 0. Then there exists a neighborhood V C U of the point z such that f (x) < 0 on V n Z. By condition, 0 p(V) = 0, whence V C X \S, which is a contradiction. above,
5.9.6. Theorem. Let p > 1. Suppose that T sends C5(X) to C ,(X) (or, more generally, Lp(p) contains a dense set of continuous functions. which are taken by T to continuous functions). Then: (i) Every function F = T f , where f E LP(p). has a quasicontinuous modifica-
tion F' such that CT (x: F* (x) > R) < R'1IIf Ilp.
dR > 0:
(5.9.2)
5.9.
Gaussian capacities
245
(ii) For any sequence { fn} convergent to f in LP(p), there exists a subsequence of {(Tfn)'}, which converges quasi-everywhere to (Tf)'; (iii) For any set B, there exists a unique element uB in the set
FB = {u = Tip: p E LP(p),
> 0, u` > 1 quasi-everywhere on B}
with the minimal norm. In addition, CI-(B) = IIT- 'uBIIP.
PROOF. Let {fn} C Cb(X) be a sequence convergent to f in LP(t). Put F" = T fn. Note that, for every r > 0 and any function po E Cb(X ), one has
r) < r-'II;oII,
CT(x:
since Tcp/r > 1 on the open set {x: TV(x) > r}. Hence, for every r > 0, we get
n,k-+oo.
CT(IFn - FtI > r) Therefore, for any n, there exists an integer kn such that Cg-(IF)
- F,j > 2-") < 2"-",
Vi, j > kn.
Let us verify that the sequence {Gn} = converges to F quasi-everywhere. It suffices to show that this is a Cauchy sequence quasi-everywhere. Let £ > 0. Let us choose n1 such that 1/2n' < E. Put
XE = n {x: IG3 - G3.i I < 2-7 }. By virtue of the subadditivity of CT and the choice of G", one has
CT(X\Xo) < E 2-j < 2-n, < £. )>n,
Obviously, on the set XE, one has ICj+I - Gj I < 2-J for all j > ni. Therefore, the o(C3. - Cj), Go = 0, converges on X(, which proves the convergence of {Gn} by the equality E," (G3+I - Gj) = Gn. Since e is arbitrary, we get the quasi-everywhere convergence. This reasoning shows that, for any £ > 0, there exists a closed set Xe with CT(X\X,) < e, on which the convergence is uniform. Hence the limit F' of the sequence {G,,} is quasicontinuous. Moreover, it follows from our reasoning that, for any fixed r > 0 and £ E (0, r), there exist a closed set Xr with CT(X \X,) < e, on which F' is continuous, and a function g E Cb(X) series
such that If - 9IIP < e and IF' - TgI < £ on X. Then CT(F` > r) < CT(X, f1 {F' > r}) +£
r-e})+£r-£)+£ <- (r-£)-'II9IIP+£ <
(r-e)_'(IIfIIP+£)+£.
Since £ is arbitrary, we arrive at (5.9.2). Let now {fn} be a sequence convergent
to f in LP(y). Put F. = T f,. For any n, there exists an integer k" such that Ilfk - f IIP < 4'". Making use of the estimate CT(IFF, (x)
- F'(x)I > 2-") < 2 ",
we get, as above, the quasi-everywhere convergence of {F; (i) and (ii) are proved.
to F. Thus, claims
Chapter 5.
246
Sobolev Classes
It remains to prove claim (iii). Suppose first that B is an open set. Recall that the space LP(p) for p > 1 has the Banach-Saks property: every bounded sequence {y;n} C L"(p) contains a subsequence {z/in} such that the sequence Sn _
n
converges in LP(p) ([195, Chapter 3, §71). Let us apply this property to the sequence of nonnegative functions V. chosen in such a way that TV, > 1 p-a.e. on B and nlim IIVnIIp = C-r(B). Let g be the limit of {Sn} in LP(p) and uB := Tg. It is clear that g > 0 and TSn > 1 M-a.e. on B. Hence uB > 1 p-a.e. on B. In addition, CT(B) <- IIgIIp 5 limsupllVnllp = CT(B).
In the general case, let {Bn} be a sequence of open sets containing B such that CT(Bn) > CT(B)+2-n. Put un = uB = Tcpn, where nonnegative quasicontinuous modifications of un are chosen. Then the sequence {Sn} of the arithmetic means of some subsequence IV,,,) of {V,} converges in LP(IA) to a nonnegative function g.
Setting uB := Tg and passing to a subsequence once again, we may assume that the convergence takes place quasi-everywhere. By Lemma 5.9.5, un > 1 quasieverywhere on B. Hence uB > 1 quasi-everywhere on B. Obviously, IIgIIp 5 lim sup IIcpn Il n
,=
The uniqueness of a quasicontinuous function u, for which u > 1 quasi-everywhere on B and IIT-'ullp = CT(B), follows from the uniform convexity of the space Lp(p). Indeed, if v is another function with the same properties, then w = (u + v)/2 > 1 p-a.e. on B, whence Cr(B) <- IIT-'wllp <_
IIT
't'llp +IIT
' ullp
= CT(B)-
2
This means that the vectors T-'u, T-v and their half-sum have equal norms in Lp(p), which is only possible if they are equal.
In order to prove the minimality of the norm of uB, it suffices to show that for every quasicontinuous function h E T(Lp(p)) such that h > 1 quasi-everywhere
on B, one has IIT-'hlip > CT(B). Suppose first that h is nonnegative a.e. and let c > 0. One can find a closed set Z, on which h is continuous, such that Cr(X \Z) < E and h > 1 on B n Z. Note that the set
G=(Zn{h>1-E})U(X\Z) is open and B C G. As we have already proved, there exists a nonnegative function
ho = ux\z such that ho > 1 on X\Z a.e. and IIT-'hollp = CT(X\Z) < C. Since the functions h and h.o are nonnegative a.e., we get h + ho > 1 - e a.e. on G. Therefore,
CT(B)
IIT-1h +T-'hollp <
1E
IIT-'hIl + E
1-E
Since E is arbitrary, we get CT(B) < IIT-'hll,. Finally, if h is not supposed to be nonnegative a.e., one can take the function h1 = (Tlgl)', where g E LP(IL) is such that Tg = h. Clearly, h1 > h a.e., since IgI > g. Hence h1 > h quasieverywhere by Lemma 5.9.5. Thus, h1113 > 1 quasi-everywhere and IIT -' h 1 l I p=
IIT-'hllp = CT(B). The uniqueness result implies that h1 = h. hence h > 0 quasi-everywhere.
0
5.9.
Gaussian capacities
247
The function ug from the previous theorem is called the equilibrium potential of the set B.
5.9.7. Corollary. Suppose that the conditions in Theorem 5.9.6 are satisfied. Then, for any increasing sequence of sets Bn, one has lim CT(Bn) =CT(Un==1Bn) nx
PROOF. Let un be the equilibrium potential of B,,. The sequence {T-'u;} is bounded in L%µ). Hence it has a subsequence {T-tank}, whose sequence {Sn} of the arithmetic means converges in LP(p) to some function g. Put u = Tg. Clearly, the union D of the sets Bn n fu,, < 1} has capacity zero, since each of these sets
has zero capacity. On the set B\D, one has Tg > 1. Therefore, Tg > I quasieverywhere on B, whence CT(B) <_ IIgIIp by virtue of claim (iii) of Theorem 5.9.6.
On the other hand, Ilgllp
lim sup IIT-tunllp = l msupCT(Bn) < CT(B), n
n
whence the claim follows.
The role of the topology of the space X has not yet been clarified, although the open sets are explicitly involved in the definition. Surprisingly enough, in many cases capacities are invariant under weakening the initial topology.
5.9.8. Lemma. Let C be a nonnegative subadditive monotone set function on the family of open subsets of a Hausdorff topological space X. For any set A, we extend C by the formula
C(A) = inf{C(U): A c U. U is open}. Suppose that exists a sequence of compact sets Kn such that lim n-x
0.
Let r be a weaker Hausdorff topology on X and let C, be the extension of C from the class of r-open sets to all sets by means of the foregoing formula. Then C(A) _ C, (A) for any A.
PROOF. It suffices to verify that C(U) = C,(U) for any set U that is open in the initial topology. Let us fix e > 0. Let K be a compact set with C(X\K) < e. Note that K is compact in the topology r, hence the initial topology coincides with
r on K. In particular, there exists a -r-open set V such that K n V = K fl U. Therefore,
C,(U) < Cr(U n K) +Cr(X\K) = Cr(U n K) + C(X\K)
whence C,(U) < C(U). The opposite estimate is trivial. Not every capacity has the property mentioned in this lemma, which is called the tightness of capacity (see Problem 5.12.41). However, the capacities connected with the Sobolev classes are tight.
Chapter 5. Sobolev Classes
248
5.9.9. Theorem. Let X be a locally convex space X with a centered Radon Gaussian measure y and let Cp,r be the capacity generated by the operator Vr on LP(-y), r > 0, p > 1. Then, for any e > 0, there exists a metrizable compact set K, such that Cp.r(X\Ke) <e. In addition, Cp,r(B) = sup{Cp,r(K); K C B is metrizable compact} for any Bored set B (the same is true for any Souslin set B).
PROOF. First we consider the case, where X is complete. In this case, as we know, there exists a metrizable absolutely convex compact set K such that y(K) > 1/2. Denote by g the Minkowski functional of the set K defined by zero outside the linear span E of K. Since y(E) > 0, one has y(E) = 1 by the zero-one law. By Fernique's theorem, one has g En, LP (-I) - Put G(x) := Jg(x) := Tiog2/29(x) =
Jg(±i) y(dy), x
where (TL),>o is the Ornstein-Uhlenbeck semigroup. Let Kn be the closure of the
set {G < n} n E. Note that I v'2-J9(x)
- g(x) 1:5 f 9(y) -y(dy) = d,
x E E,
x whence
{Jg2n+d. Therefore, K,, is a metrizable compact set. By virtue of Proposition 5.4.8, one has G E HP.r(y) for any p, r > 1. Since n-1G > 1 a.e. on the open set X\Kn, we have
Cp,r(X\Kn)
n-'IIGIIHP.r,
whence the tightness of Cp,r follows. Let now X be an arbitrary locally convex space and let Z be its completion. We shall consider the measure y on Z. Let D be a compact subset of X of positive -y-measure and let E be the linear span of D. The set E, as it has already been noted above, is a-compact. In addition, by the zero-one law, it has full measure. The indicator function IM of the set M = Z\E equals zero a.e. and JIM = IM pointwise. Since JIM E HP,r(y), we get JIMIIHp.r=0. Therefore, G" (M) = Cp,r(IM = 1) = Cp,r(JIM = 1) <_ II JIM 11 y'.r = 0. Hence, for any e > 0, there exist a metrizable compact set Q C Z and an open set
U C Z such that M C U,
Cp,,.(U) < e,
Cp,r(Z\Q) < C.
Then Z\U C E. Put K = Q n (Z\U) and notice the following:
(i)KCECX;
(ii) the set K is compact and metrizable, being a closed subset of Q; (iii) Cp,r(X \K) < Cp,r(Z\K) < 2e. The second claim of the theorem follows from the general properties of capacities on compact metric spaces (see [541, Ch. III]).
5.10.
249
Measurable polynomials
5.9.10. Corollary. Let E be a y-measurable linear subspace in X such that y(E) > 0. Then Cp,r(X\E) = 0. PROOF. One applies the reasoning used above to show that Cp,r(M) = 0.
5.9.11. Corollary. Let T: X -. Y be an injective continuous linear mapping to a locally convex space Y. Denote by CP r the capacities associated with the measure v = yoT`'. Then, for any Borel set B C Y. one has
Cp,r(B) = Cp(T-1(B)). More generally, the same is true if T is an injective linear mapping which is (B(X).y,B(Y))-measurable, provided y oT` is a Radon measure. PROOF. In fact, we shall use the following weaker condition of injectivity: T is injective on a full measure linear subspace X0 and T(X\Xo) C Y\T(Xo). Since the capacities corresponding toy and y o T-' are concentrated on Souslin linear subspaces which can be continuously and linearly embedded into 1R", it suffices, by
virtue of the previous results, to prove our claim in the case where X = Y = 1R". Theorem 3.6.5 and the previous corollary reduce this case to the case where X and Y are separable Banach spaces. Moreover, since there is a measurable linear
mapping S: Y - X satisfying the foregoing injectivity condition such that y = (yoT-' )oS"' (see Theorem 3.7.3), it suffices to show that CT r(B) _> Cp,r(T-' (B)) for every Borel set B. Finally, due to Proposition 3.11.28, there exists a stronger separable norm !) 11E on X such that T is continuous on (X, ii - !!E). Applying again the previous corollary and the invariance of capacities we can deal further with E. Suppose that B is open in Y. Let cp E LP(yaT-') be a function such that 1 y o T-1-a.e. on B. Then Vr(yo o T) > 1 a.e. on the open set T-' (B) and 1), whence Cp,r(T-'(B)) < CT r(B). Hence the same is ]IV true for any Borel set B in Y. -
Finally, note that the Gaussian capacities Cr,,. satisfy the condition in Theorem
5.9.6 (since T=(FC") C FCI). Therefore, any function f E Hp.r(y) has a Cp,rquasicontinuous version. According to Problem 5.12.44, any function f E H'(1) version, i.e., a version that is Cp,r-quasicontinuous for has a all r > 0 and p > 1. Note that any proper linear -y-measurable functional f is C,-quasicontinuous. Indeed, one has f = cVf, hence, taking a sequence of continuous linear functionals f, convergent to f a.e. we get the functional fo = lim n,c fn = lim n-xcVr f,,, which is defined on a full measure linear subspace L and is quasicontinuous (since the set X\L has Cp.,-capacity zero). It remains to note that the complement of the linear space If = fo) has Cp,r-capacity zero. More generally, if T is a proper linear mapping with values in a locally convex space Y such that T is (B(X),,, B(Y)) -measurable and y o T-1 is a Radon measure, then T is C,,-quasicontinuous (Problem 5.12.45).
5.10. Measurable polynomials A homogeneous polynomial of degree d on a locally convex space X is defined
as a function of the form F(x) = V(x.... ,x), where V: Xd
1111 is a function
that is linear in every argument (a polynomial of degree 0 is a constant). By analogy one defines homogeneous polynomial mappings with values in locally convex spaces.
Chapter 5.
250
Sobolev Classes
Using symmetrization, one can always find a symmetric d-linear mapping V generating F. The corresponding symmetric d-linear mapping V is uniquely determined and can be evaluated by the following polarization identity (see Problem 5.12.29):
V(x1.....xd) =
TV
E
(-1)d_£1_..._1dF(x0+flxl +...+ fdsd),
(5.10.1)
e,E(0,1)
where xo E X is an arbitrary element (e.g., x0 = 0). A polynomial mapping is defined as a finite sum of homogeneous polynomial mappings. The degree of such a polynomial is defined as the maximum of the degrees of its homogeneous components (so that a polynomial of degree d is a polynomial of degree d + 1 as well). The exact degree of a polynomial is the maximal degree of its nonzero homogeneous components. Continuous polynomial mappings are defined as polynomial mappings with continuous homogeneous components. A natural generalization of continuous polynomials are measurable polynomials on infinite dimensional spaces with measures. In the case of a Gaussian measure ry it is convenient to define -y-measurable polynomials of degree d as the elements of d
the spaces ® Xk. The exact degree of a measurable polynomial f is the minimal k=O
possible d.
We shall see in this section that -y-measurable polynomials can be defined in several equivalent ways (as closures of continuous polynomials of a fixed degree d or as mappings that are polynomial of degree d along H(y)). We shall also discuss some integrability properties of polynomial mappings.
5.10.1. Lemma. Let - be a Radon Gaussian measure on a locally convex space d
X and let f E ® Xk. Then f has a Borel version F such that. for every .r E X. k=O
the function h 1--+ F(x + h) is a continuous polynomial of degree d on the Hilbert space H = H(y).
PROOF. It suffices to prove the claim for f E Xd. We may replace X by its full measure linear subspace X0 that is a countable union of metrizable compact sets and put F = 0 outside X0. Further, we may deal with a Borel version of f.
Then f = edTi f a.e. and edTi f is a Borel function. We know that. for a.e. x, the function F := edTi f has the following property: h - F(x + h) is continuous on H. Let us take a full measure Borel set B C X0 with this property. Let {e,} be an orthonormal basis in H. There exists a sequence of continuous finite dimensional polynomials f,, of degree d convergent to f a.e. and in L2(-y). Clearly, T1 fn(x + h) are continuous polynomials of degree d on H for the functions h
every x. Therefore, there exists a full measure Borel set C C B (we can take for C a countable union of metrizable compact sets) such that. for every x E C and j E T, the set (t E 1R1 : F(x + te,3) = lim edTi fn (x + te,) } has positive measure. noc
Then F(x + te,) = lim edTj fn(x + te,) for all t E IR1, and t - F(x + tee) is a polynomial of degree d, since so are T1 fn (x + tee) and t - F(x + t e,) is continuous. This reasoning shows in fact that edTi fn (x + h) converge to F(x + h) for every h
in the linear span of {e,}. In addition, It '-» F(x + h) is a polynomial of degree d on the linear span of {ej}. By the continuity of this function, we conclude that
5.10.
Measurable polynomials
251
h '-+ F(x + h) is a continuous polynomial of degree d for every x E C. Noting that C + H is a full measure Borel set in X (since both C and H are countable unions of metrizable compact sets, so is their sum), we can redefine F outside C + H by zero, which yields a version with the desired properties. d
Thus, all functions from ®Xk have versions, which are continuous and polyk=O
nomial along H of degree at most d. This property of -f-measurable polynomials is characteristic. The next example shows that it would not be reasonable to define measurable polynomials as Borel polynomials (however, we shall see later that measurable polynomials could be defined as Borel mappings that are polynomial of certain degree d along H).
5.10.2. Example. Let X and Y be Freshet spaces (e.g., Banach spaces) and let F: X -. Y be a polynomial mapping. If F is Borel, then F is continuous. d
PROOF. By definition, F(x) _ E Vk(x..... x), where VA.: Xk -+ Y is a k=(I
symmetric k-linear mapping. Since Vd(x.... x) = lim n-dF(nx), the mapping Vd(x,... , x) and, hence, all the homogeneous components in the representation of F are Borel. Let us prove the claim for Fd(x) = 1d(x.... ,x). By formula (5.10.1)
for the symmetric polylinear mappings, we can write Vd(xj,... xd) as a linear combination of the mappings (XI,xd) Fd(ci xi + ... Fdxd), where ri = 1 or e, = 0. Since these mappings are Borel, then Vd on Xd is Borel as well. Hence it is Borel in every variable, whence by Corollary 3.11.13 we conclude that Vd is continuous in every variable. Since X is a Freshet space, 17d is continuous (see [670, p. 88, Ch. III, §5, Corollary 11.
5.10.3. Proposition. Let 1 be a Radon Gaussian measure on a locally convex space X, let H = H(-y), and let Y be a normed space, having a countable family r C Y' separating the points. Suppose that 41: X -+ Y is a mapping such that. for every I E F, the function l(1') is -y-measurable and h -+ 'Y(x + h). H - Y, is a continuous polynomial mapping for a.e. x. Then there exists d such that the degrees of these mappings, for a.e. x, do not exceed d. and II'PITS' E lr,>1L"(7).
PROOF. It suffices to prove our claim for centered measures. It is readily verified that the set Ald of all x such that the degree of the corresponding polynomial
mapping is at most d is measurable. In fact, A/d coincides up to a measure zero set with the intersection of the sets {x: 'I(* (x)) = 0}. I E I', where {en} is an orthonormal basis in H. Since the sets Ald are invariant with respect to the shifts to the elements of H, we get, by the zero one law, that the first of them that has nonzero measure is a set of full measure. Now we use induction in d. For d = 0 the claim is true, since in this case, for every I E I', the function 1(41) is constant along H, hence coincides a.e. with some constant. Then %P coincides a.e. with some element from Y (recall that IF is a countable separating family in Y'). Suppose the claim is true for some d > 1. Let us consider the mapping DH it: X --' C(H. Y). The mapping D 4 with values in C(H, Y) is polynomial along H of degree less than d. In addition, C(H,Y) satisfies the condition of this proposition. Indeed, let (h;} be a countable set everywhere dense in H. Then the countable family of the functionals of the form g: A - I(Ah,), I E I', separates
Chapter 5. Sobolev Classes
252
the points in C(H,Y). For every such functional, the function g(4k) = 1(8h,') is y-measurable as the pointwise limit of n[I(41(x+n-1h;)) - l(a(x))], since the functions x - 1(e(x+n-1ha)) are measurable by the inclusion h, E H. In addition, the function II4'(x)jly is measurable as well, since it coincides with sup.,1 (11(x)) for some sequence {ll} C Y' with llljlly < 1. In a similar manner one gets the measurability of the function By the inductive assumption, we get II D, WIIc(H.y) E f1P>1LP(y). This implies the claim. Indeed, let :p,,(x) = sups
sup
f
IDHScaI'dy5
flID,,'4'IJ,,.y ) d,
whe nce, by virtue of the Poincare inequality (see Theorem 5.5.1). we get the integrability of yon and the estimate sup
I
n
Ipn -
fendy Ipd'1' < oo.
1
Since lim ,;,(x) = II'y(x)jJy < oo a.e., the integrals of ipn are uniformly bounded
n-x
by Fatou's theorem.
Applying the monotone convergence theorem, we get the
claim.
5.10.4. Corollary. Let Y in Proposition 5.10.3 be a separable Hilbert space. d
Then 41 E ® Xk(Y). In particular, 'I' E H' (-y, Y). k =O
PROOF. The standard approximations 9,, of the mapping %J? constructed in Corollary 3.5.2 by means of the conditional expectations are, as one can easily see, polynomials of degree d in finitely many continuous linear functionals if we use
a basis {e,} such that e, E X. In particular, they are continuous polynomials d
of degree d along H. Therefore, '1'n E ® Xk(Y), which implies the inclusion k-0
d
41 E ® Xk(Y), since the sums of Xk(Y) are closed. k=0
5.10.5. Remark. Let f E L2(y) be such that, for some orthonormal basis {en} in H, for every n and a.e. x, the function t F-. f(x + ten) is a polynomial of d
degree at most d. Then f E ® Xk. Indeed, it suffices to note that the functions k=0
Ti f satisfy the conditions in Corollary 5.10.4 and converge to f in L2(y) as t -. 0.
5.10.6. Proposition. Let f E WP-"(-y), where p > 1. Then f E ® Xk k=0
precisely when D, ,f coincides with a constant a.e. In particular. the second order measurable polynomials are precisely the functions f E such that D" "f = A a.e., where A is a symmetric Hilbert-Schmidt operator on H.
5.10.
Measurable polynomials
253
n
PROOF. If f E ® Xk, then the equality D f = const a.e. follows from the k=0
fact that this is true for Hermite polynomials. Conversely, let D. "f (x) = c a.e. for some constant element c. Then, for every t > 0, one has
f = e-nrTjDH' f = e-ntc a.e.
Since the mapping h -# Tt f (x + h) is infinitely Fr6chet differentiable on H for almost all x (see Proposition 5.4.8), it is a continuous polynomial of degree at most n
n
n for almost all x. By Corollary 5.10.4, Tt f E ® Xk, whence f E ® Xk. An k=0
k=0
alternate way of proving is this: if {e,} is an orthonormal basis in H, and the chaos decomposition of f involves a nontrivial polynomial cHk, (e,,) . Hkm (L,) with
k, +
+ k,,, > n, then D f is not constant.
5.10.7. Theorem. Let -y be a Radon Gaussian measure on a locally convex space X, let Y be a separable Frechet (e.g., separable Banach) space, and let F: X - Y be a y-measurable mapping. Then the following conditions are equivalent:
(i) There exists a sequence of continuous polynomial mappings Fn : X Y of degree d convergent to F y-a.e. (ii) There exists a sequence of continuous polynomial mappings Fn : X -i Y of degree d convergent to F in measure, i.e.. for every continuous seminorm p on Y and every e > 0, one has lim y{x: F(x)) > e} = 0. n d
(iii) For every I E Y', one has l o F E e Xk. k=0
(iv) There exist a nonnegative integer d and a version Ft) of F such that for every x E X, the mapping h H Fo(x + h), H -+ Y. is a continuous polynomial of degree d.
If either of the conditions above is fulfilled, then one can find a Borel version of F satisfying (iv). PROOF. It suffices to consider centered measures. Clearly, (i) is equivalent to (ii) (since any sequence of mappings with values in a metric space that converges in measure, has a subsequence convergent almost everywhere), and either of these two conditions implies (iii), since 1 o F is a continuous polynomial of degree d for every functional I E Y. Suppose that (iii) is fulfilled. By Theorem 3.6.5, there exists a separable Banach space E compactly embedded into Y such that y o F-' (E) = 1. The closed unit ball of E (which is compact in Y) is denoted by K. Since there is a sequence of continuous linear functionals on Y separating the points, we may assume that
Y is embedded into 1Rx. Then F as a mapping with values in Rx can be written as F = (F,.... , F,,... ), where Fn are -y-measurable polynomials of degree d. By Lemma 5.10.1, every Fn has a Borel version G,, such that, for every x E X, h - G,, (x + h) is a continuous polynomial of degree don H. Clearly, C = (G )n t is a version of F. In particular, G(x) E E for y-a.e. x. Obviously, the polynomial mappings (C,,... , Gn, 0, 0, ...) converge to G pointwise (hence in measure) if they
are regarded as 1R"-valued mappings. Each of these mappings it is the limit of
Chapter 5. Sobolev Classes
254
a sequence of continuous finite dimensional polynomial mappings of degree d cond
vergent in measure, since G, E ® X,,. Therefore, G is a limit of a sequence of k=0
continuous finite dimensional polynomial mappings Q,,: X IR' of degree d convergent in measure. Passing to a subsequence, we may assume that G(x)
a.e. in the topology of IR". We shall show that there is a Borel version Fo of G (hence of F) such that Fo(x + h) E E for all h E H and x E X. Let us assume first that the polynomial mappings Qn are homogeneous. Let us denote by W. the corresponding continuous symmetric d-linear forms on Xd. Note that the forms Li'n are uniquely determined and can be evaluated by (5.10.1) as follows: (-1)d-c,-- -,,Qn(x0+-1x1 +...+-dXd), (5.10.2)
Wn(xl,... xd) = I E c, E{0.1}
where Xo E X is an arbitrary element. Let us put
B := {x:
lim Qn(x) E E
n
_X
where the limit is taken in 1R". It is readily verified that B is a Borel set. Clearly,
y(B) = 1. Let us put Go(x) = lim Qn(x) if x E B and Co(x) = 0 if x ¢ B. Since the mappings Q, are homogeneous, we have AB = B and Co(ax) = \dGo(x) for
nx
any x E B and A > 0. Therefore, for every rational number r, we get
1 = y(B) = y((r2 + 1)-1/2B) =I
- rx) -y(dx),
X
since the measure -y equals the image of yey under the mapping
(x,y) '- r(1 +r2)-1/2x+ (1
+r2)-1/2y.
Hence there exists a set Cr E 8(X) such that -y(B - rx) = 1 for all x E Cr. Letting C = lrCr, we have y(B - rx) = I for all x E C and all nationals r. Clearly, C is a full measure Borel set. Passing to a full measure subset, we may assume that C is a countable union of metrizable compact sets. Then we have
y(n(B-rx-sh))
= 1,
Vx E C.Vh E H.
r,s
where the intersection is taken over all rationals r and s. Let us now fix c E C and h E H. By choosing xo E lr.,(B - rx - sh), we get xo + rx + sh E B for all rationals r and s, i.e., lim Qn(xo+rx+ax) E E for all rational r and s. It follows n-.x by identity (5.10.2) that, for every j E (0,... , d), there exists the limit V,(x,... , x, h, ... h),
n-ic
where (x, ... , x, h, ... , h) stands for the vector in X d with the d- j first components x and the last j components h. Indeed, K n(x, ... , x, h, ... , h) can be written as
F,
C,n.q,)Qn(Xo + mx + qh),
m,qE{0,....d}
where the coefficients c,,,,,,) are some absolute constants depending only on m, q, j. d. Therefore, we get V, (x, ... , x, h, ... , h) =
>
m. qE {0.....d}
cm,q,jGo(xo + mx + qh ).
255
Measurable polynomials
5.10.
This shows that V, (x, ... , x, h, ... , h) E E for all x E C and all h E H. By construction, for every x E C, the mapping h H Vj (x, .. , x, h,... , h), H -+ E, is a Borel homogeneous polynomial. According to Example 5.10.2, this mapping is continuous (hence it is also continuous as a mapping to Y). Note also that
x + H C B for every x E C. Indeed, if x E C and h E H, then Qn(x + h) is a x, h, ... , h) with some absolute coefficients
finite linear combination of
independent of n. Hence lim Qn(x + h) exists in IR" and belongs to E (since n-x the limits lim 1Vn (x, ... , x, h.... , h) belong to E). It remains to redefine Go by
n-x
zero outside C + H. which gives a version Fo with the desired properties (note that C+H is a Borel set, since both C and H are countable unions of compact metrizable sets). Let us consider the general case where Qn may not be homogeneous. Then d
Q. = E Qn.k, where Qn.k is a homogeneous of order k continuous polynomial k=0
mapping. We shall assume that H is infinite dimensional (otherwise the claim is n
trivial). Let us put pn = n-1 E e,2, where {e,} is an orthonormal basis in H with =1
e, E X'. As we know, pn - 1 a.e. Therefore, there exists a sequence {in} such that (Pi.. - 1)[Qn,o +'_ + Qn.d-2] - 0 in measure. Thus, p,,, [Qn,o+ +Qn.d-2]+Qn.d-1 + Qn.d - G in measure. Repeating this procedure and noting that p,,, Qn,k is a continuous polynomial mapping of order k + 2, we arrive at the situation with Qn,k = 0, k = 0, ... , d-2. Passing to subsequences, we may assume that the corresponding mappings converge almost everywhere. In this situation, letting A be the set of convergence of Qn.d(x) + Qn,d-1(x), we note that (-A) fl A has full measure. As a consequence, Qn,d(x) - Qn,d-I(x) converges
a.e., hence {Q,.d } and {Qn.d-1 } converge a.e., which reduces the claim to the homogeneous case considered above. By Proposition 5.10.3, IIF0IIE E
L2(-y).
Suppose now that (iv) is fulfilled. Clearly, then (iii) is fulfilled as well. Let us take a version Fo constructed in (iii) and taking values in a separable Banach space E continuously embedded into Y. It suffices to approximate F in measure by continuous polynomial mappings F. taking values in E. Let {en } be an orthonormal basis in H such that en- E X. Since E is a separable Banach space and IIFoIIE is integrable as shown above, we have Fo E L'(-t, E). Hence we can take for Fn the finite dimensional approximations constructed in Corollary 3.5.2, i.e.,
Fn(x) = f F'(>e-,(x)e, + X
t=1
t=n1
F,(y)e+)'y(dy)
n x Note that Fn is a continuous polynomial mapping, F, (x)e, + E since F. (y)e, E
=1=n+1
C for every x E X and every y E C, where C is the full measure set constructed above. With this choice, we have IF,. - FIIE - 0 in all LP(-y). 0
5.10.8. Definition. Denote by Pd(y,Y) the class of mappings satisfying either of conditions (i) - (iv) in Proposition 5.10.7. Let us put Pd(-y) := Pd(-t, IRl ). The mappings from the class Pd(-y, Y) are called -y-measurable polynomial mappings of degree d.
Using this new notation, we have Pd(-y) = Xo e
e Xd.
Chapter 5. Sobolev Classes
256
Now we prove two zero-one law type results for polynomial mappings.
5.10.9. Proposition. Let y be a centered Radon Gaussian measure on a loC Pd(y.Y).
cally convex space X, let Y be a separable Fi-chet space, and let Then the following sets have measures zero or one:
L: _ {x: nx lim FF(x) exists}, Al: _ {x: lim F,(x) = 0}. n-x PROOF. Clearly, L and Al are -y-measurable. Suppose that y(L) > 0. Let us choose an orthonormal basis (e;} in H(y). Let L, be the set of all points x E L such that the set {t E R': x + te, E L} has positive Lebesgue measure. Obviously, y(L\L,) = 0. Hence, letting Lo = n,, IL one has I (Lo) = -y(L) > 0. We shall work with versions of Fn which are polynomial of degree d along all vectors e,. Note that if a sequence of polynomials of degree d on R' converges at d+ 1 points, then it converges pointwise. Therefore, for any x E Lo, one has x + te, E Lo for all t E IR' and i E V. By the zero-one law, y(L0) = 1. The same reasoning applies to the set M.
5.10.10. Proposition. Let y be a centered Radon Gaussian measure on a locally convex space X and let F: X -+ (Y,A) be a -y-measurable mapping with values in a linear space Y equipped with a a-held A. Suppose that {en) is an orthonormal basis in H(y) such that, for every n E IN and y-a.e. x. the mapping t - F(x + ten) is a polynomial. Then, for any linear subspace L C Y such that L E A. one has -r (X
EX
:
F(x) E L) = 0 or y(x r= X: F(x)
E
L) = 1.
In particular, this is true if Y is a separable A chef space. L E B(Y) is a linear subspace and F E Pd (-Y, Y) -
PROOF. Suppose that the set f2 = F-1(L) has positive y-measure. The claim is trivial if X = R', moreover, in this case S2 = R'. Indeed, if F is a polynomial on the real line with values in Y such that F(t) E L for infinitely many values of t, then F(t) E L for all t. In the general case, let us put
fln = {x E 0: mes(t E R': F(x + ten) E L) > 0 where "mes" is Lebesgue measure. Clearly, y(i2n) = y(f2) (this is easily seen from Theorem 3.10.2 about conditional measures). Letting 11o = nn , iln, we have -r(no) = y(f2) > 0. By the one dimensional case, x+ten E i2 for all x E iho, t E R' and n E IN. Hence ibo + Rlen = N. By the zero-one law, y(120) = 1. 5.10.11. Corollary. Suppose that (T, fit, a) is a separable measurable space
and that (WsET is a measurable centered Gaussian process. Let Q(t, z) _ n E ck(t)zk, where the ck's are measurable functions on T. Then k=1
either
fQ(tE.4Fa(di) < oo T
a.e.
JQ(t.t)Ic'(dt) = cc
or
a.e.
T
The latter is equivalent to f E K{(t, t)k121ck(t)I o(dt) = oc, where K; is the cor k=1 variance function of .
Measurable polynomials
5.10.
257
PROOF. Let us put e(t) _ (1 + E Ick(t)IK{(t, t)k'2)
1
and consider the mea-
k=1
sure A = p or. We shall apply Proposition 5.10.10 to the separable Banach spaces X = Ln (Jn), L = L' (a) and Y = L' (.1). According to Example 3.11.14, the process generates a centered Gaussian measure on the space X = L"(A). Let us denote
this measure by pt. The mapping F: X -. Y, F(x)(t) = Q(t,x(t)), is continuous and polynomial of degree n. The space L is continuously embedded into Y. Hence
µF o F-'(L) is either 0 or 1. In the latter case, IIFIIL E L'(p{). Let ry, be the standard Gaussian measure on the line. By the equivalence of all norms on the finite dimensional space of polynomials of degree n on 1R', there is do > 0 such n n that IlgllL'(,) ? do E Iakl, whenever q(s) = E aksk. Therefore, k=1
k=1
f IIF(x)II,. ut (dx) = IE
f
X
T
IQ(t, t) I a(dt)
f
= Tf f-xIQ (t, KE(t, t)1/23) I y1(ds) a(dt) > dok='E T
I ck(t)I KE(t, t)k/2 a(dt).
Conversely, if the integral on the right is finite, then the integral on the left is finite as well.
5.10.12. Example. Let (wt)j>0 be a standard Wiener process, let f be a measurable function on [0, 11, and let Q(s) _
cksk, where ck E IR' and m >_ 1. k=m
Then
f
1r
1
either
If (t)Q(wt)I dt < oo a.e. or
0
J0 If (t)Q(wt)I dt = oo a.e.
1
The latter is equivalent to f If (t)Itm dt = oo. 0
Let us discuss second order measurable polynomials on a locally convex space X with a centered Radon Gaussian measure ry having the Cameron-Martin space H = H(-y). It follows from Corollary 5.10.4 that the class of all measurable functions that are continuous second order polynomials along H is precisely P2(-y). In the finite dimensional case, any second order polynomial is written as Q + I + c, where Q is a quadratic form, I is a linear function, and c is a number. The same representation is valid for continuous second order polynomials in infinite dimensions. However, for measurable polynomials in infinite dimensions, there is no natural
way of separating "quadratic forms" from constants. Indeed, let {6n} C X' be an orthonormal sequence in X. Then the continuous finite dimensional quadratic forms S,, = n-' E fk converge a.e. and in L2(ry) to 1. Thus, we arrive at the class k=1
X0 ®X2 as a reasonable candidate for the space of measurable quadratic forms in infinite dimensions.
Chapter 5. Sobolev Classes
258
5.10.13. Proposition. Let F E P2(-y). Then there exist an orthonormal basis
It,,) C X. , two sequences {an } E 12 and {c,, } E 12 and a number c such that F=c+
Cntn + E a,,(yn - 1) F, n=1 n=1
y-a.e.,
(5.10.3)
where both series converge y-a.e. and in all LP(-y). Conversely, given {cn}, and c with the aforementioned properties, both series in (5.10.3) converge y-a.e. and in all LP(y) and F E P2(-y).
PROOF. By Proposition 5.10.6, DH F(x) = A a.e., where A is a symmetric
Hilbert-Schmidt operator on H. Note that D F(x) = A(x) + v a.e. for some constant vector v E H, since D,, A = A. Let {en} be an orthonormal basis in H consisting of the eigenvectors of A corresponding to eigenvalues an. Then one has (an) E 12. Put C. = en, an = a,/2, c = (F,1)L2(.,), and C. = (v,en)x. Then E 12. Let us define F0 by means of the right-hand side in (5.10.3).
Since lE(l;n - 1) = 0 and E
112 < oo, the corresponding series converge
n=1
almost everywhere by a classical result due to Kolmogorov and Khintchine (see [697, Ch. IV, §2, Theorem 11). By Corollary 5.5.8, we have convergence in all LP(y).
Clearly, D Fo(x) = A(x) + v a.e., for one has rOe Fo(x) = cn + 2ant n(x) a.e. and anon = Ae,. Since F and Fo have equal integrals, we get F = F0 a.e. The last claim has already been proved. Note that F = c + v - z6A. 0
5.10.14. Remark. It follows from the results above that the elements of W2.1 (-Y) X0 G X2 ("measurable quadratic forms") are precisely the functions F E orthogonal to X1 in L2(y) such that D, F(x) = A a.e. for some symmetric HilbertSchmidt operator A on H. This corresponds to c = 0 in (5.10.3).
5.10.15. Remark. Note that. if F E Xo 6X2 is nonnegative, then in (5.10.3) one has X
an > 0.
>0.
Ean <00, n=1
n=1
Indeed, the conditional expectation of F with respect. to the a-field generated by m equals c+ E an (fin - 1) and is nonnegative, whence the claim follows n=1
immediately. since the polynomial m
m
m
n=1
n=1
C+an(xn - 1) = Cn=1
on lR'n is nonnegative almost everywhere precisely in the case, where the numbers
an and c - E an are nonnegative. In particular. D,22 is a trace class operator. n=1
This shows that an element F E Xo X2 with D, ,2F not of the trace class cannot be written as F, - F2 with nonnegative F1, F2 E XoDX2; in particular, this is true 1). Clearly, the condition E 4(1)(H) is also sufficient for F = E n=1
for the existence of the decomposition into a difference of nonnegative second order
5.10.
Measurable polynomials
259
polynomials: we take
x
x
F, =Io(F')+-n +n S.1 C
n=1
n=1
where a,+, and an are nonnegative and negative eigenvalues of D,2F corresponding
in the eigenbasis {en} and n = en. 5.10.16. Proposition. Let Q be a sequentially continuous quadratic form (in the usual algebraic sense) on a locally convex space X with a centered Radon Gaussion measure -y. Then S := DH Q is a symmetric trace class operator and them exist two nonnegative -t-measurable quadratic forms Q1 and Q2 such that Q = Qi - Q2.
More precisely, there exists an orthonormal basis {tn} in X. such that, letting an be the eigenvalues of 1S, one has Q=
anSn
a.e.
n=1
In addition, the restriction of Q to H = H(y) has the form -1 (Sh, h),,. and the following equality is valid: JQ(s)"Y(dx) = 2trace D,2Q.
(5.10.4)
X
PROOF. There exists a symmetric bilinear function T on X such that Q(x) _ M
basis in H. Then lim E en(x)en = x for
%P(x,x). Let {en} be an
m-x n=1 y-a.e. x E X according to Theorem 3.5.1. By the sequential continuity of Q, we get m
Q(x) = mime Q( E en(x)en)
(5.10.5)
a.e.
n=1
Clearly, m
m
Q(en(x)en) = FQ(en)en(x)2+2 n=1
n=1
E 41(en,ek)en(x)ek(x) nvik.n,k<m
By Corollary 5.5.8, the quadratic forms in (5.10.5) converge not only almost everywhere, but also in L2(-y); then one has Q E P2(y) and is a symmetric Hilbert-Schmidt operator on H, which we denote by S. Now let {en} be the eigen-
x
basis of S. Then Q(x) _ E Q(en)en(x)2 a.e. Integrating this equality term by n=1
term (which is possible by its convergence in L2(y)) and taking into account the relationship (Semen),, = aeQ = 2Q(en), we conclude that the series E (Se,,,e,,)H converges and its sum is the same for all n=1
permutations of the eigenvectors en of S, which shows that E (Se.,en)H < oc, n=
i.e., S is nuclear. In particular, one gets (5.10.4). Note that we could deduce the first claim from Proposition 3.7.10, which shows that the operator R, o AIH is 2R, o Al H. 0 nuclear, making use of the equality
Chapter 5. Sobolev Classes
260
It is worth noting that unlike the case of a Hilbert space, not every continuous quadratic form on a general Banach space can be decomposed into a difference of two nonnegative continuous quadratic forms (see Problem 5.12.46). Another example is this: let Y be a reflexive Banach space that is not linearly homeomorphic to a Hilbert space (e.g., Y = L'4[0,1]); then the continuous quadratic form Q(x, x') = x' (x) on XxX' cannot be represented as a difference of two nonnegative continuous quadratic forms. This follows from [800, Ch. III, §1, Exercise 3d].
5.10.17. Example. Let -y be a Radon Gaussian measure on a locally convex space X and let Q E P2(y) such that expQ E L1(y). Then the measure
µ=
(JeQ&v)'exPQ . y x
is Gaussian.
PROOF. Suppose first that expQ E 1P(y) with some p > 1. Let Qn be the finite dimensional approximations of Q constructed in Corollary 3.5.2. Then Q,, is an exponentially integrable second order polynomial in s , ... , t,, (a finite sum of
second order Hermite polynomials in ti, i = 1,... , n). Note that a second order polynomial on R has the form F(x) = (Ax, x) + (x, v) + c, where A is a symmetric operator, v E IR", c E IR1. Such a polynomial is exponentially integrable with
respect to the standard Gaussian measure y,, if and only if A < 1/2. Clearly, in this case the measure II exp FII ti (yn exp F yn is Gaussian. This shows that the measures vn = II exp Q II L, t-,) exp Q,, y are Gaussian. Since the sequence { Q } converges to Q a.e. and the sequence {exp Qn } is uniformly integrable by the estimate
f exP(pQ,) dy < f exp(pQ) dy which follows by Jensen's inequality for the conditional expectations, the measures vn converge to µ in the variation norm. Hence p is Gaussian. In the general case, let us put cn = I - n-1. Then the measures µ, = II exP(c,Q)IIL, (.,) y are Gaussian. It follows by Lebesgue's dominated convergence theorem that converge to expQ in LI(y), whence the claim. 0
5.10.18. Example. Let y be a Radon Gaussian measure on a locally convex
space X and Q E P2(-y). Then there exists e > 0 such that exp(rQ) E LI(y). Moreover, this is true for any e < IID, Q(IC('H).
PROOF. This is readily seen from Proposition 5.10.13 and equality (4.8.5). 0
5.10.19. Example. Let -y be a Radon Gaussian measure on a locally convex space X and let e E L1(y) be such that the measure p: = p y is Gaussian. Then logp is a second order measurable polynomial, i.e., log Lo E P2 (-f).
PROOF. The claim is obvious in the finite dimensional case. Let
be an
orthonormal basis in H(y), where n = en E X', let P,,(x) _ E {i(x)ei, and let p, be the conditional expectation of p with respect to the a-field generated by P and the measure y (see Corollary 3.5.2). Note that p,, = r o P,, a.e., where rn is the density of the measure p o PI I with respect to y o P,,- 1. By the finite dimensional case, rn = exp F,,, where Fn is a second order polynomial on IRn. Recall that {pn }
5.11.
Differentiability of H-Lipschitzian functions
261
is a martingale convergent to p in LI(ry) and almost everywhere. Therefore, the continuous second order polynomials F,, o P converge a.e. to log o (note that o > 0
0
a.e.). Hence log B E P2(-y).
Suppose we want to investigate the distribution of a continuous quadratic form Q on a locally convex space X with a centered Radon Gaussian measure -y. An algorithm suggested by the preceding discussion is this: we write the restriction of Q to the Cameron-Martin space H = H(-) as (Alt. h)5, where A is a symmetric Hilbert-Schmidt operator and find the eigenvalues of A. To be more specific, let X = C[0,1]. -y = Ptt', and let I
Q(x) = fp(t)x(t)2dt.
where p E L1 [0. I].
0
Then the restriction of Q to H(PII) = ll o'' [0.1] is given by (Ah, h)H(pw). where t
t
Ah(t) _ ff p(u)h (u) du ds, which is verified by the integration by parts formula (recall that ( , ti)N(pit i = (pp'.t:")t. o,ij). Hence we arrive at the boundary value problem
Ah"(t) = -p(t)h(t).
h(O) = 0, h'(1) = 1.
Measurable polynomials on the classical Wiener space can be described as the multiple stochastic integrals defined as follows. We shall take the Wiener space (C[O.1]. P") as a probability space. Let U E L2([0. 1]"), where [0.1] is equipped with Lebesgue measure. Since Z"(u) E X,, for simple kernels it. the same is true for arbitrary kernels (see the definition of the multiple stochastic integral Z"(u) in Section 2.11). Thus. 1, (it) is a measurable polynomial on the space (C[O,1]1 Ptt ). Conversely, any P't'-measurable polynomial can be written as a sum of multiple stochastic integrals. This can be seen in several different ways. For example, one can verify that the linear span of the multiple integrals of the simple functions is where the dense in X. and then use (2.11.10) to show that if a sequence u,'s are simple kernels, converges in L2(Ptt'). then {u,} converges in L2([O.1J"). Another possibility is to show that, given an orthonormal basis {yP,} in L'2[0' 1], the H ermite
,(t) dwt and n I +... + n,. = n, f where u is expressed as a certain product of
polynomial f1 H", (, ). where
=
coincides up to a factor, with the functions ,%",(t,) (see [399, §6.6]).
5.11. Differentiability of H-Lipschitzian functions The classical Rademacher theorem states that any Lipschitzian mapping F from IR" to IR k is Frechet differentiable almost everywhere. This result has no direct extensions to the infinite dimensional case. The principal reason is not the lack of infinite dimensional analogues of Lebesgue measure, but just the existence of Lipschitzian mappings between Hilbert spaces that have no points of the Frechet differentiability at all (although as shown in [625], any real-valued Lipschitzian function on a Hilbert space has a point of the Freshet differentiability). However, the Rademacher theorem can be reformulated (in the finite dimensional case) in
Chapter 5. Sobolev Classes
262
equivalent ways that admit infinite dimensional generalizations. We discuss here one of such possibilities. The proof of the following theorem is completely analogous to the proof of Theorem 5.11.2 below.
5.11.1. Theorem. Let X be a separable normed space and let F. X -- Y be a locally Lipschitzian mapping with values in a Banach space Y with the RadonNikodym property. Then F is Gateaux differentiable (and Hadamard differentiable) everywhere, except. possibly, at the points of some Borel set, which is zero with respect to every nondegenerate Radon Gaussian measure on X.
In particular, this implies the F chet differentiability along any compactly embedded normed space E. According to Problem 5.12.23. the Gateaux differentiability in Theorem 5.11.1 cannot be replaced by the Frechet one along the whole space X. However, admitting to consideration the differentiability along a smaller subspace. it would be quite natural to impose the Lipschitz condition only along this subspace. This leads to the following question. Let F be a mapping from a locally convex space X to a Banach space Y with the Radon-Nikodym property, measurable with respect to a nondegenerate Radon Gaussian measure p on X, such that, for IL-almost all x. one has the estimate 11 F(x + h) - F(x)I),. < CIhIc,
V h E E.
where E is some normed space continuously embedded into X. Is F differentiable along E p-a.e.'. It is shown below that this question is answered positively for the Gateaux differentiability and negatively for the Frechet differentiability.
5.11.2. Theorem. Let p be a Radon Gaussian measure on a locally convex space X. H = H(p). let Y be a separable Banach space with the Radon Nikodym property, and let F: X - Y be a measurable mapping such that p-a.e. one has I{F(x + h) - F(x)II,. < CIhl,.
Vh E H.
(5.11.1)
Then:
(i) there is a set n with fl + H(p) = 1 and p(1) = I such that (5.11.1) holds true for every x E !l: in particular, there exists a modification of F satisfying (5.11.1) for all x: (ii) p-a.e. there exists the Gdteatcr derivative D, ,F and II D F(x')II c(H.v) S C
for p-a.e. x. (iii) F E GP-' (1, Y) for all p E (1. z).
PROOF. Put H = H(p). Assertion (i) has already been proved in Lemma be an orthonormal basis in H and let H. be the linear span of
4.5.2. Let
its first n elements. Let X. be any closed subspace in X algebraically complementing H,,. On the finite dimensional subspaces y + H,,, y E X,,, one can choose conditional Gaussian measures absolutely continuous with respect to the natural Lebesgue measures on these subspaces. Therefore, by virtue of the finite dimensional Rademacher theorem (which applies due to the Radon-Nikodym property of Y). the Gateaux derivatives DH F exist p-a.e. Let M be the set of all those points x at which all Gateaux derivatives DH F(x), n E IN, exist. Let us show that F is Gateaux differentiable along H at any point a E Al. Note that on the linear span L of all subspaces H,, we have a well-defined linear mapping G: h - OhF(a), which is continuous by virtue of (5.11.1), hence extends uniquely to an operator
5.11.
Differentiability of H-Lipschitzian functions
263
G E G(H, Y). Let h E H. Let us choose a sequence {hn } C L with Ih,, - hl, The estimate
F(a + thn) - F(a)
limit
x, the vectors
F(a + th) - F(a) t
F(a + th) - F(a) < C(h II
t
t
implies that, as n
0.
F(a + thn) - F(a)
- 8r,, F(a) converge to the
L
- G(h) uniformly in t. Therefore. im t-0
F(a + t h) - F(a) t
= G(h).
which means the Gateaux differentiability. Clearly,
C a.e. c(H.y) Let us prove (iii). By the integration by parts formula, it suffices to show that F E LP(ry,Y), because then the Gateaux derivative X -- C(H.Y) serves as the generalized derivative. Note that the mapping OhF is measurable for every h E H, since it coincides I-a.e. with the limit of the measurable mappings n(F(x + n-1 h) - F(x)). Since Y is separable Banach, it suffices to show that the Y-norm of F is in LP(-y). This follows from Theorem 4.5.7, since the function x -' IIF(x)IIy is H-Lipschitzian. It is clear from the proof given that the statement remains valid if H is replaced by an arbitrary normed space E that is linearly embedded into X in such a way that E contains a countable everywhere dense set from H. 5.11.3. Corollary. If the conditions in Theorem 5.11.2 are satisfied. then. for any normed space B compactly embedded into H. the Frechet derivative D. F exists p-almost everywhere.
In general, the last corollary is not valid for the space H itself. X
5.11.4. Example. Let X = IR", it = ® JA,,. where p,, is the standard Gaussn=1
ian measure on the real line, let H = 12. and let F: X 12, F(x) = where the fn's are 21_"-periodical functions on the real line such that fn(t) = t
if t E (0,2-"[, f,. (t) = 21-" - t if t E 12-n. 21-i(. Then H = H(µ) and F is Lipschitzian along H. but at no point is Frechet differentiable along H. This example can be easily modified to make F everywhere Gateaux differen-
tiable along H. Certainly, we could take some Hilbert space X instead of IR". In [86) there is an example of a probability measure p on IR" such that it is quasiinvariant along H = lz and the function f (x) = sup,, Ix,, I is p-a.e. finite and Lipschitzian along H, but a-a.e. is not Frechet differentiable along H. It was conjectured in [230) that a similar example exists also for a Gaussian measure. Although this conjecture seems likely to be true, we have no such examples.
5.11.5. Remark. Recall that, by Corollary 4.5.4, if condition (5.11.1) is fulfilled for every h E H on a full measure set dependent on h, then F has a modification, for which (5.11.1) is fulfilled for all x E X and h E H simultaneously. The following example is borrowed from [230).
5.11.6. Example. Let '1 be a centered Radon Gaussian measure on a locally convex space X and let S C X' be compact in the topology o(X'. X). Then, for -y-a.e. x, the function f - f (x) on S attains its maximum at a unique point.
Chapter 5. Sobolev Classes
264
PROOF. Put V(x) = suppEs f (x). The set S is bounded in X; by Problem 4.10.18, hence supf£S f(h) < CIhI for some C. Therefore, the function yp is H-Lipschitzian. Let x belong to the full measure set of the points where there exists the Gateaux derivative of
along H. We observe that there is a unique g E S
such that g(x) = cp(x). Indeed, suppose that there is k E S such that k g and k(x) = a(x). Then g(h) > k(h) for some h E H. We have ,;(x + th) > g(x + th) and ,.(x - th) > k(.r - th), whence 8h;p(x) > lim t-' [g(x + th) - g(x)] = g(h)-
On the other hand, 8hya(x)
5.11.7. Theorem. Let y be a centered Radon Gaussian measure on a locally convex space X. H = H(y). let Y b a separable Banach space, and let F E GP,'(y,Y). where p > 1. Suppose that JIDnF(x.)IIc,N.y, < C a.e.. for some constant C. Then F admits a modification Fu such that
IIFo(x+h) - Fo(x)II, < CIhI dx E X,Vh E H. In particular, exp(o1IFII;.) E L'(y) for all a < (2C2)
.
PROOF. Let It E H. By Proposition 5.4.2, there is a modification 1j) of F such that the mapping GJ : t +- Fo(x + th) is locally absolutely continuous. In addition, for a.e. x, the derivative of this modification in t coincides with 8hF(x. + th) (the generalized derivative) for a.e. t. Since IIahFII,. < C A.P., we conclude that Gr is Lipschitzian with constant CIhI,,, in particular, II F0(x + h) - F)(x)JI,. < CIhI,,.
Since Fi, = F a.e. and It E H(y), we obtain that F)(x+h) = F(x+h) a.e.. whence JIF(x + h) - F(r) II1. < CIhI , a.e. It reinains to apply Lemma 4.5.4. 5.11.8. Corollary. Let f E be such that I D f I < C a.c. Then f has a modification fo with I fo(x+h) - fo(x)] < CIhI for allx E X and h E H(y).
5.11.9. Remark. It should be noted that JID FIIK need not be uniformly bounded a.e. even if F E Ii'21(y. H) is H-Lipschitzian. Indeed, let be the countable product of the standard Gaussian measures on R' and H = 12. Put F(x) = (2-"fn(xn))
where f, E Co (R' ), 0 < f., < 1, If I S 2". the set {t: f'(t) = 2") contains a nontrivial interval J. for every n E N. and Ilfn IIL1 .., r < 1. where y, is the standard Gaussian measure on IR'. Then F is H-Lipschitziau and belongs x 6i'2.' (y H). since I I D F(x)IJx = to 2 '"J f,' (x") 2 is y-integrable. However, n-1
is unbounded on any full measure set, since IIDF(x)lll > k on the set { x : (X,,... . xk) E J1 X
X JA. } having positive -,-measure.
Now, following (420), we discuss an interesting modification of the classical problem of extending a Lipschitzian mapping f, defined on a subset A in a normed (or just metric) space X and taking values in a normed space Y. One of the first results in this direction was obtained by McShane 1539], who proved that any real
Lipschitzian function f on an arbitrary subset of a metric space X extends with the preservation of the Lipschitz constant to the whole space (the corresponding
5.11.
Differentiability of H-Lipschitzian functions
265
extension is given by a simple explicit formula). The situation is more complicated for multidimensional mappings. For example. it may happen (see [1831) that X and Y are Banach spaces, but some mapping f : A -- Y has no Lipschitzian extensions
at all (even without any restriction on its Lipschitz constant). One of the best known positive results is Valentine's theorem (see [183)), which states that any Lipschitzian mapping f. defined on a subset of a Hilbert space X and taking values in a Hilbert space Y. has an extension to all of X with the same Lipschitz constant.
5.11.10. Theorem. Let X be a Souslin topological vector space (e.g.. a separable Frechet space). E C X a linear subspace equipped with a norm II IIE such that the ball CITE = f h E E : II h II E < 11 is a Souslin set in X. let A C X be some Souslin set. and let f : A IRt be a function. which has the following properties:
a) the sets {x E A: f (x) > c} are Souslin. b) for all h E E and r E A such that x + h e A. one has (5.11.2)
I f(x + h) - f(x) I< AIIhIIE.
Then the function f extends to a function F: X -+ lRt , hating property a) (in particular. universally measurable. i.e. measurable with respect to every Borel measure
on X) and satisfying inequality (5.11.2) for all r E X. h E E. Put A,. = (x + E) n A. Obviously. the function defined by the formula
F(x) = sup [f (y) - allx - YIIF], yE.a,
whenever Ar is nonempty. and F(r) = 0 if Ar is empty. has property (5.11.2) and, for each x E A. coincides with f (x) (since f (x) > f (y) - .llx - yll E if x, y E A). We have to prove that the sets {x: F(x) > c} are Souslin (whence it follows that the function F is universally measurable). Note that the foregoing formula is a straight forward generalization of McShane's formula [539]. It is clear that in the definition of F, when taking sup. one can replace A,. by A. Therefore, F(x) = 0 if r does not belong to the set Xo = A + E, which is Souslin as the image of the Souslin set A x E in X x X under the continuous mapping (r. y) -+ r + y (note that the fact that the unit ball in E is a Souslin set in X implies that so are all the balls in E, hence also E itself). Let us extend II 11E to X. letting I'IIE = +x if r ¢ E. For any number r, the set {r E X: IIZIIE < r} is Souslin. since it coincides with the ball of radius r in E. By the continuity of the mapping X xX - X, (r, y) - x - y, the set {(x, y) E X x X : llx - yllE S r} is Souslin for any number r. Hence, for all c E IR', the set {(r. y) E Xo x A: f (y) - ally - yII E > c} is Souslin as well, since it is the union over all rational r of the Souslin sets {(x. y) E XoxA: f(y) > r}n{(x.y)
E XvxA: r > c+allx - yIIE}
Put G(x, y) = f (y) - allx - YII E X EXo, y E A. It is clear that {x E Xo : F(x) > c} = p/l { (x, y) E Xo x A: G(r, y) >
c}lI ,
where p: X xX - X is the natural projection onto the first factor. Therefore, the set {x E X,): F(x) > r} is Souslin. 9
Chapter 5. Sobolev Classes
266
A related but weaker statement was proved in [792J, where it was shown that if f is a function, measurable with respect to a Gaussian measure µ on a separable Banach space X. defined on a p-measurable set A and satisfying on it the Lipschitz condition along the Cameron-Martin space H, then there exists a it-measurable function on X, satisfying the same Lipschitz condition and equal f p-a.e. on A. This statement follows from the theorem above, since f has a Borel modification, satisfying the same condition on some Borel set B C A with µ(B) = p(A). In fact. due to existence of Souslin supports for Gaussian measures. this result extends to general locally convex spaces X. Let us mention several open problems related to the results presented above.
(i) Does there exist a Lipschitzian function f on a separable Hilbert space, whose set of all points of the Frechet differentiability is zero with respect to all nondegenerate Gaussian measures? (ii) Let it be a centered Gaussian measure on a separable Hilbert space X and let f be a real Borel function on X, which is Lipschitzian along H = H(µ). Can it happen that the set of all points of the Ichet differentiability of f along H is ii-zero?.
(iii) Let B be a Borel set in a locally convex (say, in a separable Hilbert) space X equipped with a Gaussian measure ry with H(y) = H. What can be said about the points of Frechet differentiability of the function dB. defined in Example 5.4.10, along H?
5.12. Complements and problems Generalized Poincarh's inequalities In Chapter 1, several generalizations of the Poincare were presented. All those results extend immediately to the infinite dimensional case. For the reader's convenience, let us give the corresponding formulations. The next result follows from Proposition 1.10.3 (certainly, it can be proved directly by the same reasoning).
5.12.1. Proposition. Let -t be a centered Radon Gaussian measure on a loThen, denoting by E the integrals with
cally convex space. X and let f E respect to y. we have
(E!)2=
1,)-
(l)AE(IID,,f
( l}i}, J 2e2NtE(IITTD.fII;dt.
k=n
o
5.12.2. Corollary. Suppose that f E W2.2"(7). Then 2n-1
((k+11
2n
E -1) k=1
A.1
lE(IIDN !I'}(k)
(f ) \- (E!)2 C E -11 (
,k-11
//
_)
k=1
5.12.3. Corollary. Suppose that f E
IR
(II DN III itR).
1 and klIIDH lllVo-wk) -, 0.
Then
(ffd)2 = .t
k,yk IIDH f11 2 k=p
The following is the infinite dimensional version of Proposition 1.10.6.
5.12.
Complements and problems
267
5.12.4. Proposition. Let y be a centered Radon Gaussian measure on a locally convex space X and let f E 11'4 2(y) be positive a.e. Then, for every s E [0, 1]. one has l' r f (Lf)2dy+s 1 f IDHfI2 dy-s(1-s) 1 IDf21
f X
X
f
dy,
X
X
provided all the integrals exist. If f E
a
11'2.2(7) is positive
a.e.. then
dy 5 f (Lf)2dy+sf /IDfl. dy,
X
X
X
provided the last integral exists.
Beckner's finite dimensional result [47) mentioned in Chapter 1 extends automatically to the infinite dimensional case and yields the following generalized where y is a centered Radon Gaussian Poincar6 inequalities: let f E measure on a locally convex space X and let 1 < p 5 2, e-t = . Then
f If 12dy-f X
fIfI2dy.\
le-t1. fI2dy 5 (2-p) f ID.fI2dy, X
X 2iP
(f IfIPd-) X
<(2-p)J ID"fI2dy. X
Compactness in Sobolev classes Embedding theorems for Sobolev classes play a fundamental role in the classical
analysis. A typical example of an embedding theorem is the fact that the natural over a bounded region D C IR" into the embedding of the Sobolev space space L2(D) is compact. In the case D = lR" the same is true for many weighted Sobolev classes. Another typical result of this sort states that any element of %'(W) has an infinitely differentiable modification. Neither claim is valid in the infinite dimensional case. For example, if the measure y on X = 1R" is the countable product of the standard Gaussian measures on 1R', then, for any nonzero smooth function with bounded support on the real line, the sequence of functions f"(x) = y,(x") is bounded in W2 t(y). but is not precompact in L2(y), since the mutual distances between f" are all equal. A measurable linear functional which has no continuous modification gives an example of a function from 14''(y) without continuous modifications. We shall present two results concerning the compactness conditions in the Sobolev classes: more detailed proofs can be found in [175], [176). More general results are found in [271).
5.12.5. Theorem. Let y be a centered Radon Gaussian measure on a locally convex space X, let H = H(-y), and let K be an injective symmetric compact (y) such that operator on H. Then, for any c > 0. the set F of all f E D, ,f E K(H) y-a.e. and IIfIIL-(,) + IIK-'DMfIIL7(,.H) 5 c. is relatively compact in L2(y). PROOF. Let {e" } be an orthonormal basis in H formed by the eigenvectors of
K, corresponding to the eigenvalues k". Put l;" = e". Denote by A the set of all finite sequences (gj,g2.....q".0.0,...) with nonnegative integer q,. As we know,
268
Chapter 5.
Sobolev Classes
x
the functions H. = n Hq,(E,,). q E A. where Hq, is the q,-th Hermite polynomial. ,=1
form an orthonormal basis in L2(1). Suppose that F =
cgHq satisfies the qE.\
condition II F II L=r, , + II K-' D FII L2{,.H) < c. Since
D F = F E cq
'q H, t (e) [f Hq,
qE.\ ,: q,>O
J
,
we get the equality x
III
1
D,,L'i,.H1 111 2
= Eeq [E qk2 qE.\
The estimate E c2 [l+
,=1
t
q,/kz, < c implies; that.F is precornpact in L2(-,) (recall
that a subset A of Hilbert space E with an orthonormal basis {:,2A} is precompact precisely when it is bounded and the series F_,, .(a. :,X.)2 converges uniformly in
0
aEA).
This theorem and Problem A.3.39 in Appendix yield the following statement.
5.12.6. Theorem. Let , be a centered Radon Gaussian measure on a locally ronvex space X and let the Sobolev class If"2 X(y} = n,,,1 be equipped with its natural topology of a Frechet space by means of the seminorms II ' Ilz..,
The
set F C 1 l"2-' (y) is relatively compact if and only if it is bounded in L2(-) and, for every n _> 1. there exists an injective symmetric compact operator K on ?-1 such that
Vf E F D," f E K,(N,,) 1-a.e. and sup IIK 1D,'fj/.Ji'..N,- < x.
fEF Let the class l1'X(1) be equipped with its topology of a Frechet space by means of the seminorrns II p. n E W. The set F C ll'X (7) is relatively compact if and only if the following two conditions are satisfied: (i) sup l[f llp., < x. Vp > 1. r > 1: fEF
(ii) For any n > 1, them exists an injective symmetric compact operator K on
?t such that V f E F D,;' f E
1-a.e. and
sup fEF
< x.
It should he noted that the Sobolev classes can be defined for derivatives along Hilbert subspaces E C X different from H(-f), i.e.. one can consider the completions
of FC' with respect to the norm IIfIIN.x.E::_
.e,l, provided
the closability condition is satisfied. i.e.. any sequence that is Cauchy with respect to the norm II IIp.k.F: and converges to zero in La(y). converges to zero with respect
to the norm I. ilp.k.F Embedding theorems for these more general classes are obtained in [2711. The corresponding capacities are of interest as well (in general, such capacities are not tight). These objects arise, e.g., in connection with linear stochastic differential equations. In general. there is no log-Sobolev inequality for such classes. For more details, see 12711, [276).
5.12.
Complements and problems
269
Negligible sets Let us recall several concepts of a "zero-set" in the infinite dimensional case. Since in this case there is no reasonable substitutes for Lebesgue measure (as well as any preference in the choice of, say. a distinguished nondegenerate Gaussian measure among the continuum of mutually singular measures), one has to introduce this concept without making use of any specific fixed measure. One of the definitions of this sort is due to Christensen [165], who suggested to call a Borel set A in a Banach space X universally zero if there exists a nonzero Borel measure µ such that µ(A+x) = 0 for all x. Certainly, this definition applies also to locally convex spaces, so that in this subsection we deal with a locally convex space X. Another definition is due to Aronszajn [22]. who introduced the following class Ao of exceptional Borel sets. For every vector e in a locally convex space X, let
us denote by Ao the class of all Borel sets A such that mes(t: x + to E A) = 0 for every x E X, where mes is Lebesgue measure. For any sequence in X, let As{en} be the class of all sets of the form A = where A. E AB, for all n. 5.12.7. Definition. A set A is called exceptional (A E A6) if it belongs to the for every sequence with the dense linear span in X. The corresponding class is smaller than that of Christensen, but both coincide with the class of all Borel sets of Lebesgue measure zero in the finite dimensional spaces. Then Phelps (599] introduced the class go of Gaussian null sets. class
5.12.8. Definition. A Borel set A is called a Gaussian null set if it is zero for every n.ondegenerate (i.e., having full support) Radon Gaussian measure on X.
According to [599], A6 C Cg. but it remains open whether this inclusion is strict. Finally, in [69] the following definition was introduced. 5.12.9. Definition. A Borel set A is called negligible if it is zero for every Radon measure. which is differentiable along vectors from a dense set. The class of all Borel negligible sets is denoted by PG. The classes introduced so far can be extended in such a way that in the finite dimensional case they will embrace all sets of Lebesgue measure zero (not necessarily Borel). To this end, let us denote by C the class of all sets in X, which are measurable with respect to every Radon measure on X which is differentiable along vectors from a dense subspace (dependent on the measure). Then Definitions 5.12.7
- 5.12.9 extend naturally to C (in particular. in the definitions of A, and the sets from C are now admissible). Let us denote the classes obtained in this way by A, G, and P, respectively. Then the following relationships hold true (see [70], [71], [72] for the proof).
5.12.10. Theorem. One has As C 9' = Po and A = B = P. Note, in particular, that go C A. however, it is open whether the sets A in the corresponding decomposition can be chosen in 8(X) (and not only in C). Similar classes A' and 9c; arise if. instead of C, we consider the class CG of all sets measurable with respect to all nondegenerate Radon Gaussian measures on X. Then, by virtue of the same reasoning as in [71], (72], one has A' = C9G. We have no examples distinguishing the classes C and CG (or the classes GG and P). The class of negligible (or Gaussian null) sets is invariant with respect to affine isomorphisms of X and possesses a lot of other useful properties of finite dimensional Lebesgue zero sets (see Theorem 5.11.1). However, it is not stable with
Chapter 5.
270
Sobolev Classes
respect to nonlinear diffeomorphisms (see Chapter 6). Let us mention an interesting open problem posed in [834]: is it true that the image of any Borel negligible set under a Lipschitzian mapping in a Banach space is universally zero in the sense of Christensen?
Laplacian AH Let X be a locally convex space with a centered Radon Gaussian measure y. In applications, besides the Ornstein- Uhlenbeck semigroup, one encounters the semigroup (P,),>o defined on the Banach space C,,(X) of uniformly continuous bounded functions on X (with the sup-norm) by the formula
Pif(x) = ff(x + V y),7(dy)5.12.11. Proposition. (PP),>o is a strongly continuous sernigroup onC,(X). PROOF. The semigroup property follows from Proposition 2.2.10 and equality
a2 + 32 = 1, where o = t112(t + s)-1I2, 0 = st/2(t + 3)-1/2. Let f be a bounded uniformly continuous function on X and let e > 0. There exists an absolutely convex neighborhood of zero V such that If (x) - f (z)l < e whenever x - z E V. Clearly, IP f (x) - P, f (z)I < e. Further, there is n such that -y(nV) > I - e. If 0< t< n-2, then f y E V for every y E nV, whence, for every x E X,
If(x)-P=f(x)1 <
rIf (x)-f(x+rty)l (dy)<_r+2supIfJE. .x
Therefore, (P,)t>o is strongly continuous.
0
Note that the situation is different for the Ornstein-Uhlenbeck semigroup (see Problem 5.12.27). On the smooth cylindrical functions, generator g of this semigroup is given by the equality 9f = ! AH f , where A,, := trace, D,? f . Choosing an orthonormal
basis {e,,} in H, we get AH f = E 8.f. The Laplacian OH, unlike the Ornst.ein-
n1
Uhlenbeck operator L, is not closable in L2(-Y), which makes it a more complicated object for the investigation. It is easy to deduce from the results above that if a
Borel function f is such that fIf(x + f y) I P y(dy) < coo for some p > 1, then the function h
f f(x + h + f y)y(dy)
is infinitely Frechet differentiable on H and its derivative of order n is an n-linear Hilbert -Schmidt mapping. Precise estimates of the Hilbert-Schmidt norms of these derivatives are found in [474).
Measures of finite energy The collection of sets of capacity zero is much smaller than that of measure zero. For example, a closed hyperplane in a locally convex space X with a nondegenerate Gaussian measure y has measure zero, but positive capacities. On the other hand, there exist measures mutually singular with y, but vanishing on the sets of capacity
zero (we shall see in Chapter 6 that surface measures generated by sufficiently nondegenerate Sobolev class functions have such a property). Therefore, it is of interest to describe all Radon measures vanishing on every set of C,.,-capacity zero
5.12.
Complements and problems
271
(such measures are called measures of finite Cp,r-energy). Measures of finite energy can be characterized by means of positive generalized functions. More precisely, we have the following two results.
5.12.12. Theorem. Let v be a nonnegative Radon measure on X such that v(A) = 0 for each Borel set A with Cp,r(A) = 0. Then. there is a strictly positive bounded Borel function p on X such that the functional
f -. J f (r) p(r) v(dx) is continuous on HP.' (y). X
PROOF. Let h(t) = sup{v(A): Cp,r(A) < t). Note that limh(t) = 0. Indeed, otherwise for some c > 0 there is a sequence of Borel ' sets 0 A. such that Cp,r(An) < 2-" and v(An) > c. This leads to a contradiction, since for the set B = limsupA,, = n,,>, Uk>,, Ak one has v(B) > limsupv(Ak) > c and Cp.r(B) < r Cp.r(Ak) < 21n11 for any n, whence Cp.r(B) = 0. k>n
We shall apply the following result due to Maurey (see, e.g., [800, Lemma 5.5, §5, Ch. VI]). Let C be a convex set of v-measurable nonnegative functions which is bounded in the space L°(v) of all v-measurable functions equipped with the metric If (X) - g(x)I v(dr). r 1+If(x)-g(x)I
!
Boundedness of a set Al means that, for every ball V centered at zero, there is A > 0 such that Af C W. Then there exists a strictly positive measurable function p such that sup f(x) 9(x) v(dr) < 1. fEC
J
Let us take for C the set
C = { f E H'(): IIf
1. f > 0}. Note that all functions in HP-'(-t) are v-measurable. Indeed, -y-equivalent functions coincide as elements of HP,'(-r), and every function f E HP-'(-y) possesses a quasicont.inuous modification g. By condition, given e > 0, there is 6 < e such that v(A) < e provided Cp,r(A) < 6. Since there is a closed set Z with Cp,r(X\Z) < 6 on which g is continuous, we get v(X \Z) < e, whence the measurability of g with respect to v. Note that v(x: f (x) > r) < h(r) if IIf Ilp.r << r2. since
Cp.r (x: f (x) > r) < r-' IIf II,.r < r. Thus,
j 1 + f(x) v(dx) < h(r) + r, f (x)
Vf E r2C,
x
which, together with the continuity of h at zero, means that C is bounded in the space L°(v). Let p be a function from Maurey's lemma cited above. Finally, note that, for any function f with IIfIIp,r < 1, the function g = Vr(IV, 1(f)I) is nonnegative, IIgIIp.r = Ilf1Ip.r < 1, and If I < g quasi-everywhere. Hence, ff()o(x)v(dx) < 1 g(x)p(x)v(dx) < 1.
Therefore, the functional f ' - f f(r)p(x) v(dx) is continuous on Hpr(y).
Chapter 5. Sobolev Classes
272
5.12.13. Theorem. Let '' be a linear functional on FC' continuous with respect to the norm of HP,'(-y) and nonnegative in the sense that (1I'.,') > 0 for every non negative smooth cylindrical function y.. Then there exists a nonnegative Radon measure v p on X such that (I1+, 0) = I ;p(x) vy(dx).
V
E .FC" .
X
See [7401, 14121 for the proof of this result and its extensions.
5.12.14. Remark. A Borel set B has C,,,,-capacity zero if and only if every Radon measure of finite (p. r)-energy vanishes on B (see 14121 for a proof).
More on measurable polynomials The arguments used in the proof of Theorem 5.10.7 yield the following result relating measurable polynomials to the classes Sd(y) introduced in Section 4.3. Note that according to the results above, every real-valued y-nteasurable polynomial of degree d has a version F,1 that belongs to Sd(h) and Ad (h) coincides with OFII(.r) (which is a constant).
5.12.15. Corollary. Let 7 be a Radon Gaussian measure on a locally convex space X. let Y be a separable FHchet space. and let F E Pd(-i.Y) Suppose that and .,(Ax) = ) (x) for : Y - (-x.+-lc] is such that ,(x + y) <- w(x) + all x, y E Y and A > 0 and that p(F(x)) + ;rs(-F(x)) < +x v-a.e. Then F has a version Fe such that ;p o Fe E Sd(,) and A°= Y o Fd, where Fd is the d-homogeneous part of the polynomial mapping h
JF(x + h) ti(dy). X
Finally, one has 1
!
lim t- E'2 log 7(,,; o F > t) = - ( sup Fd(h)) +a-.
(5.12.1)
2 hEtH
I
exp(aIir'oFI2!d) E L1(7),
ba < (2 sup IFd(h)I) hEt'H
.
(5.12.2)
Let us give a formulation of the results on the small ball comparisons obtained in Chapter 4 in terms of the second derivative along H.
5.12.16. Proposition. Let .t be a centered Radon Gaussian measure on a where q,, e X_ . locally convex space X and let Q = E n=1
a , < oc. Then n=1
fQr)Y(d.r)
_ trace(5.12.3)
X
In particular, this is true if Q is a sequentially continuous quadratic form on X.
5.12.
Complements and problems
273
PRoor. Recall that the last claim has already been shown (see (5.10.4)). Clearly, for any n E IN and h E H, one has Olin = 2(rin.h)y.),). Let {e,} be an orthonormal basis in H. Recall that D Q is a nuclear Hilbert --Schmidt operator (see Remark 5.10.15). Then (5.12.3) by follows by the absolute convergence of the series x x x aalltln E : an(rln, FOL. (.) = 1=1 n=l
n=1
where the Parseval equality was used.
5.12.17. Theorem. Let y be a centered Radon Gaussian measure on a locally convex space X, let q be a 7-measurable seminorm such that V. = {q < e} has has summable positive measure for all e > 0. and let Q E P2(y) be such that (possibly, to -oc or +x) trace trace
X
X
,=1
n=1
2a; +2ro,,.
where 2a ; and 2an are, respectively, the positive and negative eigenvalues of DN'Q. and at least one of these two series converges to a finite number. Then liml
y(V)
JexPQ(z)'(dx) = exp(Io(Q) - trace
(5.12.4)
1.
The limit in (5.12.4) is 1 if Q is a sequentially continuous quadratic form.
As it follows from the results in Chapter 4. if Q does not satisfy the aforementioned condition, then, for a suitable norm q, there is no limit in (5.12.4).
5.12.18. Example. The assertion in Theorem 5.12.17 (hence also in Theorem 4.8.3) is valid if Q = c + f + Qo, where c E 1111, f E X; , and Qo is a sequentially continuous quadratic form on X. Let us now discuss the case where q is a seminorm which may not be a norm
on H. To this end, let us denote by F the a-field, generated by the sequence 1f, ,j that determines the seminorm q, and by IE- the conditional expectation with respect to F. In addition, let us denote by 0 the a-field, generated by the orthogonal complement to if,) in X. and by lE-' the corresponding conditional expectation. Note that if q is a norm on H, then IEFf = f and 1E° coincides with the expectation with respect to 7. In terms of the restriction of q to H, the a-field F is the one generated by h, h E Z. where Z = Kerq, and IE" is the one generated by h. h E Z. The proof of the following result is found in (84).
5.12.19. Proposition. Suppose that the conditions in Theorem 5.12.17 are satisfied except that now q is a y-measurable seminorm which may not be a norm H. Then
linJ.(7+Q) = 11eXp(lE Q)IIL (.i exp(-Ztrace
(5.12.5)
This limit is finite and positive if is a nuclear operator. In particular. if p is a Gaussian measure on X such that dµ/dy = exp Q, then there exists the limit lira p({q 5
s}).
Chapter 5. Sobolev Classes
274
According to (569], for any strictly positive absolutely continuous function i2 I
such that
has bounded variation on [0.1) and J ,p(t)-2dt = 1, one has u
Pit' (x: Ix(t)l < -ap(t), ''t E [0, 1]) lim E_0
- ,tl)
Ptr x: sup Ix(t)] < e)
(0)
(5.12.6)
This result can be deduced from Theorem 4.8.3. To this end. note that the nominator on the left can be written as Ptt' (x: q(Tx) < <), where
rr(t)x(r(t)).
Tr(t) = and r is the inverse function to
3(t) _
J(s)_2ds.
(5.12.7)
Indeed, the inequality ]x(t)] < tap(t), Vt E [0.1]. is equivalent to Ix(r(t))I
r'(t) < Vt E [0, 11. since Y(r(t)) _ ep(T(t)), Vt E [0,1], hence to r'(i) by virtue of (5.12.7). According to Example 6.5.2. the measure PII' oT- I is equivalent to P" and TT' - I E N(H). Then T = (I + K)U, where U is orthogonal and K is symmetric Hilbert -Schmidt. The operator U preserves n. Hence Theorem 5.12.17 is applicable if K is nuclear. Observing that K` is nuclear
we arrive at the following condition: the quadratic form (Th,Th) - (h, h) on H is generated by a nuclear operator. The calculations in Example 6.5.2 yield the following representation: I
(Th,Th)" - (h, h),,
f(t)h(t)h'(t)dt + J0(t)h(t)h(t)dt. 0
0
where
T" (3(t))
r'(3(t))2
B(t) =
1 T"(3(t))2 4
,(,3(t))'
The second integral on the right defines a quadratic form generated by a nucle operator, since it is a restriction of a continuous form on C(0,1]. Observe that
r'(3(t))t?'(t) = 1 and r"(3(t))3'(t)2 + r'(3(t))3"(t) = 0. whence r"(3(t)) = 2,p'(t)ap(t)3. If ap' has bounded variation, then ti does also. Hence the first in-
tegral is a restriction of a continuous quadratic form on C[0,1). which implies that it is given by a nuclear operator on H. However, according to [558]. the limit in (5.12.6) exists under the weaker condition gyp' E L2]0,1). Let us now describe one more widely used way of introducing measurable polynomials. Let E be a separable Hilbert space and let 7 (H. E) be the space of
n-linear Hilbert-Schmidt mappings from H' to E. Denote by J,,,, the mapping that associates to every' E 11 (H, E) the operator Jn,r41 E
Nr(H,fn-,(H. E)),
5.12.
275
Complements and problems
IJ41(h,....h,)](k...... k"-r) = 41(ht.....hr,kl.... ,k"-,) By induction, one defines the mappings 3,: fl (H. E) - X"(E) by the relationships
31(A)(x) = A(x) If E = IRl and the functional g on H is defined by the vector h, then 31(g) = h. 5.12.20. Theorem. The mappings 3" are well-defined and surjective. and one has
d- =n!(W,e)H..(H.E).
(5.12.8)
In addition, if the mapping is symmetric with respect to the permutations of its arguments. then one has D [3"('F)j = n3i_1[(,In.n-141)]-
In order to prove this statement, it suffices to apply the integration by parts formula to verify the validity of the announced equalities for the finite dimensional
mappings, which enables us to extend the definition of 3 to all mappings from 9.1"(H, E) with the preservation of these equalities. The details of the finite dimensional case can be found in [178. Ch. 11, §31. Having proved that the mappings 3n are surject.ive in the finite dimensional case, we get the surjectivity in the infinite dimensional case from relationship (5.12.8).
Besides polynomials, one can consider polylinear measurable mappings, i.e.,
the mappings %P: X' -. R, measurable with respect to the measure y" = y, such that, for 7 1-a.e. fixed y = (yl,... , y1), the function x 'y"_ 1, x) is a measurable linear functional and the same is true for all permutations of its variables. Such functions can be described as the limits in L2(y") of finite dimensional continuous polylinear functions. The simplest examn -1 x"yn on the product IR." x R' equipped with n.1 the measure 7Fy, where y is the countable product of the standard Gaussian measures on the real line. For y-a.e. y, the function x -+ Q(x, y) is a measurable linear
ple: the function Q(.r, y) _
functional. since E
n_2yn
< oc a.e.. which follows by the monotone convergence
n=1
theorem. In particular, the domain of convergence of the series defining Q has full
measure. Note that Q(x, x) _
x
rr-l
n -1 xn = oc a.e. We do not know whether there
exists a function Q1 on IR" x IR" that is bilinear in the usual algebraic sense and Q1 = Q a.e. with respect to -y . This question is equivalent to the question posed by H. von Weizsiicker about. the existence of a bilinear function Ql on the space CIO, 1) x C[O. l[ such that 1
Q1 (.r. U.)
= fr(t)dw(t)
for a.e. (x. uw) with respect to the square of the Wiener measure (the integral is understood as the stochastic one).
Chapter 5. Sobolev Classes
276
Problems 5.12.21. Show that the mappings (5.1.1) and (5.1.2) are everywhere Hadamard differentiable, but nowhere Freshet differentiable.
5.12.22. Let F be a Lipschitzian mapping from a hall in a normed space X to a normed space 1'. Prove that if F is Gateaux differentiable at a point x. then it is Hadamard differentiable at this point.
5.12.23. Let X be an infinite dimensional separable Hilbert space and let p be a 2-"diet (z, C ). where C" are absolutely convex Borel measure on X. Put f (x) _ ".1
compact sets such that p(X\ U 1 Cr,) = 0. Show that f is convex and Lipschitzian. but is not Freshet differentiable p-a.e. Hint: prose that the distance function to C,, is not Freshet differentiable on C"; see [6251. [6271.
5.12.24. Let a Radon measure p on a locally convex space X be differentiable along a vector h. (a) Show that for every smooth cylindrical function f one has the equality ,
f [f(x+th)- f(x))p(dx)=-f f f(x+sh)dhp(dx)ds. \ oX (b) Deduce the estimate 11p,,, - pll S It[ IldhpII implying the continuity of p along h.
5.12.25. (cf. [129). [1551). Let. y be a centered Radon Gaussian measure on a locally convex space. H its Cameron-Martin space. f E W2-1(-.). (i) Show that for every h E H one has f 8h f dry < IhIH Ilf
(ii) Show that
f
1IL01
_<,!f-d-,'
Hint: (i) note that the left-hand side equals f fhdy: (ii) use the same idea as in (i) and apply Bessel's inequality to
where {e,, } is an orthonormal basis in H.
5.12.26. Suppose that a -y-measurable function f has the stochastic Gateaux derivative DH f which belongs to Lt'(y, H) for some p > I. Show that f E LP (y), hence f E D"-'(-Y). Hint: consider the sequence of bounded functions f = g,, o f, where 9,, (t) = min(max(t, -n).n). 5.12.27. Let be a centered Radon Gaussian measure on a locally convex space X and let (T,), >n be the Ornstein-Ublenbeck semigroup. (a) Show that T,p is a bounded uniformly continuous function for every bounded uniformly continuous function w. (b) Show of bounded uniformly conthat (T,)t=o ?s not strongly continuous on the space tinuous functions with the sup-norm if dim X > 0. (c) Let X be a normed space. Show that (T,),>,, is strongly continuous on the Banach space C,.,,(X) of all bounded uniformly continuous functions f on X such that f (x) - 0 as IIxil - oo equipped with the sup-norm.
Hint to (b): consider the semigroup Srf(x) = f(e`x). prove the claim for (S,),> and note that suplT,f(x)-S,f(x)I 0. f 5.12.28. Construct an example of a function F E f l G1''" (-,) that does not belong p."
to the class lt.2 2(y), Hint: use Example 5.4.18 and consider F(x) suitably chosen c and p": see [596).
5.12.29. Prove identity (5.10.1). Hint:
sec [68).
with
5.12.
Complements and problems
277
5.12.30. Show that for a continuous function F on a Banach space X, the following conditions are equivalent: (i) F is a polynomial: (ii) for every fixed a. b E X. the function t F(a + tb) is a polynomial. Hint: apply Baire's theorem to the sets M. of those pairs (x, y) E X x X, for which the degree of the polynomial t - F(x + ty) does not exceed n: see [681.
5.12.31. Let -y be a centered Radon Gaussian measure on a locally convex space X and let f be a measurable polynomial of degree d. Show that the function
g(h)=J f(x+h)y(dx) is a continuous polynomial of degree don H(y) with the norm I
5.12.32. Let H in the previous problem be infinite dimensional. Show that f can be represented in the form f = fd + fd-1. where fd is the limit of a sequence of continuous homogeneous polynomials of degree d convergent in measure y, and fd_ 1 is the limit of a sequence of continuous homogeneous polynomials of degree d -1 convergent in measure y. Hint: see the proof of Theorem 5.10.7.
5.12.33. Let {{, } be a sequence of independent standard Gaussian random variables a,.,. Show that the and let i, j E N. be real numbers such that 0 and
sequence S.
converges a.e. if and only if
a2 < oc. Hint: evaluate
1E[Sn], apply Theorem A.3.5 in Appendix and use the fact that the convergence of {S"} a.e. implies its convergence in L2.
5.12.34. Let y be a centered Gaussian measure on a separable Hilbert space X and let Q(x) = V (x, ... , x), where V is a continuous k-linear function on X. Prove that the Fourier transform of the measure v = Q y has the form A(`q). where A is a differential operator of order k. Find an explicit expression for A via V. 5.12.35. Let y" be the standard Gaussian measure on IR" and let f be a nonconstant polynomial on IR" such that its degree in every coordinate does not exceed m. Show that
e" f(( )-y.(dx)c(f)Itr
I:.,, .
III
R^
Hint: see [2951, where an explicit expression for c(f) is found.
5.12.36. Let f E L2(R1) and F(x, w) = r f (2: + u<<(W)) dt, 0
where w1 is a standard Wiener process. Prove that for a.e. w, the function F(-,.;) belongs to the Sobolev class (in particular, is locally absolutely continuous). Obtain
a multidimensional analogue. Hint: consider first the case f E CU (R1) and estimate lE
J
y2 l F(y, w) I2 dy, where F( , w) is the Fourier transform of F( -
. v j).
5.12.37. Complete the proof of Corollary 5.4.14. Hint: see 17401. 5.12.38. Let y be a centered Radon Gaussian measure on a locally convex space and let f E W2.1 (-Y). Suppose that the Poincare inequality becomes an equality for f. Show
that f = c + 1, where c is a constant and I E X. (in the one dimensional case this was noted, e.g., in [158]). Hint: use the expansion f = E I^(f) in the Wiener chaos and note ^=0
Chapter 5. Sobolev Classes
278
that
whence it follows that every function /n(f)
n=n
also satisfies the equality in the Poincar6 inequality; use that II D,r9II izc,.N) =
for anygEX". W2.1 (1. H) 5.12.39. Show that for every f E L2(7) with f d-y = 0. there is v E J such that by = f. Hint: show that for every finite linear combination f of nonconstant Hermite polynomials on R", there is v E W2"(ryn,R") such that by = f and IIv[I22.1 < 2IIJII2. To this end, notice that if k, > 1, then Hk,.....k" equals bu with n 1
(
1l
,=l
k,)
IInIi2.1
2-
,=1
5.12.40. Let 1 be a centered Radon Gaussian measure on a locally convex space X and let .4 and B be two symmetric Hilbert-Schmidt operators on such that AB = 0. Show that 6A and 6B are independent random variables on (X. -y). Show by example that this is not true for non symmetric operators. Hint: note that A and B have a common orthonormal basis {e,,}, hence 6A and 6B are simultaneously represented as c + E a (e - 1). Note that if bA and 6B are independent, then .4B = 0 (see [413, n=1
15.13] or [786. Theorem 6.4]).
5.12.41. Give an example of a capacity on the real line. which is not tight.
5.12.42. Prove that Gaussian C2,1-capacity of a point in R1 is positive, but is zero in R3 (however, the C2.2-capacity of a point is positive also in R3).
5.12.43. Show that the Gaussian capacities of the Cameron-Martin space H equal zero if dim H = oc. Hint: take an orthonormal basis {fn} in X, such that fn E X' and show that C;_(SR) - 0 as n oc for every R > 0, where SR = {x: E f,(x)2 < R}.
5.12.44. Show that every function f E W' (-y) has a modification f' that is Cy,.quasicontinuous for all p > 1, r > 0. Hint: use Theorem 5.9.6. 5.12.45. Let 1 be a centered Radon Gaussian measure on a locally convex space X and let T be a proper linear mapping with values in a locally convex space Y such that T is (B(X ), , B(Y)) -measurable and -, o T-1 is a Radon measure. Prove that T is CD,.quasicontinuous for all r > 0, p > 1. In particular, this is always the case if Y is Souslin. Hint: in the case where Y = R", the claim follows from the analogous statement for linear functionals discussed above. Reduce the general case to the case where Y is Souslin taking a Souslin linear subspace of full measure for Is = -y oT-1. Then find a continuous
linear injection S: Y -» R'"; verify the statement for S-1: S(Y) -» Y making use of Corollary 5.9.11 and the continuity of S-1 on S(K) for every compact K c Y. 5.12.46. Construct an example of a continuous quadratic form on a Banach space, which cannot be represented as a difference of two nonnegative continuous quadratic forms. Hint: find a sequence of two dimensional Banach spaces (X,,, II II n) with quadratic forms
Qn such that 1 on the unit ball U,,. but the positive part of Qn takes value 2" on U,,: consider the space of all sequences x = (x,,). xn E X. IIxII = supIIxnlln, and the
form E n-2Qn. n=1
CHAPTER 6
Nonlinear Transformations of Gaussian Measures That which changes changes either from subject into subject. or from non-subject into non-subject. or from subject into nonsubject, or from non-subject into subject. Aristotle. Metaphysics
6.1. Auxiliary results This section contains a collection of well-known facts from the measure theory on Souslin spaces. Although these facts are used below only for the spaces which are countable unions of metrizable compact sets, the proofs are given in the general case,
since this does not influence the corresponding reasoning. To simplify notation, our discussion concerns nonnegative measures, but, by virtue of the Hahn-Jordan decomposition, the same results hold true for signed measures as well.
6.1.1. Lemma. Let (bl,M,µ) be a measurable space and let T: M M be a µ-measurable mapping such that the sets T(N) and T-'(N) have measure zero for every set N of measure zero. Suppose that there exists a µ-measurable mapping
S such that T(S(x)) = S(T(x)) = x for µ-a.e. x. Then there exists a set f2 of full µ-measure such that T maps Il one-to-one onto itself (and S is its inverse) and, in addition, T(M\S2) C 161\Q.
PROOF. It follows from the condition that µ o T-' ti u. Denote by 110 the
set of all points x such that T(S(x)) = S(T(x)) = x. The mappings T and S are obviously injective on 1 o. Let A = M\00. By assumption, the set T(0) has measure zero. Further, the set T-' (T(0)) has measure zero as well. Hence T is a one-to-one mapping of the full measure set f21 = ilo\T-' (T(0)) onto the full measure set T(fl1). In addition, T(M\f 1) C dl\T(S21). Since it - It o T'' and p (T-' (S21)) = µ o T-' (f21), we see that the set T-' (521) has full measure. Put Zo = 521 n T-'(521). On the set Zo of full measure, the mapping T is injective, S(T(Zo)) C S21 and S(T(x)) = x. Hence, for any B C Zo with B E M, one has T(B) = S-1(B)nZo E M,,. Since T takes all sets of measure zero to sets of measure zero, it follows that T takes all it-measurable sets to µ-measurable sets. By the equivalence of the measures µ and µ o T-'. we conclude that, all sets of full measure
are taken by T to sets of full measure. For all integers k, we define inductively the sets Zk by the equalities Zk = Zo n T(Zk_ 1) if k > 1, Zk = Zo n T-' (Zk+l) if k < -1. It follows from what has been said above that the sets Zk have full measure.
Let us now put 52 = nk" ,c Z. This set has full measure and it is straightforward to verify that T takes ft one-to-one onto itself, whereas T(M\S2) C M\52. Indeed, T and S are injective on Q. Let w E Q. Then T(w) E Zo n T(Zk) for all k > 0, since w E T-'(Zo) and w E Zk. In addition. T(w) E Zo n T-'(Zk) for all k < 0, since w E Zk_2. On the other hand, w E Z1, i.e., w = T(u), u E Zo, whence u E 1, since Tk(u) = Tk-'(w) E ci, k > 1, and T'(u)nZo = Tk-t(w)nZo, k < 0. Finally, 279
Chapter 6.
280
Nonlinear Transformations
if T(x) = w for some x, then x E 521i since T(AI\521) C AI\T(521) C M\T(52). By the injectivity on 521, one has x = S(w) E Q.
The following statement has been obtained in the proof above.
6.1.2. Corollary. The mapping T from Lemma 6.1.1 takes all p-measurable sets to p-measurable ones.
6.1.3. Lemma. Let T be a one-to-one mapping of a space with a measure p such that the mappings T and S = T-1 are measurable and p o T-' ti p. Then
d(paS`')(x) =
1
g(T(x)) PROOF. Since T = S-1, we have dp
where p= d(poT-1) du
poS-'(B)=p(T(B))= f !d(PoT-') = f IT(B)oT T(B)
I
dp'
X
whence the claim follows, since IT(B) a T = IB.
6.1.4. Lemma. Let X and Y be two Souslin spaces, let p be a Borel measure on X, and let F: X -» Y be a p-measurable mapping, i.e., F-'(8(Y)) C B(X),,. Then there exists a Borel mapping Fo : X -. Y, which coincides with F p-a.e.
PROOF. Since the spaces X and Y are Souslin, we have that the measures p and p o F-' are concentrated on countable unions of metrizable compact sets (see Appendix). Hence the claim reduces to the case where X and Y are compact metrizable. In this case F is uniformly approximated by p-measurable mappings with finitely many values. For such mappings the claim is obvious. This gives a sequence of Borel mappings convergent to F uniformly on a Borel set of full measure, whence our claim follows.
6.1.5. Lemma. Assume that p is a Borel measure on a Souslin space X and F: X - X is a Borel mapping such that
p(B) = p(F(B))= p(F-'(B)), dB E B(X).
(6.1.1)
Then there exists a Souslin set Xo C X of full p-measure such that F is one-to-one
on Xo, F(Xo) = Xo and F(X\Xo) C X\Xo. PROOF. A well-known theorem of Lusin states that the set of all values y with a unique preimage under a Borel mapping F on a Souslin space X is the complement to some Souslin set A (see [507, p. 232, Ch. 4[ or [344, p. 119, Ch. III, §6, Theorem 2)). Therefore, the set N = l y E X : card F-' (y) > 1 } is Souslin in X, since it equals An F(X). In fact, Lusin proved his theorem for subsets of the real line, but this yields the general case. Indeed, there exist a continuous injection of X into lR" and a Borel injection of IR" into 10, 1]. Therefore, their composition is a Bore] isomorphism of X with a Souslin subset of 10, 11. In particular, it would
be enough to prove our claim for X C [0, 11. Let us show that p(N) = 0. By the measurable selection theorem, there exist a Borel set No C N with p(No) = p(N) and a Borel mapping 4i: No -. F''(No) such that F(4i(y)) = y, Vy E No (see, e.g., [217, Appendix 3, p. 254]). Removing from F-1(No) the set 40(No), we get a measurable set A, for which F(A) = No according to our choice of N. By condition, have the same measure p(No), since both are taken the disjoint sets A and
6.1.
Auxiliary results
281
onto No by the mapping F. The image of A U I'(No) under F is No as well. Hence the measure of A U 4 (No) equals µ(No), whence p(No) = 0, since A and 4'(No) are disjoint. The set Y = X \F-' (N) has full measure, and the mapping F is injective on Y. Let us choose a Borel set Z C Y of full measure. As in the proof of Lemma 6.1.1, put Zo = Z n F-1(Z) and, for every integer k, define inductively the sets Zk of full measure by the relations Zk = Zo n F(Z-_ 1). k > 1, Zk = Zo n F-1(Zk.1), k < -1. Then the intersection Xo = n+", Zk is a Souslin set of full measure, F is injective on X0 and F(Xo) = X0, which is readily verified. Since F is injective on Y and Y is disjoint with F-1(N), we see that F(X\Xo) C X\X0. Indeed, if x % X0
.
and F(x) E X0, then there is xo E Xo with F(xo) = F(x), whence xo E F- I (N), which is impossible, since Xo n F-1(N) C Y n F-1(N) = 0. 6.1.6. Corollary. The conclusion of Lemma 6.1.5 remains valid also for any p-measurable mapping F such that F(B) is p-measurable for all B E 8(X) (which is the case for any Bore! mapping), provided condition (6.1.1) is satisfied. PROOF. In order to apply Lemnia 6.1.5, it suffices to find a Souslin set Y of full
measure such that F is Borel on Y, F(Y) C Y and F(X\Y) C X\Y. By Lemma 6.1.4, there exist a Borel set No of p-measure zero and a Borel mapping F0 which coincides with F outside No. The set F(NO) has measure zero, hence it is covered by a Borel set At() of measure zero. Put Co = X\A1o. Then F-1(Co) C X\No. Thus,
we get Borel sets Co and B0 := F-'(Co) of full measure such that F(Bo) = Co. F(X\Bo) C X\C.o and F is Borel on B0. By analogy with the proof of Lemma 6.1.5, we show that the sets N := {y E Co : card F-' (y) > 1 } and F` 1(N) have measure zero and find a full measure Borel set Yo C Bo\F-1(N). As above, we put
Yk=F(Yk_1)nYoifk> 1, Yk = F-1(Yk+i)nY0 ifk<-1. LetY=nk",,Y,. All the sets Yk are Souslin (since F is Borel on Bo) and have full measure. Therefore. Y is a Souslin set of full measure. By construction. F(Y) C Y. Finally, using that F is injective on Bn\F-1(N), one readily verifies that F(Bo\Y) C C.o\Y, whence
F(X\Y) C X\Y. 6.1.7. Remark. Let (Af, M, p) be a measurable space. A mapping F: Al -Af is said to satisfy Lusin's condition (N) (or have Lusin's property (N)) if F takes every set of p-measure zero to a set of p-measure zero. It is known from the standard course of the theory of functions of real variable that even a homeomorphism of 10, 11 may fail to satisfy Lusin's condition (N). On the other hand, the reader can verify that any everywhere differentiable function on the real line satisfies Lusin's condition (N). Lusin's condition (N) is fulfilled for any injective p-measurable map-
ping F on a Souslin space such that poF-1 - p. This is seen from the equality µ(B) = uoF-1(F(B)). In order to justify this equality, note that if F is Bore], then F(B) is measurable for B E 8(X)1 since X is Souslin. The same is true for measurable F. since there exists a Borel mapping Fo, which coincides with F outside some Borel set C of measure zero, and, in addition, F(C) has measure zero by
virtue of the inclusion F(C) C X\F(X\C), where F(X\C) = Fo(X\C) has full measure with respect to paF-', hence also with respect to p. Let p be a Borel measure on a Souslin space without points of positive measure. Problem 6.11.11 suggests to prove that a Borel mapping F: X X satisfies Lusin's condition (N) if and only if it takes every p-measurable set to a p-measurable set.
Certainly. Lusin's property (N) may be lost if we replace a mapping by its modification.
Chapter 6.
282
Nonlinear Transformations
6.1.8. Lemma. Let X be a completely regular topological space with a Radon probability measure p, let T : X X be a p-measurable mapping such that the measure p o T-' is Radon. and let TT : X X be a sequence of p-measurable mappings convergent p-a_e. to T. Suppose that the measures poT,-' are absolutely continuous with respect to p and that their Radon-Nikodym densities p,, form a uniformly integrable sequence. Then the measure p o T-' is absolutely continuous with respect to p as well. In addition, its Radon-Nikodym density p is the limit of the sequence { p } in the weak topology of the space L' (p).
PROOF. Let K be a compact set of p-measure zero and e > 0. The uniform integrability ensures the existence of b > 0 such that
f
pn(x) p(dx) < e,
do E 1N.
for any measurable set A with measure not greater than b. Let us find an open set U D K with p(U) < b. By Lemma A.1.2, there exists a continuous function f : X [0.1], which equals 1 on K and 0 outside U. Then one has
fl(s) )t o T '(dx) = f f (T(x)) p(dx) = X
liym
= lim
f
f (T. (x)) p(dx)
x
x
f (x) pn(x) p(dx) < SUP n
f
pn(x) p(dx) < s,
U
whence p o T -'(K) < F. Therefore, p o T -' (K) = 0, which implies p o T` << p, since both measures are Radon. Letting p := d(p o T-')/dp, one has for any bounded continuous function f:
f f o dp = f f o T dp = lim
f
f pn dp.
(6.1.2)
According to a well-known result from functional analysis (see [214, p. 294, Corol-
lary IV.8.11] or 1541, Ch. II, §2, Theorem 23)), the sequence {en} is relatively sequentially compact in the weak topology of L' (p), i.e., every subsequence of this sequence contains a weakly convergent subsequence. However, (6.1.2) shows that all such weakly convergent subsequences may have only one limit p, whence we get the weak convergence of [Lo,,} to p.
6.2. Measurable linear automorphisms In the finite dimensional case all invertible linear transformations send nondegenerate Gaussian measures to nondegenerate ones (hence to equivalent ones). As it was noted above (see Example 2.7.4), even the simplest linear mappings, such as x --. 2x, transform every Gaussian measure with infinite dimensional support to a non-equivalent one. In this section we discuss this question in more detail. Throughout this section we assume that ry is a centered Radon Gaussian measure on a locally convex space X and H = H(y) is its Cameron-Martin space with
the natural Hilbert norm I [,,. For any operator A E C(H), we denote by A the (B(X ) B(X))-measurable linear extension of A constructed in Theorem 3.7.6. More precisely, A is an equivalence class of (B(X )-,, B(X ))-measurable mappings. For example, we can deal with an arbitrary proper linear version of this extension;
6.2.
Measurable linear automorphisms
283
recall that such an extension need not be unique pointwise, although the equivalence class is uniquely determined by A.
6.2.1. Lemma. Let A, B E C(H). Suppose that ,& transforms the measure 7 into an equivalent one. Then AB = AB y-a.e. PROOF. First of all, note that the mapping AB is -t-measurable and, up to a set of _measure zero, does not depend on our choice of modifications of the factors,
since A is measurable and B-1(E) has measure zero for any set E with y(E) = 0. Indeed, it suffices to verify this for Bore[ sets E, but then, by condition, one has
y(B-1(E)) = - o B-1(E) = 0. Therefore, we can deal further with proper linear versions of A and B. This reduces the proof to the verification of the equality of AB and AB on H. The first of these
two operators equals AB on H. Let {e } be an ort honormal basis in H. Then, using Theorem 3.5.1 and the fact that the domain of convergence of the series defining A contains H, we get
x A(Bh) = A(Bh) _
x en(Bh)Ae _ >2(en.Bh)HAen = ABh, Vh E H. n=I
n=1
The lemma is proven.
6.2.2. Theorem.
(i) Suppose that T: X
mapping such that
yoT-1
X is a -y-measurable linear = y. Denote by To any proper linear modification
of T and by U the restriction of To to H. Then U E L(H) and U' is an isometry (i.e., U' preserves the inner product). In particular. if the mapping To is injective on H. then U is an orthogonal operator. (ii) Conversely, for any operator U : 11 H such that U' is an isometry, there exists a -y-measurable proper linear mapping T. preserving the measure y and equal U on H. PROOF. (i) We may assume that T = To. According to Theorem 3.7.3(i), one has T(H) = H. In addition, U := TIH E L(H). Since T preserves the measure y, one has Q(f)2 = o(f oT)2 for any f E X', which can be written as
IR',fIH = IR,(foT)IH. Note that R., (f oT) = U' R.I f , where U' is the adjoint to U on H. Indeed, the inner product with any vector v E H for both sides of this equality gives f (Tv). Thus, IU'hI H = lhIH for all h E R,(X'), which, due to the density of X' in X;, implies that U' is an isometry. (ii) Conversely, an operator U E C(H), for which U' is an isometry, can be extended to a --y-measurable mapping T (see Theorem 3.7.6). The isometry property
of U' implies that the induced measure yoT-I has the same covariance as y (this is clear from the reasoning in (1)). Therefore, yoT-I = y. Note that in infinite dimensional spaces an operator T can preserve the measure y without being injective on H(-y). For example, let us take the countable product
y of the standard Gaussian measures on IR'. Then the mapping T: IR" -y 1R", (Tx),, = (x2.... , x , . . ) , transforms y into y, although it is not injective on 12. However, as Theorem 3.7.3 shows, a given mapping T can be replaced by a mapping F, which is injective on H(y) and has the property yoF-1 = yoT-'. .
284
Chapter 6.
Nonlinear Transformations
6.2.3. Definition. Let (Al, M, µ) be a measurable space. A measurable automorphism of (At, M, p) is a mapping T : Al - Ai with the following properties:
(i) There exists a set AIo of full measure such that T is one-to-one on A10, T(Alo) = M0 and T(AI\Afo) C Al\Alo. (ii) For every B E M. the setsT`'(B) andT(B) are p-measurable (i.e.. belong
to M0) and p(T-'(B)) = µ(7'(B)) = µ(B) If y is a Radon Gaussian measure on a locally convex space. then a measurable linear automorphism is defined as a y-measurable linear mapping T which is a measurable automorphism of (X,B(X),1). It is clear from the definition that the property to be a measurable automorphism may be lost if we replace a mapping by its modification (i.e., the identity mapping on RI can be redefined on a measure zero set in such a way that it will map that set onto X0,1]). By definition, a measurable linear automorphism is an automorphism which is equivalent to a proper linear mapping (but need not be proper linear itself). We shall see below that any measurable linear automorphism has a proper linear version which is a measurable automorphism. 8.2.4. Proposition. Let y be a centered Radon Gaussian measure on a locally convex space X and let T : X -. X be a -r-measurable linear mapping. The. following conditions are equivalent: (i) The mapping T is a measurable linear automorphism;
(ii) T takes all sets of -f-measure zero to sets of 1-measure zero and a proper linear version of T is an orthogonal operator on H(y): (iii) The mapping T takes all measurable sets to measurable ones and
y(B) = y(T`(B)) = 7(T(B)).
V B E 13(X) -
(6.2.1)
PROOF. Suppose that (i) is fulfilled. Denote by T any linear version T and put U = To!,,. By the previous theorem, the operator U' is an isometry (note that the measure ^y o T-' does not depend on a version of T). Hence, in order to prove that U is orthogonal, it suffices to verify that U is injective. Let us take the set Xo of full measure, which T maps injectively onto itself. Using Lemma 6.1.4 and the fact that the measure y has a Souslin support, we can choose in X0 a Souslin subset Y of full measure, on which To coincides with T and is Borel. Then T, maps injectively Y onto T0(Y) = T(Y) C X0. Since Tn(Y) is measurable, we
have y(T(Y)) = 7 o T-' (T(Y)) = I (Y) = 1. Suppose that there exists a unit vector h E H such that Toh = 0. Put B = (x E Y : h(x) > 0), where h is a proper linear version. The sets B - nh increase and their union covers 1'. Hence the union of the sets To(B - nh) contains T0(Y) = T(Y) and has full measure. However, To(B - nh) = To(B) and y(To(B)) = y(T(B)) = y(B) = 1/2, which is a contradiction. Thus, (i) (ii). Suppose that (ii) is fulfilled. Let To be a proper linear version of T. By Theorem 6.2.2, TO preserves y. Letting U be the restriction of To to H, we get by the same theorem that the mapping S = U-l preserves y as well. According to Lemma 6.2.1, ST = TS = I a.e.. since U-'U = UU-' = I. Clearly, T-'(N) has measure zero for every measure zero set N. By the assumption in (ii), the same is true for the images of measure zero sets. Therefore, by Lemma 6.1.1. there exists a set X0 of full measure such that T is one-to-one on Xo. T(Xo) = X0 and T(X\Xo) C X\Xo. Since both T and S are it follows
6.3.
Linear transformations
285
that the images under T of all Borel sets are measurable (in fact, we get that the images and preimages with respect to T of all measurable, not necessarily Borel, sets are measurable, since T-'(N) and S-1 (N) are measure zero sets for all Borel measure zero sets N). Since y oT-' = y o S-1 = y, we have (6.2.1). Thus, we get both (i) and (iii). Suppose now that (iii) is fulfilled. According to Theorem 6.2.2, the restriction U of any proper linear version of T to H(y) has the property that U' is an isometry. The condition -y(B) = 7(T(B)) implies, by the same argument as above, that U is injective. Hence U is an orthogonal operator. Obviously, (6.2.1) includes Lusin's condition (N). Therefore, we get (ii).
6.2.5. Corollary. Suppose that T satisfies Lusin's condition (N) and that a linear version of T is injective on H(y). Suppose, in addition, that -y o T-' = y. Then T is a measurable linear automorphism. PROOF. It suffices to note that any injective operator U on H, for which U' is an isometry, is orthogonal.
6.2.6. Corollary. A mapping that is inverse to a measurable linear automorphism T (i. e., a mapping S such that S oT = T o S = I a. e.) is a measurable linear automorphism as well. PROOF. Clearly, S is a measurable automorphism. In addition, it is equivalent to a y-measurable linear mapping U-1, where U is the restriction to H of any proper linear version of T (recall that this restriction is an orthogonal operator).
6.2.7. Remark. It follows from the proof above that in the definition of a measurable linear automorphism we could replace the condition of existence of a full measure set mapped onto itself by the existence of two full measure sets X1 and X2 such that T : X1 X2 is one-to-one and onto and T (X \X l) C X \X2. Note that not every measure preserving transformation takes measure zero sets to measure zero sets. For example, one can redefine the identity mapping of lR' with the standard Gaussian measure on a measure zero set E of power continuum in such a way that it will map E onto J' (in addition, this mapping will be even injective on a full measure set). The same can be done with a linear mapping on an infinite dimensional separable Hilbert space X with a Gaussian measure. Indeed, let us take a linear subspace Xo of full measure such that its algebraic complement L has a Hamel basis of cardinality continuum. Then one can find a linear mapping
T such that T(L) = X and extend it to X by setting Tx = x on Xo. 6.3. Linear transformations Recall that the symbol 1{, as before, denotes the space of all Hilbert-Schmidt operators on H (see Appendix). We shall say that an operator A on a Hilbert space H has property (E) (associated with "equivalence") if (i) A is invertible; (ii) AA' - I is a Hilbert-Schmidt operator on H.
6.3.1. Lemma.
(i) An operator A E C(H) has property (E) precisely when
A has the form
A = U(I+ K), where U is an orthogonal operator and K is a symmetric Hilbert-Schmidt operator such that the operator I + K is invertible.
Chapter 6. Nonlinear Transformations
286
(ii) If an operator A has property (E), then the operators A' and A-
have
properly (E) as well. In addition, the composition of two operators with property (E) has this property. (iii) Let A E L(H) be such that A(H) = H. Then AA' - I is a Hilbert-Schmidt operator precisely when A = (I + S)WW', where S is a symmetric HilbertSchmidt operator, the operator I + S is invertible and the operator it" is an isometry.
PROOF. Let us note first that if A(H) = H. then the symmetric operator AA' has zero kernel, since the equality AA'h = 0 implies (A'h, A'h) = 0. whence h = 0 (otherwise the range of A is smaller than H). Therefore, the range of AA' is dense
in H, hence the operator V from the polar decomposition A' = V AA' is an isometry (although its range may be a proper subspace).
If AA' - I E 1i and A(H) = H, then, using the polar decomposition A' V AA' indicated above, we get A' = where Q E I-l is a symmetric operator. In addition, ,17-+--Q - I E it, since, taking the eigenbasis of the operator Q with eigenvalues q we see that the operator S = +--Q - I has the eigenvahies 1 _+q; - 1. Thus, A = (I + S)W, where R' = V' has the adjoint which is
an isometry. Note that the operator I + S is invertible. Indeed, by the HilbertSchmidt theorem, it suffices to verify that the kernel of this operator is trivial. This follows from the equality A' = (I + S), since A' has zero kernel by the equality A(H) = If. Thus, we get one implication in (iii).
Now let A have property (E). Since A is invertible, then V is an orthogonal
operator and A' = 11"(1 + S). Hence A'A - I E R. As explained above,
K: =
A'A - I E 1i, which, by virtue of the polar decomposition, yields the desired representation A = U(I + K). Conversely, if A = U(1 + K), where U is orthogonal and K EH is symmetric, then A' A = (I + K)2 = I + 2K + K2. where 2K + K2 E 11. In addition, A turns out to be invertible if so is the operator I + K. Thus, (i) is proven. If A = (I + S)W', where S and W have the properties mentioned in (iii), then A' = IV* (I + S), whence
AA' - I = (I+S)WIl"(I+S)-1 =(I+S)2-I=2S+S2 Eli. As above, the Hilbert-Schmidt theorem implies that the range of AA' coincides with H, whence A(H) = H. It remains to show (ii). Suppose that A and B have property (E). Then the operators A', A-1 and AB are invertible. Assertions (i) and (iii) yield that A`
has property (E). Since the inverse to U(I + K) from (i) is (I + K) `U* and (I + K)-1 - I E f, we get property (E) for A-1. Writing A = U1(I + K1) and B = U2(1 + K2) as in (i), we readily get the inclusion ABB'A' - I E R.
6.3.2. Theorem.
(i) Let T: X - X be a y-measurable linear mapping
is equivalent to the measure y. Denote by T, such that the measure a proper linear modification of T. Then A := Tt)lq maps H continuously on yoT--1
H and AA' - I E R. (ii) Conversely, for any operator A E L(H). satisfying the conditions A(H) = H and AA' - I E 1i, there exists a y-measurable proper linear mapping T such that Tay = A and the measure yoT-1 is equivalent to y. PROOF. We shall start with claim (ii). By Lemma 6.3.1, one can write A = (I + S)W, where S E 71 is a symmetric operator. W' is an isometry, and I + S is
6.3.
287
Linear transformations
invertible. By Theorem 6.2.2, the operator i-4' preserves the measure y. Hence, by Lemma 6.2.1, we get A = (I + S)W a.e., and it suffices to consider separately the operator I + S, which has already been done in Example 2.7.6. In order to prove (i) note that, according to Theorem 3.7.3, the operator To maps H onto the space H(y o T-') which coincides with H(y) = H by the equivalence of the two measures. We may assume further that T = To. Let us write the
polar decomposition for A' in the form A' = V AA', where V is an isometry on the closure of the range of AA' and zero on its orthogonal complement. In our case it turns out that the closure of the range of AA' coincides with H (i.e., V is an isometry). To this end, it suffices to verify that the range of AA' is dense. Indeed, otherwise there exists a nonzero vector h 1 AA'(H), whence A'h = 0 and h 1 A(H), which is a contradiction. Thus, A = CU, where C = AA', U = V*. Sin_ce_ U' = V is an isometry, we get, by Theorem 6.2.2 and Lemma 6.2.1, that_To = CU a.e., where the mapping U preserves the measure y. Therefore,
y o C-1 - y. Let us pass to the symmetric operator C. Since A(H) = H, then also C(H) = H, whence KerC = 0. Hence C is invertible and, by the Banach theorem, IIC-'IIc(H) = A < oc. Let e = (2.X)'1. Using Theorem A.2.15 in Appendix, we find a diagonal operator D and a Hilbert-Schmidt operator G such that
C = D + G and IIGIIc(n) < s. Note that the operator D = C - G is invertible, since IIGIIc(H) < IIC-' IIC('x) Hence one can write
C = D(I + S),
S = D-'G,
where S E 7'i. As we proved above, the y-measurable linear extension of the operator A = I + S transforms y into an equivalent measure. By Lemma 6.2.1, one has C = DA a.e. Letting p := y o A-', we arrive at the relationship y
yo C-1 =pob-'
yob-1.
We have already investigated in Example 2.7.6 the equivalence conditions for the diagonal operators. It follows from that example that the eigenvalues d of the
operator D have the form 1 + o,, or -1 + o,,, where En 1 an < oo and d 34 0. Therefore, D = Uo(I + R), where R is a symmetric Hilbert-Schmidt operator, Uo is a symmetric orthogonal operator with the eigenvalues equal 1 in the absolute value, and the operator I + R is invertible. Collecting all the information we gained, we get
A = CU = D(I + S)U = Uo(I + R)(I + S)U = Uo(I + r)U, where 1= S + R + RS E 1 {. Then
AA' - I = Uo(I + r)UU'(I + r')UU - I = Uo(r + r' + This brings the proof to the end.
E fl.
0
6.3.3. Corollary. Let T be a y-measurable linear mapping. The following conditions are equivalent: (i) A linear version of T maps H into H and has property (E), and, in addition, T satisfies Lusin's condition (N), i.e., y(T(N))= 0 for every set N of -ymeasure _'ero;
(ii) There exists a set H of full y-measure such that T maps fl one-to-one onto itself, T(X \f2) C X\ft, and y o T-1 - y.
Chapter 6.
288
Nonlinear Transformations
In this case there exists a -y-measurable linear mapping S which is inverse to T,
i.e.,TS=ST=I a.e. PROOF. If condition (i) is satisfied, then the operator. which is inverse to the restriction of a linear version of T to H, has a measurable linear extension S. In addition, according to the theorem just proven, the operators T and S transform -y into equivalent measures. Hence, by virtue of Lemma 6.2.1, one has TS = ST = I a.e. According to Lemma 6.1.1, there exists a set 12 with the desired properties. Let condition (ii) be satisfied and let To be any proper linear version of T. By the previous theorem, the operator A = ToIH maps H onto H and AA' - I E ii. In order to verify property (E), it remains to prove the invertibility of A. As in the case of orthogonal operators, we find a Souslin subset Y C ) of full measure, on which T = To is a Borel mapping. Assuming that h E H, h 36 0 and Ah = 0, we get a sequence of measurable sets B - nh, where B = {x: h > 0), which is increasing to Y. Then the sets To(B) = To(B - nh) increase to the set T0(Y). Since y o T-1(To(B)) = y(B) = 1/2 and y o T- (To(Y)) = y(Y) = 1 by virtue of the injectivity of To = T on Y, we get, by the equivalence of measures, that y(To(B)) < I and y(To(Y)) = 1, which is a contradiction.
As it has already been noted, one cannot omit Lusin's condition (N) in (i). However, if this condition is not imposed, it follows from the proof that T has a modification with Lusin's property (N) (which thereby satisfies condition (ii) in Corollary 6.3.3).
6.3.4. Corollary. Let T be a -y-measurable linear mapping such that its proper linear version has property (E) on H. Suppose that f is an H-Lipschitzian function.
Then f o T is H-Lipschitzian as well and D,, (f o T)(x) = T'D. f (Tx). If f takes values in a separable Hilbert space E and is H-Lipschitzian. then f o T is also and
Dx(f oT)(x) = DNf(Tx)T. PROOF. The function f o T is measurable, since T is an absolutely continuous transformation of y. By condition and property (E), we get If o T (x + h) - f o T(x) 1 < CIThI H <
ccl hI H
V h E H.
Let h E H. Differentiating F(T(x + \h)) in \, we get a.e. 8h(F o T)(x) _ SrhF(Tx) = (DHF(Tx),Th),,. The vector case is analogous. 6.4. Radon-Nikodym densities The method of proving above enables one to derive explicit formulas for the Radon-Nikodym densities of equivalent Gaussian measures. In order to get a compact representation of such densities, we shall need the notion of the regularized
Fredholm-Carleman determinant for operators of the form I + K, where K is a Hilbert-Schmidt operator on a separable Hilbert space H. Recall that the space of all Hilbert-Schmidt operators on H is denoted by 1i(H) or simply by W. A more detailed information about regularized determinants can be found in (296, Ch. IV, §21. The basic idea is easily seen in the case, where the operator K is diagonal and
has the eigenvalues k,. Then the product det(I+ K) := f;_, (1 + k,) may diverge if K has no trace. However, the product det 2(1 + K) := f;°,(1 + k,)e-k', as one readily verifies, converges. In addition, det 2(1 + K) = det(I + K) exp(-trace K) if K is a nuclear operator.
6.4.
289
Radon-Nikodym densities
6.4.1. Definition. Let K E N be a finite dimensional operator with range K(H). Let us put det 2(I + K) := det((I + K)IK(H)) exp(-trace KI K(H)) Note that traceKIK(H) coincides with the trace of K in the sense of nuclear operators (see Appendix). Indeed, choosing any orthonormal basis e1, ... , e in K(H) and complementing this finite collection to an orthonormal basis {e;} of H, we get (Ke,, e,),, = 0 whenever j > n. In the investigation of the function det2 the crucial role is played by the following Carleman inequality: rr
Idet2(I+K)l <exp12IIKIIit,.
(6.4.1)
It is straightforward to verify that, for any finite dimensional operators A and B and I + C = (I + A)(I + B), one has the equality det 2(1 + A) det 2(1 + B) = det 2(1 + C) exp(trace AB).
(6.4.2)
Now one can extend the function det 2 to all Hilbert-Schmidt operators. If the operator I + K, K E N, is not invertible (which is only possible if -1 is an eigenvalue of K), then we put det 2(1 + K) = 0.
6.4.2. Lemma. Let K E N be such that the operator I + K is invertible. Then, for any sequence of finite dimensional operators K. convergent to K in the Hilbert-Schmidt norm, the sequence det 2(1 + K,,) converges to the limit denoted by det 2(I + K) and this limit does not depend on our choice of an approximating sequence. The function K - det 2(1 + K) on N is locally uniformly continuous on the set of all operators whose spectra do not contain -1. In addition, det 2 satisfies relationships (6.4.1) and (6.4.2).
is an arbitrary orthonormal basis in H, then, for any In particular, if operator K E N, one has
det2(I+K)= lim det(b;j +(Ke;,ej))
rt
r
n
l
i=1
1
PROOF. Let A, B and D = A - B be finite dimensional operators. Since the operator I + K is invertible by assumption, the operators I + A and I + B are invertible as well, provided A and B are sufficiently close to K in the operator norm. By (6.4.2) we get
det2(1+B)det2(I+(I+B)-'D) =det2(I+A)exp(trace B(1+B)D). Therefore,
det2(I+A)-det2(I+B) = det2(I+B)ldet.2(1+(I+B)-'D)exp(traceB(I +B)-'D)
- 1,.
In addition,
II(I+B)-'DII, <- AJIDIIN,
trace B(1 +B)-'D!
AIIBIINIIDIIn,
where the number a dependent on the norm of (I + K)-' is common for all B sufficiently close to K. Carleman's inequality implies the aforementioned local
290
Chapter 6. Nonlinear Transformations
uniform continuity of det 2. The remaining claims follow from the finite dimensional case by virtue of the continuity of det 2. 0
6.4.3. Remark. Let K be a Hilbert-Schmidt operator on H. Denote by {kr } the eigenvalues of the complexification KG of the operator K (considered on the complexification He of H) counted with their multiplicities (i.e., every eigenvalue k with multiplicity v enters this sequence v times). Recall that the nonzero eigenvalues have finite multiplicities. The Carleman-Fredhohn determinant of I + K can be defined equivalently (see [296, Ch. IV, §2]) as
det2(I+K) = fl(1+ke)e-k
(6.4.3)
n=1
If K(' has no nonzero eigenvalues, then we put det 2(I + K) = 1. For a compact operator A, the absence of nonzero eigenvalues of At' is equivalent to the equality a(AC) = 0, where a(AC) is the spectrum of Ac. This is also equivalent to lim IIAII1 = 0: such operators are called quasinilpotent. Certainly, the nix H) equality det 2(I + K) = 1 does not imply that K is quasinilpotent. However, if det2(I + AK) = 1 for all A from some nonempty open set in IR1, then, by the analyticity of this function in A, this is true for all A and K is quasinilpotent (see [839] for the proof of this fact and other equivalent conditions). Let 7 be a centered Radon Gaussian measure on a locally convex space X and
let K be a Hilbert-Schmidt operator on H = H(y) such that the operator I + K is invertible. Put
T=I+K.
where k denotes as above a measurable linear extension of K (defined up to a modification and taking values in H, since K is a Hilbert-Schmidt operator: see Chapter 3). Let
AK(x) := Idet2(I+K)I exPlhk(x) - 2IK(x)IH]. Since I+K has property (E), the measure ^)oT-1 is equivalent to 1. In addition,
letting S be the measurable linear extension of (I + K)-1, we have T(S(x)) = x for a.e. x.
6.4.4. Lemma. Let K be a Hilbert-Schmidt operator on H without eigenvalue -1. Then for any p > 1 such that I + pK + pK' + pK' K > 0. we have
''.
IIAKIILP(,)=eXp(-2'IIK112) det2(I+K) det2(1+pK+pK'+pK'K) 1
(6.4.4)
PROOF. By Proposition 3.7.10, the integral of IK(x)12 equals IIKIIx. Hence 2 Q(x) := 6K(x) - 2IK(x)IH2 + 21IKIIH
an({ - 1), where the 2an's are
belongs to X2. By Proposition 5.10.13, Q = n=1
the eigenvalues of D, 2Q and {{n} is some orthonormal basis in H. According to
6.4.
Radon-Nikodym densities
291
equality (4.8.5), we have Jexp(pQ)
dy = H "P-Pan
- Pan
n=]
]/2
if 2pa < 1. This is exactly Idet
On the other hand, DA6K(x) _
-Kx - 0x - K'Kx
-Kx - 0x according to equality (5.8.7). Hence and
6.4.5. Theorem. Let S = (I + K) -1. Then one has AK(ISx) d(yo S'1) d(7oT-1)(x) = d7
(x) = AK(x).
dy
'
(6.4.5)
PROOF. The existence of densities is already known. In order to show (6.4.5)
suppose first that y is the standard Gaussian measure on IR". Then tiK(x) = trace K - (Kx, x). Now the expression on the right in (6.4.5) for (AK o S)-1 can be written as
det7'
exp[(KT-1x,T-'x)+ 1(KT-1x,KT-1x)] exp[2(x,x)
= Idet7'I
-
2(T-1,,T-1x)l
J, which is the expression for d(yT-1)/dry given by the classical Ostrogradsky-Jacobi formula (see the calculation in the proof of Theorem 6.6.3 below, where nonlinear
transformations are considered). In the infinite dimensional case, let us take an orthonormal basis {en} in H and put K" = PKP". The finite dimensional operators K" converge to K in the Hilbert-Schmidt norm. For all sufficiently large n,
the operators T,, = I + Kn are invertible. Put S. = Tn 1. Then the densities pn = d(y oT,,-1)/dy and r" = d(-y o Sn I)/dy are given by (6.4.5) and converge a.e. to the expressions on the right-hand sides in (6.4.5) for T and S, respectively. It remains to note that these densities are uniformly integrable. Indeed, by (6.4.4), there exists p > 1 such that the sequence {rn} is bounded in LP(-y). The same is 0 true for {pn}, since T = S-1, and S satisfies the same condition as T.
6.4.6. Theorem. Two centered Radon Gaussian measures p and v on a locally convex space X are equivalent precisely when H(p) and H(v) coincide as sets
and there exists an invertible operator C E C(H(p)) such that CC' -I E f(H(p)) and
for all h E H(p).
IhIH(v) =
(6.4.6)
If C - I E 7-t(H(p)), then one has dv (x) =
dp
1 Ac._I(C-Ix)
.
(6.4.7)
Finally, if p - v, one can find a symmetric operator C with the aforementioned properties.
PROOF. Suppose that p - v. Then H(p) and H(v) coincide as sets and there exists an invertible operator C E G(H), where H: = H(p), such that (h, h E H. Then v coincides with the image of p under the
Chapter 6. Nonlinear Transformations
292
measurable linear mapping C. By virtue of the equivalence of these two measures,
CC' - I E R. Clearly, we can always take for C a nonnegative operator: just replace C by
C
I E 71, hence formula (6.4.5) applies. The existence of
an operator C with property (E) on H(µ) such that (6.4.6) holds true yields that
v=µoC'I, whence v-.p. 6.4.7. Corollary. Let u and v be two equivalent centered Radon Gaussian measures on a locally convex space X. Then there exist an orthonormal basis {en}
x
in H(p) and a sequence {an} of real numbers not equal to -1 such that E An < 00 n=1
and, for an arbitrary sequence of standard Gaussian random variables C. on a probability space (1, P), one has
µ=
Po
x
x
nen)
and v= P o (E(1 + An)ynen) n=t
n=1
/
PROOF. It suffices to take a symmetric operator C in the previous theorem x and use the eigenbasis {en } of C - I with (C - I )en = \nen. Then E an < oc, the n-1
aforementioned series converge in X almost sure and their distributions coincide with u and v, respectively.
6.4.8. Corollary. Two centered Radon Gaussian measures A and v on a locally convex space X are equivalent if and only if there exists an invertible symmetric
nonnegative operator T on H(µ) such that T - I E 7i(H(p)) and (f,9)L2(.) = (TR,f, Rp9)a(p).
df, 9 E X'.
(6.4.8)
An equivalent condition: the norms IIfIIL2(,.) and IIfIIL2(O are equivalent on X- and (f, f)L2(,) on XN is generated by a Hilbert-Schmidt the quadratic form (f, operator on X, .
PROOF. Let u - v. Then there exists an invertible symmetric operator C E C(H(µ)) with C - I E 7((H(µ)), and v is the image of µ under the mapping C. By Lemma 3.7.8, we have {f>9)L2cv) _ (f 0e,90(5)L2(p)
= (Rv(fa
),R,(9oC))H(p) =(C'RNf,C'R,g)H(p).
(6.4.9)
Hence we can put T = CC' = C2. It is readily seen that T - I E 7f(H(p)). Clearly, T is invertible. Conversely, suppose that (6.4.8) holds, where T is invertible nonnegative and T - I E 7I (H(µ)) . Put C = v T. Then writing (6.4.9) backwards, we get (f o a, g o C)t2(,,) = (f, 9)L2(v), which implies that v =,u o C-1, whence we
get v -,u.
6.4.9. Remark. If µ and v are not centered, then u ap - a E H(p) and the centered measures Ep_, the theorem above: otherwise u 1 v.
v precisely when and v_a satisfy the conditions in
Let us give a coordinate representation of the Radon-Nikodym densities of equivalent Gaussian measures.
6.4.
Radon-Nikodym densities
293
6.4.10. Corollary. Let p and v be two equivalent Radon Gaussian measures on a locally convex space X. Then dv/dµ = exp F, where F is a µ-measurable second order polynomial which admits the following representation:
(x) + E an (l;. (X)2
F(x) = c + E c,, n=1
- 1)
u-a.e.,
(6.4.10)
n=1
x
x
n=1
n=1
where c E JR'. > c < Co, E a2 < oo. an < 1/2, {t;n} is an orthonormal basis in X.;, and both series converge a.e. and in L2(µ). Conversely, if F has such a representation, then exp F E L1(p) and the measure 11 exp F1l exp F ,u is Gaussian.
PROOF. Follows from the results above combined with the description of measurable second order polynomials obtained in Chapter 5. Note that the integrability
of exp F is equivalent to the condition an < 1/2. The necessity of this condition is obvious by the one dimensional case and Fubini's theorem. The sufficiency is readily seen from the fact that 1)) = e-°(1 - 2a) -1/2 for a stan2an)-1/2 dard Gaussian random variable t; and that the product nn 1 e-°^(1 converges if an < 1/2 and
x
oc. n=1
In the case, where X is a separable Hilbert space and u and v have covariance operators K,, and K,,, we get H(µ) = K,,(X). Assuming that K,, and K have dense ranges (which can always be achieved by passing to the closure of
H(µ) in X), one can write C in the form C = v1K, K -1. On the other hand. C = K,,Co K -1, where Co = K,, -1 K E £(X) is an invertible operator. Since the operator K is an isometry of the Hilbert spaces X and H(p), we conclude that C on H(p) has property (E) precisely when Co does on X. Therefore. the equivalence of the measures µ and v is characterized by the continuity and invertibility of the operator K -1 fW together with the condition
K K K 1 - I E I{(X ).
(6.4.11)
Obviously, one can interchange the roles of p and v. Let us summarize our observations as follows.
6.4.11. Corollary. Let X be a separable Hilbert space and let µ and v be two Gaussian measures on X with covariance operators K,, and K and means a and a,,. respectively. Then y - v precisely when a,-a, E and there exists an invertible operator C on the space X such that CC' - I is a Hilbert -Schmidt operator on X and K = V"K-, C.
(6.4.12)
Otherwise p 1 v. The existence of such an operator is equivalent to the existence of a Hilbert-Schmidt operator D on X without eigenvalue -1 such that K - K,, _ K,,D K,,. If a = a,, = 0, then lnn-logan]), n=1
(6.4.13)
Chapter 6. Nonlinear Transformations
294
where the a 's are the eigenvalues of the symmetric operator (I + G)(I + G') corresponding to the eigenbasis {ipn} and t) is the element in X; generated by the vector
Kµy'.,,.
PROOF. We know that the claim reduces to the case where a , = a,, = 0 (see
Chapter 2. in particular, Proposition 2.7.3). Let p - v. We may assume again -1 that both p and v are nondegenerate. Then the operator C = KN fK is H(p) is a proper invertible and CC' - I E 7t(X). If the closure of subspace X0 in X, then we put C = I on the orthogonal complement of X0, which is consistent with (6.4.12), since Kµ = VT; = 0 on X' by the symmetry of these two operators. Let us prove (6.4.13). Since
D:= (I + G)(I + G') - I x
is a symmetric Hilbert-Schmidt operator on X, we have E (a - 1)2 < oc. The sequence
KN;pn is an orthonormal basis in H(p), hence {rmn } is an orthonormal
basis in X. Together with the convergence of the series
x
n-l
Iloga., - (an - 1)an 1I
this yields the convergence in L2(µ) of the series on the right in (6.4.13). It is straightforward to see that the function p defined by the right-hand side in (6.4.13) is in L' (p). Let us show that the measure A = !t p coincides with v. Note that
(%, 17,4-(,,) = ((I+D)ipj,y,)x =an(VJ, ,)x.
(6.4.14)
then, denoting by I(x) the element of X,, generated by Indeed, if u. v E K;x. we get (cf. Remark 2.3.3)
(1(u).I(v))L,(,,} _ (K,,KN1/2u,Kµ1"2v)r = ((I+D)u,v)c.
(6.4.15)
1(u) We observe that if u2 - u in X, then I(uj) - 1(u) in L2(p), hence 1(u,) in L2(p). Therefore, (6.4.15) holds true for any u, v E X. in particular, we get x x (6.4.14). Now let _ E cnl1n, where E cn < oo. Then by the independence of I
n=1
the rp,'s on (X.pr) and (6.4.14), simple calculations yield
J exp(i)edp=exp(-2ancn) = J x
n=1
x
0
whence p` = v.
Note that if K is injective, then the operator D introduced above equals D=K4-112K,.KNt12-I.
Then the equivalence condition can be restated as the inclusions D E 7{(X) and together with the invertibility of I + D. a - a4 E In the just considered Hilbert case, there is a sufficient condition for the equivalence which does not involve the square roots of the covariance operators. Suppose
that H(p) = H(v) and, in addition, that K. = (I+Q)K,, where Q E 7{(X) and the operator I + Q is invertible. Then p v. Indeed, let D = K,, - I K K,, -' - I.
6.5.
Examples of equivalent measures
295
Since Q = 1, then Q = K whence, in the eigenbasis {en} of the operator K, we get
x
oc
E(D2en,en) =
DZ
K,.
x
1en,
K- en = E(Q2en,en) < 00, n=1
n=1
n=1
K
i.e., (6.4.11) is fulfilled, since D is symmetric.
6.5. Examples of equivalent measures and linear transformations Let us consider the case where one of the two equivalent Gaussian measures is the Wiener measure Pu' on the space C[0,1] or L2[0,1].
6.5.1. Example. A Gaussian measure v on L2[0,1) is equivalent to the Wiener measure PW if and only if a,, E H(Pw) and its covariance operator R is an integral operator with a kernel K (the covariance function of the corresponding Gaussian process) of the form a
tr
J o
Q(u,v)dudv,
Ju
(6.5.1)
where Q E L2([0,1]2) is a symmetric function such that the corresponding integral
operator has no eigenvalue -1. In this case, for a.e. (t, s), one has the equality Q(t, s) = s). PROOF. It suffices to consider the case where a = 0. Suppose that v - Pw Let us apply Corollary 6.4.7 and take the corresponding orthonormal basis {en} in H(Pµ') = W02" [0, 1] and the sequence {an}. Then the functions 7yn(t) = e'. (t) form an orthonormal basis in L2[0,1]. Put cc
+2An)0n(t)wn(s)
Q(u,v) _ n=1
and note that this series converges in L2([0,1)2). The integral operator with kernel
Q has no eigenvalue -1, since )12 + 2)n A -1 due to our condition an 0 -1. Let {&n} be independent standard Gaussian random variables. Since the series 00
x
E n(w)en(t) and E (1+A )tn(w)en(t) converge in L2(P,CIO, 1)) and, for almost n=1
1
every fixed w, converge in C[0,1], we get that the covariance of v is given by the kernel 1: (1 + An)2en(t)en(s),
n=1
whereas the function min(t, s) (the covariance function of the Wiener process) is obtained if an - 0. Therefore. s
t
min(t,s) = ffQ(uv)dudv. o
Conversely, suppose we have (6.5.1), where the integral operator with kernel Q on L2[0,11 has eigenvalues A,, -76 -1 and eigenbasis { pn}. Then the functions
Chapter 6. Nonlinear Transformations
296
e
(t) =
(s) ds form an orthonormal basis in H(P1t) = 141[0,1] and
J
oc
ac
Anen(t)en(s).
Een(t)en(8) +
(6.5.2)
n=1
n=1
where both series converge uniformly. Let {&n} be a sequence of independent standard Gaussian random variables. Let us observe that An + 1 > 0. Indeed, by condition, the quadratic form Qo with kernel K. is nonnegative on L2[0,1]. It is readily seen that Qo extends to a continuous nonnegative quadratic form Q1 on the space C[0,1]' (identified with the space of all signed measures on [0, 1]) given by
=
f J min(t, a) m(ds) m(dt) + Jill Q(u, v) du dv m(ds) m(dt). 0
0
0
0
0
0
Hence, for any h E Y17o'1 [0, 1], defining a functional f on C[0,1[ by
x'--' -(x,h')L21o.1i +x(1), we have Q, (f) > 0. This functional coincides P11'-a.e. with the stochastic integral
of h'. Hence the quadratic form with kernel min(t, s) on C[0,1]' evaluated at f gives h' f L, :0.1 . Integrating by parts in the term involving Q, we obtain I
1
Q,(f) = Ilh'I1i(o.1) + f J Q(t, s)h'(t)h'(s) dt ds. 0
0
Taking h' = Vn , we get 1 + An ? 0. Clearly, the measure v0 obtained as the x is equivalent to the Wiener measure distribution of the sum E 1 + A,, n=1 (the latter corresponds to A. a 0). Now it remains to note that v = v0i since by (6.5.2) the covariance of v0 is given by the same kernel R as the covariance of v. If s is fixed, then the function t s K,(t, s) is absolutely continuous and its derivative for a.e. t equals a
1+IQ(u,t)du if t<s, 0
J0
Q(u, t) du if t > s.
The result is absolutely continuous in s on the intervals (0, t) and (t, 1) and its derivative is Q(s, t) for a.e. s in these two intervals.
6.5.2. Example. Let r be a continuously differentiable function with strictly positive derivative on [0, 11 such that r(0) = 0 and r(1) = 1. Put
Tx(t) =
r (t)x(T(t))
Then the measure v = P' oT`1, i.e., the distribution of the process w,(t)/ r'(t), is equivalent to the Wiener measure PW if and only if the function T' is absolutely continuous and r" E L2[0,11.
6.5.
Examples of equivalent measures
297
PROOF. Let 'r' E W2.1[0, 1] and g(t) = 1/ r'(t). Then g E W2"1[0,1]. Denote by 9 the inverse function to r. Let us show that the operator T on H(P') _ K0.1
[0,1] has property (E). /If h E 140" [0,1/], then Th E W02" [0,1]/ and (t))112h'(r(t))
Th'(t) = -2r"(t)(T'(t))-312h(T(t)) + (T = 9 (t)h(r(t)) + (T'(t))112h'(r(t)). In addition, given y E
140.1
[0, 1], it is easily verified that
T-ly(t) =
r'(9(t))y(9(t)) =
1Y(0(t))
is a function in Ii'2' 1 [0, 1]. Therefore, T is invertible on %,,02,1[0,11. Since the operator V on L2[0,11 given by Vx(t) = r'(t)x(r(t)) is orthogonal, the operator U given by t
Uh(t) _ / Vh'(s) ds 0
Pu', it remains to show that U - T is a Hilbert-Schmidt operator on H(P" ). Indeed, then TT' - I is a is orthogonal on HV2 [0,1]. In order to see that. v O
Hilbert-Schmidt operator. We have
[(U -T)h]'(t) = -g'(t)h(r(t)), vh E H(Pu'), whence
Ilg'IIi21o,t1 max Ih(t)12, which gives the desired inclusion
U - T E 7 t(H(PIS.)), since we can take an orthonormal basis
with E
in H(P1L')
oc. Conversely, assume that v - P. Then the covariance
n=t
function of the process w7(t)/ r'(t) equals min(r(t),r(s))/ r'(t)r'(s) and has the form indicated in Example 6.5.1. This shows that the function 1/ r'(t) is in W2.1 [0, 11, which is equivalent to the inclusion r' E WV2.1 [0, 11, since r' is continuous O
and strictly positive.
Let us evaluate the Radon-Nikodym density of the measure induced by the standard Ornstein-Uhlenbeck process { on [0, 1] with to = 0 with respect to the Wiener measure Pu'. Certainly, Girsanov's theorem discussed below yields immediately the equality 1
dm(
1
dP"'(w)=exp(-2 rwtdwt-8 fwdt) 0
=exp
1
0
2 wt+2-8 1
1
f w,dt 2
0
since by Ito's formula one has
f t
0
W. dw, = 2W2 _t.
(6.5.3)
Chapter 6. Nonlinear Transformations
298
However, in order to have some exercise, we shall follow a longer way based on our general theorems. The measure A, on C[O,1] is the image of Pug under the linear mapping T defined from the integral equation c
Tx(t) = x(t) - 2 JTx(s)ds. 0
This equation is uniquely solvable and the inverse linear operator S is given by
= x(t) +
Is(s)
It is easily seen that the operator Q = S - I is nuclear on H = H(Pw) and its complexification has no eigenvalues. It remains to note that [Qx[H = IIzI 2Io.1I/4
and that
/
/
5Q(x) = -2 E(Qx,en)xen(x) 1
n=1
x
1
r1
_ -2 E(x,en)L'[O.lJ J n=1
n(s)dx(s) _ -2 Jz(s) d z (9),
0
0
where {en } is any orthonormal basis in H. Now (6.4.7) yields (6.5.3).
6.6. Nonlinear transformations Let y be a centered Radon Gaussian measure on locally convex space X and let H = H(-y). In diverse theoretical problems and applications one encounters the problem of investigating the images of the measure y under nonlinear mappings
T: X -. X. One of the best studied is the situation where T has the following special form: T(x) = x + F(x),
where F: X H is a sufficiently regular mapping. It is instructive to have in mind the situation, where y is the countable product of the standard Gaussian measures on the line and F: 1R°° l2 or y is the classical Wiener measure and F(x)(t) = fJ u(x)(a) ds, where u: C[0,1] --, L2[0,1]. In fact, all the results below are invariant under measurable linear isomorphisms, so that it would be enough to consider one of these two concrete cases. As the investigation of the linear case shows, one should expect that certain conditions connected with Hilbert-Schmidt operators on H will arise. It is intuitively clear that the smooth mappings of the form above behave locally as linear mappings I + D F. Therefore, a natural candidate to fit Hilbert-Schmidt type conditions is the derivative This expectation is justified. The principal results of this section state that a mapping I + F transforms y into an equivalent measure if F satisfies certain technical conditions and I + D F is invertible on H. We shall start with a lemma which is of independent interest. In this lemma and some subsequent results, we make use of the following trivial observation: if y is a Radon Gaussian measure with the Cameron-Martin space H and F: X H is a y-measurable mapping, then y o T-l, where T = I + F, is a Radon measure concentrated on a linear subspace of X that is a countable union of metrizable
6.6.
299
Nonlinear transformations
compact sets. Indeed, let Y be a full measure linear subspace that is a countable union of metrizable compact subsets of X. Then Y has full measure with respect to y o T-'. Clearly, any Borel measure on Y is Radon (see Appendix).
6.6.1. Lemma. Let T = I+ F : X -+ X, where F: X -. H is a -y-measurable mapping such that
IF(x+h)-F(x)lH
(6.6.1)
where A < 1. Then there exists a Borel set I2 of full -y-measure such that 1?+H = 1),
T: Q -+ 11 is one-to-one and onto, T(X\I2) C X\1, and the inverse mapping S has the form S = I + G, where a Borel mapping G: X -+ H satisfies condition replacing A. In addition, an inverse mapping of the (6.6.1) with the constant As j
foregoing form is unique up to a redefinition on a set of measure zero.
PROOF. According to Corollary 4.5.5, there exists a full measure set 11, which is a countable union of metrizable compact sets, such that l + H = f2, F is Borel on I2, and (6.6.1) is fulfilled for every x E I2, i.e., one has
JF(x+h)-F(x+k)jH
(6.6.2)
Obviously, T maps x + H into x + H, hence T(X\II) C X\11. Note that the inverse mapping S can be given constructively by the aid of the sequence of iterations
-F(x+Gn(x)),
Go(x) = 0,
in the spirit of the fixed point theorem. Indeed, for all x E 0 the mapping T,: h
-F(x + h) is a contraction with the Lipschitz constant A. Hence this mapping
has a unique fixed point y E H: y = -F(x + y). Then, letting G(x) = y and S(x) = x + G(x), one has T (S(x)) = x + G(x) + F(x + G(x)) = x. Since S maps x + H into x + H and T(S(x)) = x, we see that T(x+H) = x + H for every x E Q. In addition, for such x and all h E H, one has the estimate IG(x + h) - G(x)IH < 1
A
jhIH.
(6.6.3)
This estimate is easily deduced from the following inequality verified by induction: [A+...+a"+'llhl,. IGn+i(x+h) H < Alhl,, +AIG.(x+h) -G.(x)I H S By construction, G is a Borel mapping on ft. On the complement to 11, we put G = 0. Clearly, for every x E Il, there is only one possibility to choose G(x) E H
due to the uniqueness of a fixed point of any contraction. Hence there is only one
y E x + H such that T(y) = x. If x - y ¢ H, then T(y) = y + F(y) does not equal x. Hence an inverse mapping S of the form I + G, G: X -. H, is uniquely determined on fl, i.e., almost everywhere.
0
6.6.2. Lemma. Let T = I + F, n E IN, where the mappings Fn: X -. H satisfy the conditions in the previous lemma with common \ < 1. Suppose that the
mappings F converge a.e. in the norm of H to a mapping F: X -+ H. (i) If y o Tn ' << y for all n and the sequence { p } of the corresponding RadonNikodym densities is uniformly integrable, then the mapping T = I + F has the property that the measure -y o T-' is absolutely continuous with respect
to y and its Radon-Nikodym density o is the weak limit in L' (y) of the sequence
In addition, T has a version To which satisfies (6.6.1).
Chapter 6.
300
Nonlinear Transformations
(ii) Put T;' = S = I+G, . If yoS,-,' - y and the sequence A := d(7oSn')/dry is uniformly integrable. then there is a mapping G: X - H such that IGn - GI H --. 0 in measure y and I + G = To ' Moreover, T o (I + G)(x) = (I + G) o T(x) = x a.e. PROOF. Claim (i) follows from Lemma 6.1.8. By the previous lemma, the mappings Tn ' exist and have the form I + Gn, Gn : X H. Let us prove the second claim. For every n, we take the full measure set Q.
constructed in the previous lemma for T. Recall that we may assume that the Fn's are Borel mappings. Let f1 be intersection of all sets 12n with the set of all those x, for which one has the convergence Fn(x) - F(x). It is clear that the mapping Fo defined by the formula Fo(x) = lim F,(x) on the set f1 is a version of F. Let us show that Fo satisfies (6.6.1). Denote by 12o the set of all those x E Il
such that the set H fl (f1 - x) is dense in H. According to Proposition 2.4.10, one has y(I1o) = 1. Therefore, for any x E go and any h E H. there exists a sequence {h, } C H such that Ih - h, I --+ 0 and x + h, E Q. It follows from the uniform Lipschitzness of the mappings h Fn(x + h) on H combined with the convergence Fn (x + h,) - F)(x + h,) for every fixed i that the sequence F (x + h) converges. Thus, we get x+h E h o. It is clear that F0 satisfies condition (6.6.1) for all x E go. We shall deal with the version of T given by To = I + Fo (which does not influence the image measure). By virtue of the previous lemma, To is invertible
and T, ' = I +G, where G: X -+ H is Lipschitzian along H. Put Sn = T,-'. Note that Cn = -Fn o Sn, whence IGn - GkIH 5
IF"OSn-FkoSnIH+IFkoS.-Fk0 Skl,, IF, o Sn - Fk O Sn I,, +,IS, - Sk I,,.
Since Sn - Sk = Gn - Gk, we get from the previous estimate that
IFnoSn-F4.oS.IH
IGn - Gkl < 1
By virtue of the uniform integrability of {An} and convergence of {F}, we arrive at the relationship
-,(IF. oSn-FkoSnIH>c)=
r
J
Andy-»0 asn.k--.x, b'c>0.
Hence there exists a Borel mapping Go: X -. H such that IG,, - God,, - 0 in measure y. Passing to a subsequence, we get the convergence almost everywhere. Put So = I + Go and note that by Lemma 6.1.8 we have I o y. By the equivalence of the measures y and y o S;, n = 0.1..... the full measure
set Do contains a set hi, with y(h,) = 1 such that &(f11) C S1o for every n = 0,1.... On the set hl, one has IFnoSn - FooSoIH 5 IFnoSn IFnoSo - FooSoIH < AIGn - GoIH + IFnoSo - FooS0IH
0.
On the other hand, Fn o Sn = -Gn - -Go, whence Fo o So = -Go a.e., i.e., -Fo(x+Go(x)) = Go(x) a.e. Since the equation y = -Fo(x+y) is uniquely solvable for a.e. x (see the previous lemma), we have C = Co a.e. Thus, IG,, - GIH - 0 in measure. For the initial version, we have (I +G)(T(x)) = (I +G)(To(.r)) = x a.e.
6.6.
301
Nonlinear transformations
In order to show that T((1 + G)(x)) = x a.e., it suffices to note that the preimage under I +G of the full measure set SI2 = {x: T(x) = To(x)} has full measure. This 0 follows from Lemma 6.1.8, which yields the relationship y o (I + G)-' « y. In order to formulate the principal results of this section, as in the linear case, we shall need the concept of the regularized F edholm-Carleman determinant for the operators of the form I + K, K E R. Let us introduce the following notation for mappings F of the class TV1,C (y, H):
AF(x) := det 2 (I + D F(x)) I exp bF(x) -
2I F(x)I ,]
.
6.6.3. Theorem. Let F: X - H be a -y-measurable mapping such that
IF(x + h) - F(x)I < \Ihl,,, Vh E H for y-a.e. x,
(6.6.4)
where A < 1. Suppose that D,, F(x) is a Hilbert-Schmidt operator for a.e. x and that y-a.e. one has II D,,F(x)IIx < Al < oc. Then: (i)
There exists a full measure set S2 such that S1 + H = S2, T = I + F: 1 0 is one-to-one and onto, T(X\S2) C X\S2. In addition, the inverse mapping
S (in the sense that T(S(x)) = S(T(x)) = x for all x E S2) has the form S = I + G, where G satisfies the condition
IG(x+ h) - G(x)I < all - A)-'Ihl,,. `dh E H, `dx E S2, and 5 M(1 - a)-I: (ii) The measure yoT-' is equivalent toy and the density of the measure yoT-' with respect to y has the form i
d(y
)
dT
(x) = AG(x) = AF
(x)) .
(6.6.5)
(T- I
In addition,
d(yoS-') dy
(x) = AF(x).
(6.6.6)
PROOF. The existence of the inverse mapping follows from Lemma 6.6.1. We shall now verify all other claims. We shall make use of Theorem 4.5.7 about the exponential integrability of H-Lipschitzian mappings, which gives, in particular, the inclusion IFI E L2(y). Let be an orthonormal basis in H such that e;, E X. We know that the linear mapping J: x (?1(x)) identifies y with the product p of the standard Gaussian measures on the space IR". In particular, there exists a p-measurable linear mapping L with JL(y) = y p-a.e. Since J gives an isomorphism of the
spaces H(y) and H(p) = 12, the mapping To: y '-- y + JF(Ly) on IR" has the same properties as T. Indeed, IFo(y + h) - Fo(y)I y(N) = I F(Ly + Lh) - F(Ly)I H < alLhlx =.IhIH(,.), Vh E H(p).
Chapter 6. Nonlinear Transformations
302
S Al µ-a-e. Therefore, In a similar manner, IID,, ,,,Fo(y)IIo(H(µ)) = it suffices to prove our statements assuming that y = µ. First we shall consider the case, where F has the form F = (gyp,, ... , V.,0,0.... ) with gyp; E Cb (R') . In this case, by Fubini's theorem, everything reduces to the finite dimensional case, and it remains to apply the classical Ostrogradsky-Jacobi formula for the diffeomorphism
T = I + F on IR". This formula, applied to the standard Gaussian measure y with density p,, and an arbitrary smooth bounded function >!', yields
f O(T(x))pn(x)dx = f i'(y)P'(T-'(y)) IdetT'(T-'(y)) I IV
' dy.
R^
In order to derive from this formula the desired expression for the density of the induced measure, it suffices to apply Lemma 6.1.3 and notice that ITzI2 - Iz12 = 2(F(z), z) + IF(z)I2 and det 2(1 + F') = det(I + F') exp[-trace F'). Therefore, det (I + F'(x)) I exp [- (F(x), x) - 1 IF(x)I2]
= Idet2(1+F'(x)) exp[trace F'(x)-(F(x),x)- IIF(x)I21 = Idet2(I+F'(x)) exP[6F(x) - 1IF(x)I2]. Note that in the case we consider, by construction, the mapping C also has the form G = (gl, ... , g", 0, 0, ... ), where Co = (gi . , g,,) is a smooth mapping on K" with the Lipschitz constant ta Since the mapping So = I + Co on 1R" is the inverse to To = I + FO, where Fo = (p,, ... , iPn), one has SS(T0)TT = Ion 1R", whence Go (To)
V + 'rol
It is readily seen that 11(1 + FF)"'1Ic(f°) < (1 - A) -t, since [IFollc(wt^) < A < 1. Therefore, the Hilbert-Schmidt norm of the operator G'(To) is estimated by the number M(1 - A)'. Since To is a diffeomorphism, one has
IIGoIIx(J°) 5 M(1 - X)-'. Finally, in order to finish our discussion of the finite dimensional mappings, note that all the arguments given above are applicable to Lipschitzian mappings instead of smooth ones, with the only difference that the corresponding equalities and estimates involving the derivatives make sense and are valid almost everywhere instead of everywhere (see 1237, §3.2]). Certainly, this can be obtained as a corollary of the smooth case by the aid of suitable smooth approximations.
Let us now turn to the infinite dimensional mappings. Recall that we deal with X = IR" and H = 12. We may replace F by its version and assume that F is Lipschitzian along H with constant A for every x. We shall approximate our mapping F by the finite dimensional mappings of the form F. (x) = J PnF(xt, x2, ... 1 xn, yn+i , yn+2....) Y(dy), where P" is the orthogonal projection in H onto the linear span of e1,... , en . Note
that J Fn - FI - 0 in L2(y) and almost everywhere. Indeed, F. can be written in the form F. = P"IE"F, where IEnF is the conditional expectation of F with respect to the a-field generated by the first n coordinate functions. Since IF - P" FI H -. 0
6.6.
Nonlinear transformations
303
pointwise, then, by the Lebesgue theorem, the same is true also in L2(y). According to Theorem A.3.5 in Appendix, we get
IF - FfIH = IF
in L2(-y) and almost everywhere. The mappings F,, are easily seen to satisfy the same conditions as F, i.e., IIDHFnIIc(H) S A,
IIDHFnIIx S M.
It was shown earlier that the statements of our theorem are valid for F,,. The main technical step in the proof is the verification of the uniform integrability of the sequences A,, and Rn of the densities of the measures y o S,-,1 and y o T,,- 1, correspondingly, with respect to the measure y, where T,, = I + Fn, S,, = T,a 1. To this end, it suffices to prove the uniform boundedness of the integrals I A,, log IA,I d-y and I Rn log IR&I dry. We shall do this for A. making use of the
uniform estimate of the Hilbert-Schmidt norms of DH Fn. For Rn the reasoning is completely analogous, since, as noted above, one has IIDHGnIIn S M(1 - A)-1. Since y o Sn 1 = A. y, we have f A. log IAn I dy = I log IAn o S. I dry.
Therefore, we have to bound the integral of
logdet2[I+(D.F.)0S.]l+I(5F,,)0Snl+IFnoSnIH. The first term is uniformly bounded, since
Idet2(I+K)I <exp(IIKIIx), VK E ii, hence the functions I log det 2 (I +DH Fn) I are uniformly bounded. Using the identity
Fn o S,, = -G,, (recall that S,, + Fn o S,, = I = Sn - Gn), we estimate the integral of the last term by the L2-norm of IG,IH, which, by virtue of Theorem 4.5.7 about the exponential integrability of H-Lipschitzian functions, is dominated by a number dependent only on M(1 -.WI. It remains to estimate the integral of (bFn) o S,,. Note that
(bFn) o S. = -6Gn + IGnI , - trace
[(DHF. o
S.)D.G.].
(6.6.7)
This follows from the following relationships:
DH(FnoS,)=DHFnoSn+(DHFnoSn)DHGn, FnoSn=-Gn,
b(F, oS,)(x) = -(Fn(Sn(x)),x) + trace [DH F. (S. (.T)) + DH F. (S. (x)) DHG,, (x)]
Chapter 6. Nonlinear Transformations
304 n
where (u, v) stands for E u,v; if u = (ui, ... , un, 0, 0, ...) and v = (v;) E IR", which yield
(6F.) oS. = -(F. oS.,Sn) +trace [DHF. oS, = (Gn, Gn) + b(Fn o Sn) - trace [(DH Fn o Sn )DHGn] = IGn 1H - bGn - trace [(DH F. o Sn) DHGn] .
It was shown in Chapter 5 that the L2-norm of 6G,, is estimated by IIGn112.1. Further, we have lGnll2. = f IG,I2, dy + f IIDHGnll d T < f IGI dry + 12(1- a)-2.
,
The norms IGnI in L2(1) have already been estimated. It remains to note that I trace D. FnoSnDHGnl !5 sup 11D,, F. 11n sup 11D. G. 11% :5 C,
where C does not depend on n. By the finite dimensional case and Lemma 6.1.3, the application of Lemma 6.6.2 brings our proof to the end.
6.6.4. Definition. Let r be a -y-measurable function and let fl = jr > 0). H) the class of all -y-measurable mappings F: X -. H with the following property: for every x E fl, the mapping h h-. F(x + h) is Fl-echet differentiable along H at every point of the ball U,(,) = {h E H: IhIH < r(x)}, and, in addition, the corresponding derivative is a Hilbert-Schmidt operator such that the mapping h s DH F(x + h), U,(:) 71, is continuous. Denote by 7iC' (y, H) the class of all -y-measurable mappings F: X -» H such that for almost every x E X, the mapping h -. F(x + h) is Frechet differentiable along H and its derivative is a Hilbert-Schmidt operator such that the mapping h - DH F(x + h), H - . 7t, is continuous. Denote by 7iC'r
6.6.5. Remark. (i) Every mapping F E NO (y, H) has a version that satisfies the aforementioned differentiability condition for every x E X: it suffices to note
that the set Z of all x that do not satisfy that condition has the property that Z + H = Z. hence we can put F(x) = 0 if x E Z. Moreover, there is a full measure set n such that fl is a countable union of metrizable compact sets, 11= fl+H, and F is Borel on Q. This follows by Corollary 4.5.5 applied to the sequence of mappings
Fn = co,F, where s,, is an H-Lipschitzian function such that Vn = 1 on the set A,, = {x: supIhjH<2/n IIDHF(x+h)IIn 5 n} and cpn = 0 outside A,+n-'UH. (ii) If F E 7{C'r 10 (y, H), then by a similar reasoning, we get a Borel set flo C fl
such that y(fl\flo) = 0, U,(1) C f1o for every x E flo, and F is Borel on flo. Recall that TTF E RC' (y, H) for any F E LP(-t, H), where p > 1 (see Proposition 5.4.8).
6.6.6. Lemma. (ii) Let F E 71CT,1
(i) One has fC' (y, H) C %7,P.',' (y, H) for any p E [1, oo).
(), H). For any C, r > 0, let us put
Xc.r = {x: T(x) > 2r, sup [IF(x+h)IH + IIDHF(x+h)Ilx] 5 c}. IhIH
6.6.
Nonlinear transformations
305
Then, for any bounded H-Lipschitzian function tb vanishing outside the set
Xc,,+3-'rUH , the mapping V;F is bounded and H-Lipschitzian. In addition, ,,F E WP-'(-y, H) for any p > 1. PROOF. Claim (i) follows from (ii). Indeed, we may deal with a version of F that is Ftechet differentiable along H for all x. Let us take r = 1 and c = n, r = n.-1. Then the increasing sequence of the sets X = covers X; the interiors of the sets (X - x) n H in the topology of H cover H. Let us take a function v E Co (R1) such that 0 < Vi < 1, ?P(t) = 1 on [-1/3,1/3] and tb(t) = 0 outside [-2/3,2/3]. Put 0, (x) = 0(ndx (x)), where the function dA (the distance to a set A in the H-norm) is defined in Example 5.4.10(ii). Clearly, tt,,, = 1 on X,,. By claim (ii), we get H). E Let us prove (ii). If x E X, + rUH , then there exists a vector It E rU,, such that x + h E X,,,., whence I F(x)I H = IF(x + It - h)IH < c by the definition of Xe,.. Therefore, I t'F[H < c sup lt1Jl. In addition, II DH F(x + h) Il x <- c for all x E X,,r and It E rUH. By the Frechet differentiability of F on U,(.), for every z E Xe.,,, F(z + h) on the ball W. is Lipschitzian with constant c. This the mapping It implies that I F(x + h) - F(x)I., < clhlH for any x E X,,, + 3-1rUH and It with
IhL < r/3. By condition, there exists L such that 10.(.r+ h)
Llhl, for
all x E X and all h E H. Let IhlH u(x + h)F(x + h) - b(x)F(x) I H
<-
I tb(x + h)I ! F(x + h) - F(x) I H
+ IF(x)l,, Itb(x + h) - t1'(x)I S [csup I'W + cL] IhIH.
The same is true if x + It E Xe,, + 3-'rUH. If both x and x + h do not belong to X,,, + 3-'rUH, then ?P(x + h)F(x + h) = 0(x)F(x) = 0. Therefore, tbF is H-Lipschitzian. In addition, for every It E H, we get Oh (t'F) = Oh V, F+ tbOhF a.e., which yields the estimate 11DH(,tcF)II; <- 21F121DH If
IH +2sup1,01211DHFIlx
Hence OF E W'P"l (^y. H) for all p > 1.
6.6.7. Theorem. Let T(x) = x+F(x), where F: X H is a mapping of the class lice 1a such that the set fl = {r > 0} from Definition 6.6.4 has full measure (e.g., let F E RC 1 (-y, H)). Put
M = {x E f1: det 2 (I + DH F(x)) a 0}.
Then there exists a partition of M into disjoint measurable sets M such that on M one has T = T,,: = I + F , where, for every n, the mapping F E (y, H) is bounded and Lipschitzian along H, and, moreover, T is bijective and transforms y into an equivalent measure. In addition, for any bounded measurable function g, one has the equality
f 9(T, (x)) AF, (x) y(dx) = f g(x) y(dx). Ji
Y
(6.6.8)
Chapter 6. Nonlinear Transformations
306
Further, the set T-' (x) f1 M has at most countable cardinality N(x, M) for almost every x and, for any bounded function f, there hold the equalities
f f (x) y(dx) =
J f (T (x)) Ap(x) -y(dx), Al,,
J f (x)N(x, M) y(dx) = J f (T(x)) AF(x) y(dx) x X Finally, 7I f o T-' «'y and the following equality is valid:
E
d(7IMoT_1)(x)=
dy
1
yET-1(z)1A! AF(y)
PROOF. We shall derive this theorem from the previous one, showing that "locally" T is a composition of the mappings of the type considered in that theorem with a linear mapping transforming y into an equivalent measure. Note first that, as is easily verified, the composition T3 = T1 o T2 of two mappings TI and T2 of the
form T; = I + F,, F, E W2" (y, H), satisfying the condition y o Ti-' « y, has the following properties: y o T3 ' << y and, letting F3 = T3 - I, one has I + DH F3 = (I + DH F1 o T2) (I + DH F2),
AF., = AF, o T2 A F, .
Let us choose an orthonormal basis (e,) in H such that e, E X' and denote by {hj} the countable set of all finite linear combinations of e; with rational coefficients. Let us find a countable set {Km}, which consists of finite dimensional operators and is everywhere dense with respect to the Hilbert-Schmidt norm in the set of all Hilbert-Schmidt operators on H without eigenvalue -1. Moreover, one can take for K operators which are linear combinations of the one dimensional operators of the form x l(x)h, where I E X', h E H. With such a choice, the operators K are the restrictions of the continuous linear operators on X, denoted further by the same symbols K,,. Put 41,,, = I + Km. Let am := II (I + Km) ' II L(H). Let us introduce the sets An,i,,n =
x:
r(x) > 4, n
sup
IKm(x +
h{H <1/n
sup
lhl,
IF(x+h) - K,,,(x+h) - h,LH <
h)IH < n,
sup
a,. 8n
II D,,F(x + h) - Km II h :5a8
Ihlf
There is a function w E Co (R.') such that 0 < V < 1, = 1 on [-1/3,1/3], V = 0 outside 1-2/3,2/3], and Icp'I < 4. We can write T in the form
T = Rm o Tm, Rm := T o';n'. H is a mapping of the class It is clear that Rm = I + r,,,, where rm : X fC .1 (y, H). Let A be one of the sets An, j,m f1 M. Using the function dA introduced in Example 5.4.10, we can define the mapping
FA(x) := tp(ndA(TMI(x)) )[r.(x) - h)].
If'I;'(x) E A, then FA(x) = rm(x) - h,. Let us show that IIDHFAIIk < 5/8. To this end, it suffices to get the estimate jDH(FA oWm)IIx <
$m.
(6.6.9)
Nonlinear transformations
6.6.
307
Note that *ml = I - K. o ';1, hence,
Rm=%I+,.,1+Fo9,,,1=I-Kmo4jml+Fo*,nl, whence
r,,, = F o 'P.1 - Km o'Y;,,1. By virtue of this equality and Example 5.4.11, we get D. (FA o W m)(x) = D. (c(ndA(x)) [F(x)
- K,,,x - h3] )
= ,(ndA(x)) [D F(x) - Km] + ncp'(ndA(x))D.dA(x) ® [F(x) - Kmx - h,]. Therefore,
II D (FA 0 4.)(x)II <
DH F(x) - KmIITi + 4nlnd,A<213(x)IF(x) - Kmx - huIH,
where we used the estimate j,p'I _< 4, the Lipschitzness of dA along H, and the equality (x) = Km. If ndA(x) < 2/3, then there exists a vector h such
that Ih1 < 2/(3n) and x + h E A. Hence
KmIIN < am/8. In a
similar manner we get I F(x) - Kmx - hj I,, < am/(8n). This gives estimate (6.6.9). Finally, let us put TA(x) := x+FA(x)+hj. Then T = TAO1Pm on A. The mapping TA satisfies the conditions in Theorem 6.6.3. Indeed, combining Corollary 6.3.4, (6.6.9), equality D ('Yml) = (I + Km)-1, and using our choice of am, we get IIDHFAIIx =
o''m o'; )Ilx <-
*m)If7tIlD,r(Im)-1IIC(H) 5 g.
Enumerating the countable collection of the sets A defined above we get the sequence {An} covering Al up to a set of measure zero. For Mn we shall take An n(A.\u°_11 A;). Since the mappings TA o it. are bijective and transform y into an equivalent measure, we get the desired statements, except the boundedness of the mappings TA o'm - I, corresponding to F,, in the formulation of the theorem. Let us explain how to modify our construction in order to get bounded mappings F,,.
Note that
TA OTm-I=Km+PA oWYm+hj=Km+hj+tp(ndA)Irm0Tm-hjj = Km + h3 + P(ndA)tT -'m - h11 = K. + h,, + w(ndA)IF - Km - h,J.
By construction, Ihj + p(ndA)IF - Km - hjli is bounded on the set cl,,.A: _ {x: dA(x) < 1/n} and is zero outside fn,A. The function IKmxI is bounded on nn,A as well. Now we modify F as follows. Let us take a bounded H-Lipschitzian function (n.A such that (n.A = 1 on A and (n,A = 0 outside fln,A. Then we replace TA o4m by I + (n,A(TA o'ym - I). Clearly, this mapping coincides with T on A. Our new sequence of mappings can be taken for {Tn}. 0
6.6.8. Corollary. Suppose that the mapping T in Theorem 6.6.7 is bijective and that Al has full measure. Then -(o T' - -y and
d(yoT-1) dy
\ ) W
- A,(T,_1(x)). 1
Chapter 6. Nonlinear Transformations
308
If, in addition, T has Lusin's property (N), then for S = T' one has d(y
o
d-y
S-')
(x) = AF(x)
Note that in the situation of the previous corollary Lusin's property does not follow automatically, i.e., the measure 'y o T-1 may not be equivalent to y. For example, a full measure open set St C 1R1 with the complement K of cardinality continuum can be sent by a one-to-one nondegenerate smooth mapping T onto an open set Ell with the complement C of positive measure. Then T : K C is defined as an arbitrary one-to-one correspondence.
6.6.9. Corollary. Let T(x) = x+F(x), where F E fC' (y, H). If the operator DH F(x) has no eigenvalue -1 for every x from some measurable set B, then the measure yIB oT-1 is absolutely continuous with respect to the measure y. Probably, the condition of continuity of DN F along H in the formulations above can be relaxed.
6.7. Examples of nonlinear transformations In this section we discuss some frequently encountered mappings of infinite dimensional spaces from the point of view of the corresponding transformations of Gaussian measures. The aforementioned example of the homotety x F-+ 2x gives an example of a diffeomorphism that transforms a Gaussian measure into a nonequivalent measure. However, in this case the image-measure is Gaussian; thus, the following question arises: is not it possible, for any diffeomorphism F and a fixed Gaussian measure y, to find (at least locally) a Gaussian measure µ such that y o F-1 ti µ? The subsequent examples show that this is not always possible even for diffeomorphisms of a very simple form.
6.7.1. Example. Let us take two nuclear symmetric operators A and B with dense ranges in a separable Hilbert space X such that A(X) n B(X) = 0 (see Problem A.3.31 in Appendix). Let y be any nondegenerate centered Gaussian measure concentrated on B(X) (e.g., the measure with the covariance operator B4).
Put F(x) = x + (x, x)Ax. Then F is a diffeomorphism in some neighborhood of zero V, the set v n B(X) has positive 7-measure, whereas the set F(B(X)) has measure zero with respect to every Gaussian measure, not concentrated at a single point. Similar properties are enjoyed by the mapping 00
4(x) = x + E an(x,en)2en, n=1
where the en's are the eigenvectors of A with the eigenvalues an.
PROOF. Both mappings are polynomial and have I as the derivative at zero. Hence, by the inverse function theorem, they are diffeomorphisms of some neighborhoods of zero. The measure y is positive on every nonempty open set. In order to verify the last property, it suffices to make sure that every straight line meets
the set F(B(X)) at most at three points, and the set $(B(X)) at most at two points. We shall only verify the latter property. Indeed, let x, y E B(X). We shall denote by (xn) the coordinates of x in the basis {en}. Suppose that, for some
6.7.
309
Examples of nonlinear transformations
A E (0, 1), the point )4(x) + (1 - A)$(y) coincides with Cz), where z E B(X). Then z = Ax + (1 - A)y, since the vector En 1 aKhnen is an element of A(X) for any h E X, and A(X) fl B(X) = 0. Hence, for every n, we get .x + (1-A)yR = z
6.7.2. Example. Let g E CC (1R') be such that 0 < c1 < g' < c2 < oo. The mapping
G: CO. 1[ --. C[o,1], G(x)(t) = 9(x(t)), is a diffeomorphism. The image of the Wiener measure Pw under this mapping coincides with the measure induced by the solution of the stochastic differential equation b(et)dt, fo = g(6), where a(s) = g'(g-'(s)), b(s) = ,'-,g"(g-1(s)). This follows from Ito's formula applied to the process ft = g(wt) (see Chapter 2). As shown in [77] (see also Problem 6.11.14), the measure P" oG-1 has no directions of continuity if a is not a constant (i.e., g is not affine). In particular, this measure cannot be equivalent to a Gaussian measure. If the function g is real-analytic, then according to [768], as PwoG-1 is concentrated on some set that is in the previous example, the measure zero for all Gaussian measures not concentrated at a point.
Let B be a hounded Borel mapping on Rd and let pF be the measure on C([0,T],1Rd) generated by the solution of the stochastic differential equation At = dust + B(Et) dt,
where (wi)t>O is a standard Wiener process in Rd. It is known (see, e.g., [361] or [504, Ch. 7]) that pe is equivalent to the Wiener measure Pw on C ([0' TI, Rd) and its Radon-Nikodym density is given by 7'
A(w) = exp(J B(ut) dw-
f
IB(wt)12 dt).
(6.7.1)
0
0
This result is a special case of Girsanov's theorem (see Bibliographical Comments). More generally, the equivalence holds for the measures pt and it" generated by the diffusions C and 77 on Rd governed by the stochastic differential equations with one and the same diffusion coefficient A, which is a Lipschitzian matrix-valued mapping on Rd, equal initial values, and different drift coefficients B1 and B2, respectively, provided some technical conditions are satisfied, in particular, if B1(x) - B2(x) =
A(x)C(x), where C is a mapping satisfying certain conditions (see [504, Ch. 7]). As it was mentioned before Example 5.4.15, there exists a Borel transformation F of the Wiener space such that t(w) = F(w)(t). Hence µE is the image of the Wiener measure under the mapping T defined by t
T(w)(t) = w(t) +
f
B(F(w)(s)) ds,
0
which can be written in the form T = 1 +G. where G takes values in the CameronMartin space H (which consists of the absolutely continuous functions vanishing at zero and having square-integrable derivative). However, the mapping G may fail
to be differentiable along H, so the results in the previous section do not imply Girsanov's theorem.
Chapter 6. Nonlinear Transformations
310
If in the previous example a = const > 0 and g(0) = 0, then, according to Girsanov's theorem, the measure p is equivalent to the Gaussian measure jcvla_w generated by the process fwt (i.e., µy Ow is a homotetic image of Pa'). In particular, the measures µ{ and P1v are equivalent if a = 1. One can verify that in this case the measure pt is differentiable along all vectors from the Cameron-Martin space of Pu (see [608], [93]). Pitcher [608] conjectured that there is no differentiability for nonconstant or. His conjecture was proved in [74], [77] (see also Problem 6.11.14).
Let us apply Corollary 6.6.8 to derive Girsanov's theorem in the case of the identity diffusion matrix. To simplify notation we consider the one dimensional case, however, the considerations below apply to the multidimensional case as well. Let { be the diffusion governed by the stochastic differential equation
dEt = dwt +
en = 0.
Suppose first that B E Co (IR ). We shall assume that a probability space for the Wiener process is the classical Wiener space. The measure µf on C[0,11 is the image of the Wiener measure P14' under the mapping T(w)(t) = t(w) given by the integral equation
T(w)(t) = w(t) +
J B(T(w)(s)) ds. 0
This integral equation is uniquely solvable on [0, 1], since for any continuous function So, the mapping
V (x)(0 ='P(t) + J B(x(s)) ds is a contraction on C[a, b] provided lb - of sup [B'I < 1. The inverse mapping S to T is given by
S(x)(t) = x(t) - J B(x(s)) ds.
Clearly, the mapping G := S - I takes values in H = H(P"') and is infinitely Frechet differentiable. For every x, the operator D,,G(x) is nuclear. We have
8hG(x)(t) = - J B'(x(s))h(s)ds.
h E C[0,1].
0
It is straightforward to see that the complexification of the operator D,,G(x) has no nonzero eigenvalues, since the associated linear differential equation Ah'(t) = -B'(x(t))h(t), h(0) = 0, has only zero solution in the complexification of 140'1[0.1). Therefore, det s (1+ D,,G(x)) = 1 and trace,, D,,G(x) = 0. In addition,
IG(x)I = f IB(x(s))1Zds. 0
6.7.
311
Examples of nonlinear transformations
Letting {e,,} be an orthonormal basis in H, we have that {e;,} is an orthonormal basis in L2[0,11, whence the following equality in L2[0,1]:
B(x(s))
(B o x, e041 0.1]e"(s)' n=1
Hence, cc
1
0C
Je(s)dx(s)
_ -J Bn=1 n=1
_-J
0
0
B(x(s)) dz(s).
0
Now the formula for AG(x) gives the same expression as Girsanov's theorem:
/
2I2
1
P,,. (x) = expl f B(x(t)) dx(t)
- J1Bt
dt).
In a more general case, where B is, say, just bounded Borel, we take a sequence of uniformly bounded smooth functions BB convergent to B a.e. and verify that the corresponding Radon-Nikodym densities are uniformly integrable. Clearly, they converge a.e. to the desired expression.
6.7.3. Example. Let y be a centered Gaussian measure on a separable Banach space X, let H = H(-y), and let F: X -+ H be a continuously Frechet differentiable mapping (certainly, H is equipped with its natural norm). Suppose that I + F'(x) is injective on H for every x. Then y o (I + F)-1 y.
PROOF. Since F'(x) E £(X, H), then F'(x)l is a Hilbert-Schmidt operator (see Proposition 3.7.10). It follows from our assumption that I + F'(x) is invertible on H. The condition implies the continuous differentiability of the mappings h
F(x + h), H --' H. According to Remark 3.7.13, the mapping A -- AI H(.) is continuous from G(X, H) to ?{(H). By the continuity of the mapping x '- F'(x), X G(X, H), the mapping x F'(x)1 from X to 71(H) is continuous. 6.7.4. Example. Let y be the countable product of the standard Gaussian measures on the line and let T = I + F, where F: IR" --+ 12,
F(x) = (fn(xt,... ,xn-1))
n-
>2fn(xt.... .xn_1)2 < C, 1
n=1
and the fn's are Borel functions with fi = 0. Then T is one-to-one and, letting dy = e, where S=T-t one has ryoT l ry , d(y'oT-1) = o 1T -1 ' d(yoS-1) ,
dry
x
e
1 x
P(x) = exp(- > fn(xl,... ,xn-1)xn - 2 >2 fn(xl,... ,xn-1)2). n=1
(6.7.2)
n=1
PROOF. Note that for every y E IR", the equation T(x) = y has a unique solution: xl = yl, x2 = y2 - f2(y1), x3 = y3 - f3 (y1, y2 - f2(yl )), etc. In particular, S = T` l has the same structure as T. Let F,, = (fl,... . f.,0,0,.'.). If the functions ff are smooth, then the claim for I + Fn in place of I + F follows by the
Chapter 6. Nonlinear Transformations
312
Ostrogradsky-Jacobi formula employed in the proof of Theorem 6.6.3 (of course,
it is a special case of that theorem), since 8S,Fj = 0 and det2(1 + D F") = 1. By a simple approximation argument, the claim is true for any bounded Borel F,,. n
Now observe that the sequence
-1)x J is a martingale with
F, fj (x 1, ... i=1
respect to the sequence of a-fields An generated by x1,... , x", for yn = IE't" t;, where EE-t' is the corresponding expectation and t; _ - E f j (x1, ... , x,_ 1)xj (this J=1
series converges in L2). Denote by IE the expectation with respect to -y.
Our
reasoning applies to 2F in place of F, hence Eexp[21;,, - 2f,] = 1 and IEexp(2fn) < exp(2C), whence the uniform integrability of Therefore, exp(t:n - ; -i e in L1 (y). By Lemma 6.1.8 applied to the transformations (I + F") convergent to S, we get d(ry o S-1)/dry = p. By Lemma 6.1.3, one has
yoT-1
y and d(yoT-1)/dl = 1/(AoT-1).
As an application of nonlinear transformations of Gaussian measures we shall discuss an interesting construction of quasiinvariant measures on the groups of diffeomorphisms suggested in [685]. Let Sl be a bounded domain in IR" with a smooth boundary and let Diffk(Sl) be the class of all Ck-diffeomorphisms of the closure of Q. A proof of the following result can be found in 16851, [414].
6.7.5. Lemma. Let m and k be two integers such that k > 3m + 1. Then, for every i = 1,... , n, there exist real numbers co,... , c,n such that the differential operator Q defined on Diffk(f1) by "I EcjO'[(f')-lal`-,f]
Q(f) = 3=0
has the following properties: (f) Q(f) _ (f')-1O'f+ terms of lower orders in 8,;
(ii) for any,, and f in Diffk(Sl), the expression Q(1po f) - Q(f) as a differential operator in f has order less than, k - m.
Let us fix two integers k and in = 21 such that 2m > n, 2k > 3m - 2. Denote by Lk the space of those elements f in the Sobolev space 4i'2.2k+m(Sl 1R,") that vanish at the boundary of Sl together with the first k derivatives (more precisely, which belong to
(lR" 1R") when assigned zero values outside of Sl). The space
Lk has a natural Banach norm. The affine space I + Lk, where I is the identity mapping on 11, is equipped with the topology induced from Lk. One can check that
the intersection Gk = (I + Lk) nDiff2k(Sl) is an open subset of I + Lk with this induced topology.
Let E = id o.-(Sl, IR") be the Sobolev space of I("-valued mappings which vanish with the first m derivatives on the boundary of Sl. It is well-known that E is a closed subspace of the Hilbert space VS,2,m(11,IR") (with its natural norm) and that
the operator (-1)'Am has the inverse T which is a nonnegative Hilbert-Schmidt operator. Hence there is a centered Gaussian measure -y on E with covariance T. The main idea of [685] is to transport y to Diff2k(Sl) by means of the nonlinear differential operators constructed in Lemma 6.7.5. To this end, note that by virtue of that lemma, there is a differential operator Q: Gk lR") such that the principal part of Q(f) is (f')-1 L f ,where Lf = 18?k f /8x; k, and, for every W E
6.7.
Examples of nonlinear transformations
313
Diff2k(SZ), the expression Q(+po f) -Q(f) contains only those derivatives off which have order not greater than 2k - m. The map Q is differentiable and its derivative
at the point I is L. By a classical theorem in partial differential equations, L is a linear isomorphism between Lk and R"). By the inverse function theorem, It-) there is a neighborhood W of 1 in Gk such that Q: W -+ Q(W) C is a diffeomorphism.
Let us take a probability measure v defined by v = gy, where p is a smooth function on E whose support is a ball in Q(W). Put po(A) = v(Q(A n H-)).
Denote by G the subgroup of Diff2k(Q) which consists of the elements g such
that g - I E
Ijo.2k+2m(f
Rn). Let {g,} be a countable dense subset of G with
g, E Diff2k+2m+2(ft). Finally, let us put
p(A) = Ec;po(giA),
where c, > 0 and E c, = 1.
i=1
i=1
6.7.6. Proposition. The measure p on Diff2k(1?) is left-quasiinvariant under the action of the subgroup G and the measurep * A. where A(A) = p(A-') is leftand right-quasiinvariant under the action of G.
The proof is based on the fact that the mapping T = Q o L,, o Q-1, where L, , (f) _ +p o f , has the following special form:
T, f - f =
oQ-'f) - Q(Q-1 f) =T112Am 2P,(Q-1f) where P,, is a differential operator of order less than 2k, so that A"'12Pr,g E W02"(0). Thus, T, is a mapping which, for
6.7.7. Proposition. Let y be a centered Radon Gaussian measure on a locally convex space X. let H = H(y). and let F: X E be a measurable mapping with values in a complete separable metric space E such that, for some orthonormal basis
{e"} in H, the mapping t +--r F(x + to") is continuous for y-a.e. x and all n E N. Then the support of the induced measure yoF` on E is connected. In particular, this i s true i f F E l 4 " ( , " ) .
PROOF. Suppose that the support of the induced measure p (which exists, since p is automatically a Radon measure on E) is not connected, i.e. it can be represented as the union of two disjoint closed sets Z1 and Z2. Then there exists a continuous function +p: E -+ [0,1) such that Z1 =
(F-1(m,M)) = 0, y(F-'(m)) > 0, y(F-'(M)) > 0.
Then, for every n and y-a.e. x, the set. An : _ { t : x + te" E F-' (m, M) }
(6.7.3)
Chapter 6. Nonlinear Transformations
314
has Lebesgue measure zero. Indeed, let C. be the set of those x, for which An is not a Lebesgue zero set, and let Xn be a closed hyperplane in X such that X = Xn+Rle". Denote by ir: X -- X" the natural linear projection and put v = yo7r-1. We know that on the straight lines y+Rlen, y E X,,, there exist conditional Gaussian measures ryv, not concentrated at single points. Hence, for v-almost all
y E Xn, the set An has Lebesgue measure zero. Since Cn = (Cn n X") + Rle", then, by the definition of v, this is equivalent to the equality y(C,) = 0. By virtue of the continuity of F on x + RI en for a.e. x, we get that, for a.e. x, the following alternative takes place:
either F(x+ten) <m, VtER1, or F(x + ten) >M, VtER'. In addition, for every such x, if one of these two cases takes place for n = 1, then
the same case takes place for all n. In other words, the space X up to a set of measure zero is decomposed into two sets
D1 = {x: F(x+ten) <m, Vt E R1, Vn E IN}, 12 = {x: F(x+ten) > M, Vt E R', Vn E IN}. Clearly, these are measurable sets invariant with respect to the shifts along the vectors {en}. By virtue of the zero-one law, one of them has measure 0 and the other one has measure 1. This contradicts (6.7.3). 6.8. Finite dimensional mappings Let 7 be a centered Radon Gaussian measure on a locally convex space X and
let F: X -. R" be a sufficiently regular mapping. What can be said about the induced measure k = 7oF-1? Is it absolutely continuous with respect to Lebesgue measure A? Does it have a bounded density? Is it possible to choose a smooth version of this density? Certainly, some additional assumptions of nondegeneracy
of F are needed, since otherwise the measure j may have atoms. For example, if X = R', n = 1 and F is a smooth function, then a necessary and sufficient condition for the absolute continuity of k is that 7(x: F'(x) = 0) = 0. The proof of the following result is found in [237, § 3.2].
6.8.1. Theorem. Let F: R" - R" be a Lipschitzian mapping. Then, for every bounded Borel function g on R" and every measurable set A C R", the following identity
htrue:
J g(F(x)) (det F'(x)( dx = f 9(y)N(FI A, y) dy, A
(6.8.1)
Rn
where N(FI A, y) is the total number of elements in An F-1(y).
6.8.2. Corollary. Let A C R" be a measurable set and let
F = (F1, ... , Fn) : R"
R"
be a measurable mapping that a.e. on A has the first order partial derivatives. 0 a.e. on A. Then Lebesgue measure on A Suppose that det((VF=, VFj)) ti,j=1 is transformed by F into an absolutely continuous measure.
6.8.
Finite dimensional mappings
315
PROOF. In the case where F is Lipschitzian, the claim follows directly from identity (6.8.1). In fact, this identity means that the measure (f de F'j dx) o F- 1 has density N(FIA, x). In the general case, we may assume that F has the first order partial derivatives at every point of A. We shall use the well-known fact (see (237, 3.1.4]) that F is approximately differentiable at all points of A, and hence, according to 1237, 3.1.8(, A can be represented as a countable union of measurable
sets Aj such that the restriction of F to each of the Ac's is Lipschitzian. Since F can be extended from A. to IR" as a Lipschitzian mapping F., and the partial derivatives of F, almost everywhere on A; coincide with the partial derivatives of F, we get the claim.
6.8.3. Theorem. Let {a,} be an arbitrary sequence in H(y) and let F: X Rd be a y-measurable mapping with the following property: for y-a.e. x, there exist vectors vi(x),... ,Vd(x) in {a,} such that the vectors
F(x + tv, (x)) - F(x)
t-0
t
exist and are linearly independent. Then the measure y o F` on Rd is absolutely continuous. In particular, this is true if F is a mapping whose components belong to the Sobolev class 14 "(,), and the mapping DH F(x) is surjective almost everywhere.
PROOF. It suffices to show the absolute continuity of the measures 11B o F', where B is the set of all points x such that the vectors aa, F(x), i = I, ... , d, are linearly independent. Let E be the linear span of at.... , ad, and Y a closed linear subspace complementary to E. There exist. the conditional Gaussian measures y° given by densities on the planes E + y, y E Y. By Corollary 6.8.2, the measures yy(Bn(e+ ,) of-' are absolutely continuous. Hence, for every Lebesgue measure zero
set Z C Rd, we have yV (BnF-1(Z)) = 0, whence y(BnF-'(Z)) = 0. Note that, given an orthonormal basis {e,, } in H(y). the condition D F(x)(H) = Rd implies that Rd is spanned by DHF(x)(e,,) for some it,... ,id. Now the last claim follows from the existence of a modification of F that is locally absolutely continuous on
the lines parallel to e . ... , e (see Proposition 5.4.1). Unlike the finite dimensional case, the condition in Theorem 6.8.3 is not necessary. There exists an example (see (409], [410]) of a function F: C[0,11 - R, where C[0,1] is provided with the Wiener measure P's , such that F is infinitely Frechet differentiable, but the measure P"'I{F'=oi o F-' has a smooth density. In this sense, there is no direct infinite dimensional analogue of Sard's theorem (see, however, [4791 and Theorem 6.11.5 below).
6.8.4. Corollary. Let F: X -. Rd be a measurable mapping such that for a.e. x, the mapping h - F(x + h) is locally Lipschitzian on H(-y), and the set of all points x, where the Gateaux derivative DHF(x) exists but is not surjective, has measure zero. Then the measure y o F` is absolutely continuous.
6.8.5. Corollary. Let ' be a non-atomic symmetric Gaussian measure on a separable Banach space X. Then the norm q of X has an absolutely continuous distribution on (X, y). PROOF. We may assume that H is dense in X replacing X by the closure of H (which has full measure by Theorem 3.6.1). Recall that the Gateaux derivative q'
Chapter 6.
316
Nonlinear Transformations
exists y-a.e. (see Theorem 5.11.1). Clearly, q' does not vanish at all points where
it exists (note that q'(x)(x) = q(x) > 0 if q(x) exists). Then 0 at all such points x, since q'(x)(h), h E H, and H is dense in X. Hence Dxq(x) * 0 a.e.
x
6.8.6. Corollary. Let Q = E Qn, where the function Qn is a -y-measurable n=0
polynomial of degree not bigger than n, and the series converges in L2(-y). Suppose that
x
1 AnIIQnjI L?('1) < oc nO
for some A > 1.
Then either the measure y o Q-1 is absolutely continuous or
Q = const -,-a. e. PROOF. Let {e, } be an orthonormal basis in H(-y). For every n, the conditional
measures on the lines y + )Ede, have Gaussian densities for all y E Y, where Y is a fixed hyperplane complementary to We,,. Hence, for a.e. x and every closed interval (a, b}, we have Q(x + ten) = F_',, Qn (x + ten) and /b
A- J Qn(x+ten)2dt
a
Hence the norm of the polynomial t H Qn(x+th) in L2[a, b] is estimated by CA Therefore, by the condition .\ > 1, the function W(t) = Q(x+th) possesses the mean
square approximations vpn by polynomials of degree n such that the quantities < 1. According CnM2: = IIV-YnIIL'[n.bj satisfy the condition n-x to Bernstein's theorem (see, e.g., 1785, p. 399, Ch. VI, 6.9.151), this implies that V has a real-analytic modification. Thus, for any fixed n, the function Q has a limsup(en(p)2)1
modification that is real-analytic on the lines x+W WE,, and. hence is either constant or has the derivative with at most countably many zeros. The set of all x for which such a modification is constant for all n is invariant under the shifts to the vectors ten and, hence its measure is either 1 (and then Q = const a.e.) or 0. In the latter case, the measure y o Q-1 is absolutely continuous.
8.8.7. Example. Let y be a centered Radon Gaussian measure on a locally d
convex space X and let F E e Xk. Then either F is a constant or the measure k=0
y o F-1 is absolutely continuous.
6.9. Malliavin's method In this section, we discuss basic ideas of a general approach to the study of regularity of the finite dimensional images of Gaussian measures suggested by P. Malliavin and called now the Malhamn calculus.
6.9.1. Example. Let yn be the standard Gaussian measure on IR' and let f be a polynomial on lRn without critical points (i.e., V f has no zeros). Then the measure p = y o f -1 has a smooth density which together with all its derivatives decreases at infinity faster than any power of Ix1-1.
6.9.
Malliavin's method
317
PROOF. Let us consider the Fourier transform of the measure p, which, by the change of variables formula, has the form
A(t) =
J
exp(itf)d7n
Clearly, the function N is infinitely differentiable. Let us show that, for every k E V, the function tkj.(t) is bounded. The idea of the proof is to employ the vector field v = V f . Denoting by d1, the operator of differentiation along the field v and by p
the standard Gaussian density on R", by the aid of a formal integration by parts, we get
itµ(t) = Jo etI 1k-fdry = -
J
e`tf div
p vdx. (_1)
Let us now note that the function div (p v) has the form i-'Qp, where A _
atf
I V f 12 is a polynomial without zeros, and Q is some polynomial. This integration
by parts is justified if the function I-'Qp is integrable. The latter is indeed true, since, by the Seidenberg-Tarski theorem (see [347, p. 368, Example A.2.7j),
there exist two positive numbers C and a such that o(x) > Cuxi-O, whenever JxI > 1. Thus, Itµ(t)I < Jk0-'QIl1.lr,.,,. Repeating the procedure described, we get the boundedness of all the functions t'p(t). The same reasoning applies to the functions f'p, r E IN, replacing p, which completes the proof.
O
Trying in the infinite dimensional case to act according to the same plan, we face at once the obvious difficulty that the last equality in the integration by parts formula used above makes no sense due to the lack of infinite dimensional analogues of Lebesgue measure. Certainly, this difficulty is overcome if one defines the action of the differential operators directly on measures. In fact, this is the essence of the
theory of differentiable measures of Fomin and the Mailiavin calculus. However, a more delicate problem arises of finding vector fields, for which the integration by parts is possible. A simple example: the function f (x) = (x, x) on an infinite dimensional Hilbert space X. As shown in Chapter 5, there is no Gaussian measure on X differentiable along the vector field v = V f : x -- 2x. For this reason, for the Gaussian measure y with the covariance operator K, it is natural to take the field u = KV f, along which, as we know, the measure y is differentiable. Therefore, similarly to the finite dimensional case, it remains to verify the integrability of the functions (2Kx,x)-1.
6.9.2. Theorem. Let F = (F1,... , F,): X -. R" be a mapping such that F, E 141" (,Y), i = 1, ... , n. and
E np>1 Lx'(-y), where A= det ((DHF., D. Fi )H).
H = H(-y). Then the measure u := y o F-' has a density from the class S(R"). PROOF. Put ai j = (D F;, D Fj ) and denote by v'j the elements of the matrix inverse to (a,j). By condition, ii" E bt'"(y). For any smooth function W on R", we denote by 8;V the partial derivative in xi. According to the chain rule, one has the equality yik
a,'P o F =
k,j-l
n
ak) app ° F =
[1 L,k(9"(Y ° F), k[-1
Chapter 6. Nonlinear Transformations
318
where vk := D Fk. Integrating this equality and making use of the change of variables formula, we arrive at the relationship
I &w(v) µ(dy) _ R°
I
k='X
o F(x) 7(dx)
By the aid of the integration by parts formula, each term on the right-hand side i transformed into - cp o F(x)g;k(x) y(dx), where
E W'(-y)gik = v`k6vk + Therefore, the generalized partial derivative of the measure p in x; is the bounded
y) o F. Replacing atop by the partial derivatives of higher orders and repeating the procedure described, we conclude that all generalized partial derivatives of p are bounded measures. This yields the existence of a smooth density p of p. The same reasoning applies if the measure y is replaced by any measure p - y, where B E W'(-y). This implies that p E S(1R."). measure (g;&.
6.9.3. Remark. It is seen from the proof of the previous theorem that, in order to get only some finite differentiability of the density, it suffices to impose the existence of a sufficiently large number of derivatives of F and a sufficiently high order of integrability of the function A-1.
It also follows from this proof that the Sobolev norms of the density of the measure p are estimated via the norms IIF,IIp.r and IIO_,IILo(,) for sufficiently large p and r. In addition, if the measure ry is replaced by the measure P y, where e E W", then the Sobolev norms of the smooth density k, of the measure B y are for sufficiently large p and r. This estimated via IIF+IIp.r, IIPIIp.r, and IIo-' observation may be useful for constructing generalized functions on X. In addition, it will be used below in the study of the surface measures.
6.9.4. Example. Let F E W2.2 (y) be such that D F # 0 a.e. and ILFI + IIDN F(x)Ilx
E L'(y)
ID,, F12
Then the measure y o F'' has a density p of bounded variation (in particular, p is bounded). Moreover, if E La(y), then IIL-(-,)
IIV
PxooF. As above, we consider the vector field v = D F. For any cp ErC6 (IRI),
by the integration by parts formula as explained above, the integral J p o F dy can be written as the sum - poF dy + tip o F 82F dy. J 1 (avF)2 (a&F)2 Noting that 8,,F = I DH FI H , 6v = LF, and
ai F(x) = 2 (D F(x),
F(x))) H,
which is estimated by lID.2F(x)Ilx,
6.9.
Malliavin's method
319
we see that the existence of a density p of bounded variation follows from the integrability of the function
G _ ILFI +2IID,, F(x)Ilac(H) IDDFI2
Indeed, our reasoning yields the estimate if " pdxl < sup I
In addition, the total variation of the measure p' (the derivative of p in the sense of distributions) is majorized by IIGIIL1(,). It remains to be noted that the L2-norm of IID,, F(x)Ilx is estimated by cI IILFIIL2(,) due to Meyer's equivalence.
6.9.5. Example. Suppose that the conditions in Theorem 6.9.2 are fulfilled and that n = 1. Suppose that p, -+ p in W2.2(-y). Then the continuous versions of the densities kQ, of the measures (o, y) o F-' converge to the continuous version of kV locally uniformly.
PROOF. By analogy with Example 6.9.4, replacing -' by Q, y and differentiating by parts twice, one estimates v," o F p, dry by C sup IVI [email protected], where C does not
1
depend on Wand i. Hence the sequence {k,,) is bounded in W 1,2(R'), which implies that the sequence of the corresponding continuous versions is relatively compact in C[a, b) for every closed interval [a, b]. Since the measures ke, dx converge weakly to the measure kQ dx, we get the claim.
6.9.6. Corollary. Let Q be a -y-measurable polynomial o f degree d > 1. Suppose that, f o r every n E IN, there exist linearly independent vectors h1, ... , hn E H(y) such that X = Xo + R' h i + + R' hn , where Xo is a closed linear subspace which is algebraically complementary to the linear span Ln of the vectors h,, ... , hn in such a way that Q(x) = Qo(x) + ... + Qn(x), where Q, are measurable polynomials which depend only on the projections of x onto X0 and R' h,, and Qo depends on the projection onto X0 only. Assume, in addition, that Qi(xo + th,) = c,td + , xo E Xo, i = 1,... , n, ci # 0. Then the distribution of Q has a density from the class S(R' ).
PROOF. Let e1, ... , e,, be the orthogonalization of the vectors h, in H(-t). Then h, = Ae,, where A is a linear operator on L. Hence,
F (Oe.Q)2 > n
IDHQI >
n
IIAII-2 (ah,Q)2
Put G(x) _ E(Bh,Q)2. Note that Gi := (Bh,Q)2 are nonnegative polynomials 5=1
which depend only on the projections of x onto Xo and R'hi, and Gi(xo + th,) _ d2t2d-2+ C2 . Since the conditional measures on the subspaces y+Ln, y E X0, are ; nondegenerate Gaussian with one and the same covariance (see Corollary 3.10.3), then the next lemma yields the estimate y(G < e) S coast en/(2d-2)
Chapter 6. Nonlinear Transformations
320
6.9.7. Lemma. For every polynomial f on the real line which has degree d > 1 and the principal coefficient 1, the following estimate holds: mes
(t:
I f(t) I < E) < 2dE1/d,
where mes is Lebesgue measure. In addition, for any nondegenerate Gaussian mea-
sure v on IR" and every polynomial G of degree d on fit" which has the form n G(x) = c+ E g3(xl). x = (xl,... ,x"), where c> 0 and the g,'s are nonnegative J=1
polynomials on IR1 with the principal terms xd, the following estimate holds:
v(x: G(x) < e) < c(v)(2d)"e"/d, where c(v) depends only on v.
PROOF. Let us prove the first estimate by induction in d, noting that it is obvious for d = 1. If a polynomial f of degree d > 1 has a zero a, then f (t) _ (t - a)g(t), where g is a polynomial of degree d - 1, which, by the assumption of induction, satisfies the corresponding inequality. Since the set {I f I < e} is contained
in the union of the sets [a - e1/d, a+e""d] and (IgI < E1-1"d), its measure does not exceed 2E1'd + 2(d whence the desired estimate. If f has no zeros, say, is positive, then in the case f > E the estimate trivially holds, and the case where f = E has a root is reduced to the considered one by passing to f - e. In order to prove the second estimate, it suffices to be shown that it holds true with c(v) = I for the standard Gaussian measure, which is easily achieved by applying Fubini's theorem and the one dimensional estimate proven above. 6.9.8. Example. Let Q be the polynomial on C[0, 1] with the Wiener measure Pit defined by the formula 1
Q(x) = Jq(x(t))de, 0
where q is a nonconstant polynomial on the real line. Then the measure Pit, o Q` 1 has a smooth density. PROOF. Indeed, for any fixed n, we can split [0.1] into n equal closed intervals I, and choose smooth nonzero functions h1 with supports in I,. Taking for X0 an arbitrary closed linear subspace in C[0, 11, algebraically complementing the linear span of the hi's, and writing x = x0 + cl hl +... + c"hn, x0 E X0, we obtain (since the supports of the hJ's are disjoint) a decomposition of Q with the properties required in Corollary 6.9.6.
We shall say that a quadratic form Q on a Hilbert space H is infinite dimensional if Q(x) = (Ax, x), where A E £(H) is a symmetric operator such that
dimA(H) = c. 6.9.9. Corollary. Let a quadratic form Q on X (in the usual algebraic sense) be measurable with respect to -y. Suppose that Q is infinite dimensional on H(-y).
Then the measure ry o Q-1 has a density from the Schwartz class S(R'). The analogous claim is true for Q = (Q1,... , Q") : X - ' 1R.", where the Qi 's are quadratic forms whose nontrivial linear combinations satisfy the foregoing condition.
6.10.
Surface measures
321
The proof can be found in [77], (93] (see also Problem 6.11.25). It is an open problem whether there always exists a smooth (or bounded) density of the measure y o F-', where y is a nondegenerate Gaussian measure on an infinite dimensional
Hilbert space X and F is a continuous polynomial such that F' -A 0 (or, more generally, the set {F' = 0} is a finite dimensional manifold). A closely related problem is the study of the behavior of the quantity y(G < c) for small a and a nonnegative polynomial G. It is worth noting in this connection that, in the infinite dimensional case, the set of zeros of a continuous polynomial may have a rather complicated structure (see Problem 6.11.19).
6.10. Surface measures By the aid of Malliavin's method one can define surface measures generated by a Gaussian measure y on sufficiently regular surfaces. In order not to overshadow the essence of the problem by minor technical details, we shall assume that the surface S C X is given by the equation F(x) = 0, where F E W' (-y) and 1
E
I
I
L'(y),
(6.10.1)
p>1
moreover, from the very beginning we shall take for F its C,-quasicontinuous version, i.e., a version that is Cp,r-quasicontinuous for all p, r (see Problem 5.12.44
and Chapter 5). What should one mean by a surface measure on S? A natural interpretation is to take the e-neighborhood of the surface, divide its measure by 2e and let a tend to zero. Since a metric is, in general, only on H(y), then a suitable candidate for the e-neighborhood is the set S + eU". However, for some technical reasons, it is more convenient to define the surface measure ys as follows. As shown
above, the measure y o F-' has a smooth density k. Let
O = {y: k(y) > 0}.
On the sets Sy := F-' (y), y E 0, we shall choose in a special way condition measures yy and, for any bounded Borel function V, we shall put
f p(s)ys"(ds) := k(y) f w(s)ID"F(s)(ds). S"
(6.10.2)
$"
It will be shown below that this relationship defines some measure, which vanishes on the sets having zero Cp.r-capacity for all p, r > 1. Hence this measure is independent of our choice of a Cx-quasicontinuous version of F. Note also that since
the measure y has a Souslin support E, which is embedded injectively into lR" by means of a continuous linear mapping, we may assume that X = 1R°` (we may assume also that we deal with a Hilbert space). This observation may simplify some technical details. Let us find an absolutely convex compact set K such that y(K) > 1/2. We shall fix a function 9 E W"(-y), which equals I on K and is 0 outside 2K (the existence of such a function is established in Chapter 5). Put 9,(x) = 9(j-lx). We denote by g' a C,,-quasicontinuous version of g E W'O(y). For a function g E WOO(-y), let us denote by k9 the smooth density of the measure (g y) o F-I (which exists by virtue of condition (6.10.1)). We shall assume that 0 E 0, and give a construction for y = 0; for an arbitrary y E 0 the construction is completely analogous.
Chapter 6. Nonlinear Transformations
322
6.10.1. Lemma. Let us take a sequence of smooth probability densities 'Pj on the real line of the form 'pj(t) = jWo(jt), where oo is a smooth probability density with support in [-2,2), and po = 1 on [-1,1]. Let us consider the measures
vn.j = n.j ),
(F x
n.j(x) =
k(F('
.
Then, for every n fixed, as j - oo, the measures converge weakly to some measure v concentrated on S. In addition, the measures v vanish on the sets which have zero Cp,,.-capacity for all p, r > 1. Finally, for any function g E W°°(7), one has the equality
J)((fr) =
k,9..(0) k(0)
(6.10.3)
X
PROOF. For any fixed n, the measures v,,, are uniformly bounded and have common compact support. Therefore, it suffices to prove the convergence of the gdv,,,j, as j -+ oo, for all continuous g E W°°(7). These integrals are integrals J written in the form J 9(x)en(x) k(F(x))> 7(dx) = I ksen(y) k(y)) dy.
The right-hand side tends to In addition, this gives formula (6.10.3).
as j - oo by virtue of our choice of 'j.
Let now A be a set of zero C,,,.-capacity for all p, r > 1 and let e > 0. We can find an open set U D A, for which C2.2(U) < s. There exists a function ff E W2.2('() such that f, > 1 a.e. on U and Ilfc112.2 5 e. Then
(Iu'y) o f-' S (f, 7) o f-1 = kf, dx, whence v,,(U) < kf,(0)/k(0). It remains to note that kf,(0) tends to zero as e - 0, since, by virtue of Example 6.9.4, there holds the estimate Ikf(0)1 < const Ofll2,2
0
is uniformly tight and converges weakly 6.10.2. Theorem. The sequence to some probability measure 7o concentrated on S and vanishing on all sets having
zero Cp,r-capacity for all p, r > 1. In addition, for all g E W'°(1), the following equality holds true: g*(x)'Y°(dx) = k(0,)
(6.10.4)
J X
PROOF. Equality (6.10.3) shows that the sequence of nonnegative measures v,,
is bounded. In addition, it is uniformly tight. Indeed, the sequence g,,, = 1 - 9,,, tends to zero with respect to any norm 11 Ilp,r. Moreover, sup,, -f 0 as m - ac. According to Remark 6.9.3 and Example 6.9.5, sup,, k,_#, (0) 0 as oo. Since g,,, = 1 outside the compact set 2mK and g,,, > 0, then equality m (6.10.3) implies the uniform tightness of the sequence
Hence, for the proof of
its weak convergence, it suffices to verify the convergence of the integrals
g dv
k9(y) for all bounded functions g E W°°(7). It remains to note that for any y E 0. This follows from Remark 6.9.3 and Example 6.9.5. In addition, relationship (6.10.4) is established, whence it is seen that the measure 7o is a
6.10.
323
Surface measures
probability. Finally, the fact that the measure y° vanishes on the sets having zero
Cp,r-capacity for all p, r > 1, is proved in the same manner as in the previous lemma.
In a similar manner the conditional measures ryy arise. Now, by the aid of (6.10.2), we shall introduce the surface measures ysy. Let us put S = So. 6.10.3. Corollary. For any function g E W°°(-y), one has the equality J
I
9*(x)..S(dx) _ li0 2e
(6.10.5)
g(z)1DHF(x)IHy(dx)
IFI<e
S
PROOF. For all u E W'(-y), by virtue of the existence of a smooth density of
the measure (u y) o F-', one has k,.(0) = lim
'
J IFI
u(x),y(dx).
Applying this equality to u = g) DH FI H and using equalities (6.10.4) and (6.10.2), we arrive at (6.10.5).
6.10.4. Corollary. For any function g E W°°(y) and any continuous function u: 1R.' -+ IR' with bounded support, one has the equality
I u(F(x)) g(x) I
DH
F(x)I. y(dx) =
X
J IRI
u(y) [Jg-(x) rys"(dx)] dy.
(6.10.6)
S'
The proof is left as Problem 6.11.21.
6.10.5. Example. Let F E X,*, IIFIIL2(,) = 1. Suppose that for F we take a proper linear version of F. Put S = F-'(0). Then the surface measure ys is Gaussian. Indeed, putting in (6.10.5) the function g(x) = ettt(r), where t E 1R1 and 1 E X', we reduce the claim to the obvious two dimensional case, since the right-hand side of (6.10.5), by the change of variables formula and the equality I D FI H = 1, can be written as lim
1
e-o 2e
I
J r, ISe
errs'' v(dx),
where v is the image of y under the mapping T = (F,1) : X
IR2.
The proof of the following statement can be found in [11].
6.10.6. Theorem. The surface measure ys does not depend on a choice of the determining function in the following sense: if G satisfies the same conditions (0) = G' '(0) up to a set of Cp.r-zero capacity for all p, r > 1, as F and S = F"(0) then G yields the same surface measure. The foregoing results yield the following Stokes formula (see [11] for a proof).
6.10.7. Theorem. Let v E W'°(y,H). Then
f bv(x) y(dx) _ F
J
F-1(y)
(v(z), DH F(z)) H IDH
((z)IH
ys` (dz).
Chapter 6. Nonlinear Transformations
324
It was shown in [11] that, for a continuous function F. the surface measure yS is the weak limit of the surface measures on F.'(0) constructed for the smooth cylindrical functions F. from Theorem 3.5.2. Finally, note that the constructions and results described extend without any change to the case of the surfaces of finite codimension n given by the equations of the form F = 0, where F E H" (-y. R') satisfies the condition in Theorem 6.9.2 (in the corresponding formulas one should replace by JAI, and in Corollary 6.10.3 one replaces 2c by the volume of the n-dimensional ball of radius e).
6.11. Complements and problems Nonlinear transformations An interesting class of nonlinear transformations of Gaussian measures is connected with the flows generated by vector fields with values in the Cameron-Martin spaces. We shall mention a result obtained by Cruzeiro [174] under some additional assumptions, and proved in the form presented in [595], [596], [87].
6.11.1. Theorem. Let v E W2.1(y, H) be such that exp(A [ by I) +exP(AIID,, UI[c(H)) E L1(1).
VA > 0.
Then there exists a family of transformations Ut : X -+ X X. t E 1R1, such that
U(x) = x +
Jv(u(x))ds.
for all t and a.e. X.
0
In addition, y o
1
ti y and d(y o Ui 1)/dy(x) = exp(- J6v(u_8(x)) ds).
Another interesting class of transformations of Gaussian measures was introduced in [791]. In order to clarify a natural geometric structure of such transformations, we shall consider first the finite dimensional case. Let U: lR" -+ L(1R") be an operator-valued mapping such that, for every x, the operator U(x) is orthogonal. Suppose that this mapping it sufficiently regular, e.g.. is infinitely differentiable (or,
more generally, locally Lipschitzian). Put F(x) = U(x)x. The transformation F has the property IIF(x)II = I[x[I. Let u = -1 oF-1, where -y is the standard Gaussian measure on IR". When does F preserve the measure -y? Suppose that F is injective
and I det VFI > 0. In the case of a smooth mapping this means that F is a local diffeomorphism. The density of the image-measure is given by the formula (2x)-"I2ldetVF(F_1(x))t-l
4(x) =
exp(-ZIIxiil)
Therefore, the question is: when I det VF[ = I? Geometrically the equivalence of the initial condition to the latter one is obvious (since F preserves the norm, it preserves the standard Gaussian measure if and only if it preserves the spherical Lebesgue measure). The latter is equivalent to the preservation of the Lebesgue volume by F, which is precisely our condition on the Jacobian. Note that
VF(x) = U(x) + (VU(x))(x) = U(x)[1 + U*(x)(VU(x))(x)].
6.11.
Complements and problems
Since I det U(x)I
325
1, our initial condition reduces to the identity
det[I+U'(x)(VU(x))(x)]
(6.11.1)
1.
A simple sufficient condition for the last identity is the following one: for every h E IR, the operator V (U' (x)h) is nilpotent. Indeed, differentiating the identity U`(x)U(x) = I, we get U' (x)VU(x) + [VU`(x)] U(x) =_O.
Hence the operator U'(x)VU(x)(h) is nilpotent for any h. Letting h = x, we arrive at (6.11.1). Recall that an operator A E C(H) is called quasinilpotent if lim IIAn1111" = 0. n-x If such an operator is compact, then it is called a Volterra operator. It is known that a nuclear Volterra operator has zero trace (see [296, Ch. III, Theorem 8.4)).
Let U: X -. C(H) be a mapping such that, for every h E H, the mapping Uh: x U(x)h belongs to W1 (-y, H). 6.11.2. Theorem. Suppose that U(x) is an orthogonal operator a.e. and that, for every h E H, the operator DHU(x)h is quasinilpotent a.e. Then, for every orthonormal basis {en } in H, the functions b(Uen) are independent standard Gaussian random variables on (X, -y). Therefore, the mapping
x T(x) = Eb(Uen)(x)en n=1
is well-defined and -y o T-1 = y.
6.11.3. Remark. Let l; and 17 be independent standard Gaussian random variables and p a nonnegative Borel function. According to an unproven conjecture by Cantelli (see [516, p. 316]), if C + tp(1;')q is Gaussian, then
As we already know, Gaussian measures have Gaussian image measures and Gaussian conditional measures under linear mappings. However, this is not a characteristic property of linear mappings. In applications, one encounters essentially nonlinear mappings of this type. In Theorem 6.11.2 we considered a class of (typically nonlinear) transformations that preserve a Gaussian measure. Now we are going to consider another interesting class of mappings with the aforementioned properties which arise in statistics and the information theory (see [473], [773], [8181). Let y be a centered Radon Gaussian measure on a locally convex space
X and let fj:
1Ri-1 -, Xry, j = 1,... , n, be a finite collection of Borel mappings (where fl is just an element of X;). Let us consider the mapping
T: X -+ IRn,
T(x) = (91(x), 92(x),... ,9n(x)),
where the functions gj are defined recursively by 91(x) = fl (x),
9.i(x) = 6(91(x),... ,9.1-1(x))(x),
j > I.
To simplify notation, we write fj (y) instead of f3 (y,... , yj-1) (i.e., we extend fj to IRn as a function depending only on the first j - I coordinates). We shall deal with specific versions of ff. Namely, we fix Borel versions of ff (y) with the following
properties: (y,x) ,--+ fj(y)(x) is a Borel function on some Borel set D C 1R'-1 xX
Chapter 6.
326
Nonlinear Transformations
such that, for every y E R'-1, the section Dy := {x E D: (y,x) E D} is a Borel linear subspace with y(D5) = 1 and x - f,(y)(x) is linear on Du. In order to construct such a version, let us take an orthonormal basis 1C.1 in X, such that (,} C X' and define D as the domain of convergence of the series is a Borel function Clearly. ii;n: y n=1 x on IR-1-1 and E rj,n(y)2 < oc for every y. Then D is a Borel set with the desired n-l properties. With this choice, T becomes a Borel mapping. The next very interesting result was found in [473], [818] in a bit less general form, but with a similar proof.
6.11.4. Proposition. Suppose that, for every y = (yr,... , y,,) E IR", the elements fl, f2(yl),... ,f,,(yi,... ,y,,) are orthonormal in X;,. Then yoT-1 is the standard Gaussian measure on Ut". In addition, it is possible to choose Gaussian conditional measures yy, y E IR", on X, i.e., every measure ryv is concentrated on the set T-1(y), for every B E B(X) the function y" yy(B) is measurable on R" and (6.11.2)
y(B) = f yy(B) y a T-1(dy). ER.
n
Finally. yy has mean all =
yjR, fj(y) and covariance operator n
R5: f
R,f -
f (R,fi(y))R1fi(y) j=1
PRooF. The first claim is verified by induction in n. Suppose it is true for n - 1. Let us evaluate the Fourier transform of the measure y o T-1. This reduces to evaluating the integral fexP(i[Y191(x) + ... + yngn(x)]) y(dx). X
By the change of variables formula, it suffices to consider the case where X = IR'°, It is the product of the standard Gaussian measures and g1(x) = xt. If we fix xl and integrate with respect to (x2, x3, ... ), then we get exp(iylx1) eip(-[y2+...+yn]/2), which follows from the inductive assumption (recall that, for x1 fixed, 92 = fl (xl )
is a constant element of X;, g3(x) = f3(xt,g2)(x), and so on). Integrating in x1, we get exp(-[y2 + ... + y.2]/2), whence the first claim.
i
Now let us prove the second claim. Note that, although T may not. be linear, the sets T-1(y) are afftne subspaces Ty 1(y) of codimension n, where Ty is the linear mapping generated by the functionals fl, f2(yl), ,fn(yl. ,yn_1). Clearly, Ty 1(y) = ay + Ker T, since Tr(ay) = y, which follows from the relationship
f,(y)(R,fk(y)) _
6,k.
Thus, in order to conclude that the announced measure yy is concentrated on T-'(y), it suffices to note that R1(X') = KerT5. Finally, let us verify (6.11.2). To this end, given f E X', it has to be shown that.
1(f) =
f(f)oT(d). B"
Complements and problems
6.11.
327
By definition,
y"(f) = exp[iyif(Rfi(y))] exp[-2Rr(f)(f)+1: f(R.fi(y))2] i=1
Let us integrate this expression in y with respect to the standard Gaussian measure on lR". Integrating first in y,, and using that the elements f., do not depend on y,,, we get r l ll yif(Rrf2(y)) - 2Rr(f)(f)+ 2>f(Rrfi(y))2J expl-2f(R,fn(y))ZJ
exp
,=t
L ;=1
lL
= exp[i E yif (Rrfi(y))J eXp[-2R,(f)(f) + 2 Ef (R_ fi(y))2]
L j_1 j=t Integrating then in y,,'..... y' , we get exp(- R., (f) (f) /2) = y`(f ), which proves O
our claim.
Let us mention an infinite dimensional version of Sard's theorem obtained in [795] (see also [287], [447], [452]).
6.11.5. Theorem. Let F: X -. H be a Borel mapping, which belongs to the class Woo,' (y, H). Suppose that there exist a Borel set 1 with y(Sl) > 0 and a Borel function r : Il -+ (0, oo) such that, for every x E S1, the mapping h F(x + h),
H -+ H, is differentiable on the open ball {h E H: IhI < r(x)} and the mapping h -+ D F(x + h) with values in 11(H) is continuous. Put T(x) = x + F(x). Then for any Borel set B C X, one has
r
y(T(B n 0)) < I IAF(x)I y(dx).
(6.11.3)
Bnn
In particular, the image of the set {x E 11: det
0} has measure zero.
Note that if y(Q) = 1 (e.g., if F belongs to the class 7-IC' (y, H)) and if T has Lusin's property (N), then (6.11.3) yields
y(T(B)) S f IAF(x)I y(dx).
(6.11.4)
B
This is not true in general if T does not have Lusin's property (N), but T has a version which satisfies (6.11.4).
Distributions of nonlinear functionals The following result was proved in [496]. A shorter proof was suggested in [75].
6.11.6. Theorem. Let y be a nondegenerate Radon Gaussian measure on a Banach space X and M(t, x) = y(y: Ily - xfl <- t). Then the function Al is locally Lipschitzian and Hadamard differentiable on (0, oo) x X. In connection with this result, see also [757]. Note that the function Al may fail to be Frechet differentiable, in particular, its Gateaux derivative may fail to be continuous. For example, this is the case if X = C[0,1], since, as it was mentioned in Chapter 5, there is no nontrivial functions on C[O,1] that are Frechet differentiable and tend to zero at infinity.
Chapter 6. Nonlinear Transformations
328
Recall that the modulus of convexity of a norm q is defined by the equality q( x+ y) q(x), q(y) < 1, & - y) ba(c) = inf 1 2
> 0.
Let X be a Banach space whose norm q satisfies the condition
ba(e) > Ce°, (6.11.5) where C, a > 0. Suppose that y is a Radon Gaussian measure on X such that dim H(y) > a. Then q has a bounded density of distribution on (X, y) (see 1642]). For example, condition (6.11.5) is fulfilled for the spaces LP, I < p < or, with a = max(p, 2). As shown in (642], condition (6.11.5) cannot be replaced by the weaker condition of the uniform convexity. It is known [640], [641] that on every infinite dimensional separable Banach space X, for every e > 0, one can define a new norm II IIE and a centered Gaussian measure ye such that (1 - e)llxllX < IIxVI= ( (1 + `)Ilx]lx, and the function F(r) = y,(x: Ilxll: < r) has derivative F' unbounded near zero. If X is filbert, then, in addition, the norms 11 ll, can be taken infinitely Fr@chet differentiable outside the origin with bounded derivatives on the unit sphere. Let us mention one more result concerning the distribution of the norm (see (189], [780], [751). -
6.11.7. Theorem. Let X be a Banach space such that its norm q has k Lipschitzfan Frechet derivatives on the unit sphere and satisfies condition (6.11.5). Let y be a centered Radon Gaussian measure on X. If dimH(y) = oo, then the function q(x) < t) is k times differentiable and M(k) is absolutely continuAl: t --+ -Y (x:
ous. In addition, the function Q: x --+ y(U + x), where U is the unit ball in X, has k continuous Frechet derivatives. Moreover, the mapping (t, x) - y(tU + x) is k-fold continuously Frdchet differentiable. The aforementioned condition is fulfilled. in particular, for the spaces L2' [a, b], n E IN.
6.11.8. Theorem. Let F E P2(y) be such that F V P, (-y) and let A = D,2F. The following two conditions are equivalent:
(i) either the quadratic form Q: h -+ (Ah, h)H is strictly positive or strictly negative definite on some two dimensional plane in H, or F = glg2+g3+c. where g, are measurable linear functionals, c is a real number, and 93 does not belong to the linear span of gl and g2; (ii) the induced measure p := y o F-1 admits a bounded density. Moreover, in this case p admits a density of bounded variation.
PROOF. Assume that condition (i) holds true. To be specific, suppose that Q is strictly positive definite on the plane spanned by the vectors el and e2. We can always choose these vectors in such a way that Q(ei) = Q(e2) = 1 and (Ael,e2)u = 0.
Note that Be,F = fl + c1, 882F = f2 + c2, where fl, f2 are measurable linear functionals and c1, c2 E 1&. We shall deal with proper linear versions of fl and f2. Then we get 8e,8e,F = f2(et) = (D,+Fel,e2)s = (Ael,e2)H = 0 In a similar manner, 88, F = 88,F = 1 a.e.
a.e.
(6.11.6)
(6.11.7)
Complements and problems
6.11.
329
Put v =a, Fe l + 8,, Fe2 i G = &F = (fl + cl )2 + (f 2 + c2)2. We shall prove that there is C > 0 such that
'(t) p(dt) < Csup
(6.11.8)
V p E CC (IR1).
This implies the existence of a density of bounded variation. In order to get (6.11.8), we shall prove that
x
f'(F(x)) G(x) + 7(dx) < CsuPkp(t)I, V V E Co (IRl), E
(6.11.9)
with some C independent of E. Since G > 0 a.e., this yields (6.11.8). Let,p E Co (IRl) be fixed. Integrating by parts, the left-hand side of (6.11.9) can be rewritten as
l
d7=- j
O,.(`poF)G1
VoF[(G&+e2 +Gb+EJd7.
x Using (6.11.6) and (6.11.7), we get i9 ,,G = 2G and by = 2 - (f, + cl )el - (f2 + c2 )e2. Therefore, it remains to be shown that the L1-norms of the functions X
2E
(fl + cl)el
(G+E)2 -
G+E
-
(f2 + c2)e2
G+£
are uniformly bounded in E > 0. It follows from our choice of fl. f2, el. e2 that the measure v on IR2 induced by the mapping (fl + cl, f2 + c2) is nondegenerate and has a density bounded by some M > 0. Using the change of variables formula and then polar coordinates, we get
f (G +
f
E)2
X
f
2,6
dry = ,I (x2 + y2 + E)2 v(dxdy) < 2,M j (r22+E)2 dr = 27111. R2
0
Let us estimate the norm of Al := (fl +c1)el(G+E)-1. Let el = sill +a2f2+9, where a; are some reals independent of E and 9 is a measurable linear functional orthogonal in L2(-y) to fl and f2. Then g is independent of fl and f2, and since G is a function of fl and f2, we get IIAll1L'(,)
< loll ll
(f lG++ciE)f l
IIL+ la2lH
(f lG++clE)f2
fl + cl
'ILK(,) + ll911L1(,)11
G+£
IILI(j)
Since G = (fl + cl )2 + (f2 + c2)2, in order to get a uniform bound, it suffices to show that If, +cl1G-1 is integrable. Noting that the density of v is majorized by kexp(-8x2 - 8y2) for some positive k and 0, we get, using again the change of variables formula and polar coordinates, c
J l fG + El l d7 = i x2 X
R2
+i
2
y2 +
E
v(dxdy) < 2nk f rz+ E exp(- 3r2) dr. 0
which is uniformly bounded in e. In a similar manner one estimates the term (f2 + c2)e'2(G + E)-1. If the second possibility mentioned in (i) occurs, then g3 = c191 + c292 + 94, where g4 is a ry-measurable functional independent of gl and 92; hence the distribution of F is the convolution of a nondegenerate Gaussian measure with a probability measure, whence the existence of a bounded density
Chapter 6. Nonlinear Transformations
330
follows. We observe that the previous consideration could be reduced at once to x 1), where the two dimensional case if we write F = c + E C.C. + E n=1 n=1 x x c E IR', cn < oo, an < oo, and n = en for some orthonormal basis {en} n=1
n=1
in H(y). However, this would not ease the calculations above. Let us show that if condition (i) is not satisfied, then p cannot have a bounded density. First of all note that Q has rank either 1 or 2 (if Q has rank more than 2, then condition (i) is satisfied, and Q cannot have rank zero, since then F E Pl(y)). If Q has rank 1, x then, for some m, we get F = a,,, (t n, - 1) + CnCn + c, where an, # 0. Since n=)
condition (i) is not satisfied, one has n = 0 if n 96 m. Now the claim follows from the fact that for any reals k and c, the distribution of the polynomial x2 + kx + c on IR1 with the standard Gaussian measure has no bounded density. If Q has rank 2, but is not definite, in a similar manner we obtain that F =.9192 +g3 +c a.e., where g, are nontrivial measurable linear functionals and c E R1. Again, since condition (i) is not satisfied, g3 is a linear combination of 91 and g2. Hence it remains to verify that the distribution of the polynomial xy + ax + Qy + c on R2 does not 0 admit a bounded density (see Problem 6.11.24).
6.11.9. Corollary. Let Q be a -y-measurable function which is a quadratic form on X in the usual algebraic sense. A necessary and sufficient condition for the boundedness of the distribution density of Q is the existence of a two dimensional
linear subspace L in H(y), on which the form Q is either strictly positive definite or strictly negative definite (moreover, in this case the distribution density of Q has automatically bounded variation). Let us mention also the following nice result from 14501.
6.11.10. Theorem. Let y be a centered Radon Gaussian measure on a locally convex space X and let f2 be -y-measurable polynomials. Put F = (fl,... fn). Denote by . ' the class of all polynomials Q on IRn such that Q(F) = 0 y-a.e. Let Z = {z E IRn : Q(z) = 0, VQ E J}. Then the measure y o F-1 is absolutely continuous with respect to the natural Lebesgue measure on the algebraic variety Z. In particular, the measure y o F-1 is absolutely continuous if and only if .7 = {0}.
Problems 6.11.11. (Rademacher-Ellis theorem [227], 16321). Let is be a Borel measure on a Souslin space without points of positive measure. Prove that a Borel (or, more generally, p-measurable) mapping F: X -. X satisfies Lusin's condition (N) if and only if it takes every µ-measurable set into a p-measurable set. Hint: one implication follows from the measurability of the Borel images of Souslin sets (see Appendix); to get the converse, show that every set of positive measure with respect to a non-atomic Bore! measure p on a Souslin space X contains a nonmeasurable subset (use that (X, p) is isomorphic as a measurable space to a closed interval with Lebesgue measure). 6.11.12. Let y be a centered Radon Gaussian measure on a locally convex space X, let H = H(-y), and let K E 11(H) be a symmetric operator with the eigenvalues bigger
than -1. Show that
(det2(l+K))
1/2
= f exp(26K(,x))y(dx). x
(6.11.10)
6.11.
Complements and problems
331
Hint: take an orthonormal basis in H consisting of the eigenvectors of K and reduce the claim to the one dimensional case: then Kx = ax, bK(x) = at - axe, and the integral on +a)-In, which is the left side. the right in (6.11. 10) equals exp(a/2)(1 6.11.13. Let A, B E C, (1R'). A > c > 0 and let l: be the diffusion process on (0.11. B(C,)dt. Fo = 0. where governed by the stochastic differential equation dot = wt is a Wiener process. Prove that there exists a continuous mapping 4': C(0. 11 -. C[0.11 such that , = 4'(wt) a.e. Show that this may be false for a diffusion in R'. Hint: in the one dimensional case use Ito's formula to find a diffeomorphism ,p such that the process qt = p(s;t) is a diffusion with the unit diffusion coefficient and a smooth bounded drift v. Show that the unique solution qt of the integral equation qt (w) = rp+w(t)+ f, v(q,(w)) ds depends continuously on w E C(0.1). In order to construct a counter-example in R2. show that the functional
S(w) _ ! wI(t)dw2(t) - f t w2(t)dw1(t), 1
0
w = (wl.w2).
0
where w1(t) and w2(t) are two independent standard Wiener processes. has no continuous modification (use [361, representation (6.6). §6, Ch. 6]). Deduce that the same is true for each of the two integrals separately, which implies that the functional f (w) = j yo(w1(t)) dw2(t) has no continuous modification on the unit ball in C([0,11,1R2). provided p is a smooth function such that (,p[ < 1 and ,p(x) = x on [-1. 1]. Finally, consider the matrix-valued mapping A(x) = (A,,) with .411(x) = 1..42:(x) = 1. At2(x) = `o(x2)/2, and A21(x) = 0. 6.11.14. Let A and B be twice continuously differentiable functions on R1 with bounded derivatives and let Cr be the diffusion process given by the stochastic differential equation dt:t = A(Ct)dwt + B(Ft)dt, t;o = c, c E R'. Prove that if A is not constant, then the measure pE induced by t; on C[0,1) has no nonzero vectors of continuity in the sense of Problem 2.12.23 (in particular, pf has no nonzero vectors of differentiability in the sense of Definition 5.1.3). Hint: assuming that pt is continuous along h, show that h is Holder continuous of order 1/2 (see (77]): consider the function
L(t,x) = limsupfx(t + 6) - x(t)I(26loglog(1/b))-''2. t E 10? 11, x E C[0. 1).
r-o
Show that µt(x: L(t,x) _ ]A(x(t)l) = 1 for every t (see equality (7.1.4) in Chapter 7). Notice that L(t, z + Ah) = L(t,x) for every A E R' and choose tin such a way that A(x(t) + Ah(t)) qE A(x(t)) for sufficiently small positive A and all x from a positive measure set.
6.11.15. Let {t be the diffusion process in R" governed by the stochastic differential equation dl;t = A(t)dwt + [B(t)Ft + C(t)1 dt, {o = x E R". where A and B are bounded Borel matrix-valued mappings and C is a bounded Borel vector-valued mapping on (0,11. Show that the process t; is Gaussian. Hint: use that the affine transformations of Gaussian vectors are Gaussian and that the limit of a sequence of Gaussian vectors is Gaussian.
6.11.16. Let A and B be twice continuously differentiable functions on R' with bounded derivatives. Prove that the diffusion process given by the stochastic differential
equation dEt = A(tt)dwt + B(t;t)dt. o = c, c E R', is Gaussian if and only if A = crust and B(x) = ax + b, where a, b E R'. Prove the multidimensional analogue in the case where A takes values in the positive matrices (the claim is false without this restriction). Hint: use Problem 6.11.14 to reduce the problem to the case A = 1. In that case use Girsanov's theorem to show that the distribution µF of the process f in C(0,1) is equivalent to the Wiener measure P'v' and its Radon-Nikodym density is expQ(w), where Q is given
by (6.7.1). Notice that Q E P2(P'v'), since pt is Gaussian, whence OQ E Au for any h E Co [0, 11. Noting that the same is true for any interval (0. r). r E (0, 1), replacing [0, 1], and evaluating t9sQ explicitly, conclude that B" = 0.
Chapter 6. Nonlinear Transformations
332
6.11.17. Suppose that f is a diffeomorphism of R' such that yl o f-' = yl, where yl is the standard Gaussian measure. Show that f (x) = x or f (x) = -x. Hint: show that f (0) = 0; assuming that f'(0) > 0. use the change of variables formula to show that the inverse function g satisfies the ordinary differential equation g'(y) = exp(zg(y)2 - Zy2). 6.11.18. Show that there exists a nonlinear homeomorphism f of the real line such that y, o f -' = -y,, where yl is the standard Gaussian measure.
6.11.19. (S.V. Konyagin). Construct an example of a continuous polynomial F of degree 4 on a separable Hilbert space such that the cardinality of the disjoint connected closed components of the set F-1(0) is continuum. Moreover, an arbitrary Souslin set A on the real line can be obtained as the orthogonal projection of such a set F-'(0). Hint: given a closed set T C 12, whose complement is the union of the open balls with centers a(") and radii r", consider a polynomial F on 12x12, defined by the formula
F(x, y) _ E 2 " (c" (r.2 - IIx - a'"' 112 ) + y2) 2. "=1
(
i
Verify that the projection of F-'(0) to the first . factor coincides with T. Show that A can be obtained as the orthogonal projection of a
where c" = 2-" (l + r2 + IIa(n)112) closed set in 12.
6.11.20. Let y be the standard Gaussian measure on 1R' and let be a mapping such that the F,'s are polynomials, and the matrix with the entries or., _ (VF,, VF,). i. j = 1,... , d, is nondegenerate at every point. Then the induced measure y" o F-' on Rd has a smooth density from the Schwartz class S(1Rd). Hint: use that I det o,., I -' E L' (y") for all r > 1 by the Seidenberg-Tarski theorem. 6.11.21. Prove Corollary 6.10.4.
H such that F 6.11.22. Construct an example of a measurable mapping F: X is smooth along H, IIDHFIIc(H) 5 1/2, but the measure y o (I + F) ' is singular with respect to y. Hint: find F of the form F(x) on 1R" with the measure y which is the countable product of the standard Gaussian measures on the real line. 6.11.23. Show that the quadratic form Q(w) =
r1
J0
t
J 9(t, s) dw dwt 0
has a bounded distribution density if and only if there is a two dimensional plane L C Co [0,1] such that Q on L is either strictly positive definite or strictly negative definite.
6.11.24. Let l;l and {2 be two independent standard Gaussian random variables. Show that t2 - 42 has no bounded density of distribution. Show that 41 - fz has a Cb -density of distribution.
6.11.25. Let y be a centered Radon Gaussian measure on a locally convex space X and let F = (F, , ... , F.), where F. E Xo O Xi G X2. Suppose that all nontrivial linear combinations of the operators DH F, , ... , D,,2& have infinite dimensional ranges. Show that y o F-1 has a density from S(1R").
CHAPTER 7
Applications First. I believe, an introduction must be given at the beginning of the speech... Second is statement of the facts with any direct evidence to support it. third is indirect evidence, fourth is what seems probable, and I believe confirmation and supplementary confirmation are spoken of by that outstanding master-artist in words, the man from Byzantium. Plato. The Phaedrus
No philosophy will prove the connection necessary for all sciences so well as a specialized investigation of a part of a science
whatever it be. Here at every step one encounters something that cannot be understood without knowing one thing or another; and it is sometimes a long way to go for a reference. It is here that it becomes most clear that the sciences not only border one with another, but strike and penetrate one into another. However, in specialized investigations, it is a method and a direction which is principal.
N. 1. Pirogov. Letters from Heidelberg
7.1.
Trajectories of Gaussian processes
Let (WtET be a Gaussian process on a set T. The trajectories (or the sample paths) of the process t are functions on T, hence it is natural to raise questions about their properties such as boundedness or continuity (in the case, where T is a topological space). Since the distribution of the process is uniquely determined by its mean and covariance, one can expect that there exists a characterization of the boundedness or continuity of the sample paths in terms of these characteristics. The first result in this direction was obtained by Kolmogorov, who posed the problem in the general case. This problem has proved to be very difficult, and only recently the efforts of many years resulted in a solution that can be considered as a sufficiently complete one. In order to simplify the formulations we consider below only centered Gaussian processes. The principal idea in this circle of problems is to consider the semimetric
d(s, t) =
IEIC. - Ct12,
s, t E T,
and the associated metric entropy H(T, d, e). which is defined as H(T, d. e) = log N(T, d, e), where N(T, d, e) is the minimal number of points in an --net in T with respect to the semimetric d (i.e., points a.i such that the open balls of radius e centered at ai cover T). Therefore, one is concerned with the case, where T is completely bounded with respect to d. Note that normally d is a metric (i.e., 333
Chapter 7.
334
Applications
t;t = , a.e. if d(s, t) = 0), but we do not assume this. The expression
J(T,d) := r H(T d, e) de 0
is called the Dudley integral. It is clear that, for any completely bounded (with respect to d) space T, the convergence of this integral is equivalent to its convergence at zero.
7.1.1. Example. Let wt be a Wiener process on [0, 1]. Then d(s, t) = It - sl ] + 1, where [c] is the entire part of c. Hence the integrand has the logarithmic singularity at zero, and the Dudley integral converges.
and N(T,d,e) = [
Making use of the metric entropy one can estimate the supremum of the Gaussian process fit. Put
Sup(T) = sup{E(sup t:t), F C T, card F < oo tEF
In the case of a separable process there is no need to employ finite subsets of T.
7.1.2. Theorem. There exist two positive numbers CI and C2 such that, for any centered Gaussian process fit, the following inequality is valid:
C1 supeVH(T,d,e) < Sup(T) < C2J(T,d). f>0
(7.1.1)
Let us now introduce another metric characteristic. We fix a natural number q > 1. Let A = (An)n5N be a decreasing sequence of finite partitions of T into parts of the diameters at most 2q-n. For every t E T, let us denote by An(t) the unique element of the partition An, which contains t. Suppose that the elements A E An of every partition are equipped with weights an(A) such that EAEA., an(A) < 1 for every n. Put 00
BA, = su
(1 09
-
1/2
1
an (An(t)) CET n=! ` Finally, denote by 8(T) the infimum of the quantities 04. over all possible partitions A = (An ),,EN and weights a = (a(A))AEA., The relationship 8(T) < oc is called the majorizing measures condition. This condition is related indeed to measures on T. A probability Borel measure it on T is called majorizing if SUP f0 tier
1/2
r(log 1
i(B(t,e))
de < co,
where B(t, e) is the open ball of radius e centered at t. It is known that there exist two positive constants KI(q) and K2(q) such that
xr
K, (q) 8(T) < inf sup j (log 1 ) v tETJ p(B(t,e))
1/2
de
K2(q) 8(T),
0
where the infimum is taken over all probability measures µ. The following theorem characterizes the regularity of Gaussian processes in terms of the majorizing measures.
7.1.
335
Trajectories of Gaussian processes
7.1.3. Theorem. Let = (E, )eET be a centered Gaussian process. (i) For any q > 1, there exists a positive constant C(q) such that Sup(T) < C(q)O(T). If, in addition, 6A,,, < oo for some sequence of partitions A with weights a and
(log sup k-'x tETa(An 1
q"
(7.1.2 \ )
= 0,
/ lt))
then the process t; has a modification, whose almost all trajectories are uniformly continuous on (T, d). (ii) There exists a number qo > 2 such that, for any q > qo, there exists a number C(q) > 0 (independent oft), for which the following inequality is valid.
6(T) < C(q)Sup(T). If, in addition, the process C is continuous a.s., then condition (7.1.2) is fulfilled for some A and a. The following fact is very useful in applications.
7.1.4. Corollary. The convergence of the Dudley integral of a Gaussian process t; implies the existence of a continuous modification of C.
7.1.5. Example. Let t; = (Ft)IET be a centered Gaussian process on a set T C IR" such that there exist positive numbers C and b such that Eltt - SaI2 < Cllog it
-
S111-6.
Then t; has a modification with continuous trajectories on T equipped with the metric from R". al
PROOF. In this case the function H(T, d, e) is estimated by coast e- r , hence 0 the Dudley integral converges.
Proofs of the aforementioned results and a more detailed discussion can be found in [4711, [472], (491].
Concerning various interesting properties of the Brownian paths, see [369), [4811. For example, almost every Brownian path has no points of differentiability and has unbounded variation on every interval. The fact that the Brownian path has infinite variation (almost sure) on every interval [a, b] follows immediately from theorem on the quadratic variation (see Problem 7.6.12)) which states that 2^
nx t-1
lim F, (wit
- w(i-1)2
12 = 1
(7.1.3)
a.e.
Indeed, for any path of bounded variation the corresponding limit (on [a, b] replacing [0, 1J) is zero. The following more subtle result is proved in [257].
7.1.6. Proposition. Let f : R' x [0, oo)
R' be a continuous function. Sup-
pose that for every T > 0, [2-T)-1
nx lim
I f (w(,+1)2 t=0
(i + 1)2-") - f (wi2
i2-n)
2
0
Chapter 7. Applications
336
in probability. Then f (x, t) = f (0, t) for all x E IR' and t > 0. In particular, this is the case if f (t, wt) a.s. has locally bounded variation.
An extension of L6vy's theorem to more general Gaussian processes is found in [45].
Another interesting property of a Brownian path wt is expressed by the following Khintchine loglog-law (see, e.g., [369, Ch. 1]): for every fixed t > 0, one has
P(w: limsup [Wt+6(w) 6-.0+
wt(w)[
= 1) = 1.
(7.1.4)
J26 log log 3'
It is worth noting in this connection that a sample modulus of continuity of wt satisfies a.s. the condition 6,,,, (h) = O ( h 0911; . More precisely, as shown by Levy (see [369, Ch. 1]),
P(w: limsuP t-s-'o+
1w,P) - w'(w)I
= 1) = 1.
2[t - s[ log it--1.
In addition, for every t, one has
P(w: limsup [wt+h(w) h-o
- w,(w)[
[h[
> 0) = 1.
Chung, Erdos, and Sirao [168] showed that if h is an increasing continuous function such that t-1/2h(t) is decreasing for small t, then the convergence of the integral
J(h) = , J t`712h3(t) exp( h2t)`) dt 0
is necessary and sufficient for the equality
P(w: ta=c max Iwt(w) - ws(w)[ < h(e), a - 0+) = 1. Moreover, if J(h) = ac, then the probability above is zero. 7.2. Infinite dimensional Wiener processes We shall now discuss the concept of a Wiener process in infinite dimensional locally convex spaces. Note that if Wt is a standard Wiener process in 1R", then for any unit vector v E R.", the process (v, Wt) is one dimensional Wiener. Hence one might try to define a Wiener process in a separable Hilbert space X as a continuous process We with values in X such that, for every unit vector v E X, the real process (v, Wt), is Wiener. However, such a process does not exist if X is infinite dimensional. Indeed, let u and v be two orthogonal unit vectors in X. Then z E[ (u ft, W) a =t=2E[(u,wt)2]+21E[(v,44t)2J. X
Therefore, IE [(u, Wt) x (v, W, ), ] = 0. Let {en } be an orthonormal basis in X . Then
the orthogonal jointly Gaussian random variables (en,Wt)a are independent and lE[(e,,, Ilt)X] = t. By virtue of a classical result, the series 00
(Wt, 1411t)x = E(en, l'V')2 n=1
7.2.
Wiener processes
337
diverges a.s., which is a contradiction. Nevertheless, the idea suggested can be embodied as follows. Let X be a locally convex space, let H be a separable Hilbert space continuously and densely embedded into X, and let jH : X' -. H be the embedding defined as follows. For any k E X', the functional h (k, h)" is continuous on H. Hence there exists a vector j,, (k) E H such that, for all h E H, one has (7.2.1) (jH (k), h) = x, (k, h), . N
For example, if H = H(ry), where -y is a centered Radon Gaussian measure on the
space X, then jH (f) = R, f for every f E X'. 7.2.1. Definition. Let X and H be the same as above. A continuous random process (Wt)t>o on (11,.F, P) with values in X is called a Wiener process associated with H if, for every k E X' with Iju(k)I,H = 1, the one dimensional process (k,11t)
is Wiener.
Let Ft C F. t _> 0, be an increasing family of o-fields. A Wiener process (Wt)t>o is called an .fit-Wiener process if, for all t, s > r, the random vector Ht -4'. is independent of .F and the random vector Wt is ,t-measurable.
Note that in the case X = H = 111" we arrive at the usual definition. In connection with this definition, the following two problems arise.
a) Let X be a locally convex space. Is it possible to find H such that there exists an associated Wiener process in X? b) Let H C X be given. Does there exist an associated Wiener process in X? The first question has a positive answer for many spaces. 7.2.2. Proposition. A Wiener process in a locally convex space X exists precisely when there exists a separable Hilbert space E, continuously and densely embedded into X. If X is sequentially complete, then this is equivalent to the existence of a bounded sequence in X, whose linear span is dense. PROOF. The necessity of this condition is obvious. Suppose that X is a separable Hilbert space with an orthonormal basis {fin}. Let us choose numbers to > 0 such that to < oo. Let n=1
x
Jh1Hn<00 }.
H= I h = n=1
n=1
l
11
Then H is a separable Hilbert space with the norm IhjH and the natural embedding H -+ X is a Hilbert-Schmidt operator. The vectors en = t,, form an orthonormal basis of H. Let {w,, (t)} be a sequence of independent real Wiener processes. Then
x
the series E tnwn(t)2 converges a.s. Put n=1
X
Wt = E tnwn(t)vn n=1
It is readily verified that the process Wt is continuous (see the proof of the next theorem). Let v E X and vn = (v,Wn)x. Then (v, h),, = (iH (v), h)5 = E to 2(71t (v), Vn)x (h, vn)x , n=1
Chapter 7. Applications
338
whence
00
?H (v) E tnvn'Pn n=1 00
00
If I j,, (v%, = 1, then r (tnvn)2 = 1. Hence the process (v, Wt)x = E tnvnwn(t) is n=1
n=1
Wiener. Let X be a locally convex space such that there exists a separable Hilbert space E, continuously and densely embedded into X. We can take an arbitrary Wiener process in E, associated with any Hilbert space H embedded into E, and consider it as an X-valued process. Clearly, it is continuous. Let v E X' be a vector such that I?H (v) I H = 1. Denote by u the restriction of v to E. Then
&.(u,Wt)s = IiH(u)IH
=suP{(.?H(u),h)y:
IhIH <_
1}
= sup{ &.(u, h).: IhIH < 1} = sup{ x.(v, h)x : supf(jH(v),h)N: IhIH <_ II
IhIH <_
1}
=12N(v)IH = 1.
Therefore, (WW)t>o is a Wiener process in X. In order to prove the last claim, note that if {an } is a bounded sequence in a sequentially complete locally convex space X, then there exists a separable Hilbert space, containing {an} and continuously embedded into X. Such a space can be constructed by the aid of the mapping IZ - X, (h,,) - En= 2-nhnan, which is well-defined and continuous by virtue of the boundedness of {an} and the sequential completeness of X (the image of the unit ball under this mapping is bounded in X). Problem b) is more delicate, since it is connected with the problem of existence of a Gaussian measure on X with the Cameron-Martin space H (the distribution of the random vector WI will be such a measure).
7.2.3. Proposition. Let X be a sequentially complete locally convex space, on which there exists a centered Radon Gaussian measure -y with the Cameron-Martin space H = H('y) dense in X. Then there exists a Wiener process (Wt)t>o associated with H such that the distribution of W1 coincides with y. PROOF. Let K be an absolutely convex metrizable compact set of positive -ymeasure. There exists a sequence {ln} C X separating the points of the linear span
E of the compact set K. Put J: E -+ RO°, Jx = (ln(x)). The set Q = J(K) is absolutely convex and compact in R°°. By virtue of Problem A.3.27 in Appendix,
its Minkowski functional pQ has the form pQ = supi fi, where fi E (R)'. Let us take an orthonormal basis {en } in H and a sequence of independent one dimensional
Wiener processes wn(t) on a probability space (St, P). Put
Wt(W) = E wn(t)(w)en.
(7.2.2)
n=1
By Theorem 3.5.1, for every t, series (7.2.2) converges in X a.s. In addition, WI induces the measure -y. It is clear that, for every i, the real process fi(JW1) is a
7.3.
Logarithmic gradients
339
martingale with respect to the filtration generated by the processes w"(t), since it is proportional to a standard Wiener process. Therefore, the process qi = PQ(JWL) = supf+(JWt) i
is a submartingale (see Problem 7.6.19). Hence, by Doob's theorem (see Theorem A.3.6 in Appendix), for any fixed T, one has
E sup q2 <2E4=2T2 (PQ(Jx)2 y(dx) < oo tE(O.T]
X
due to Fernique's theorem. Therefore, for all w from some set l'Zo of full measure, one has sup PQ(Jlvt(w)) < 00, i E (0,'11
which means, for such w, the existence of n(w) E IN such that JWt(w) E n(w)K whenever t E [O,T[. It remains to note that for almost all w E 120, the function C,(JlVt(w)), where the t;'s are the coordinate functions on IR'°, is continuous on [0, T]. Then, for such w, we get the continuity of the mapping t'- Wt(w), since on the compact set n(w)K the initial topology coincides with the topology, defined by the countable family of the functionals {,oJ, which separate the points.
7.2.4. Remark. It is clear that the sequential completeness can be replaced by the existence of a convex compact set K of positive measure. Indeed, then the convex hull of the union K and -K is convex and compact (see [220, Proposition 9.1.9)). In addition, it is absolutely convex. Hence one can take in K a metrizable compact set of positive measure and pass to its closed absolutely convex hull. For example, the space co with the weak topology is not sequentially complete, but possesses the indicated property (since any Radon measure with respect
to the weak topology of co is Radon with respect to the norm topology as well). We do not know whether the previous theorem is valid without additional assumptions concerning the space X (in the part concerning the continuity of the process; formula (7.2.2) defines a process Wt in an arbitrary space). In the case, where X is a separable Banach space, similarly to the scalar case, by virtue of Theorem A.3.22 in Appendix, for any 6 E (0, 1/2), the Wiener process in X has sample paths that are a.s. Holder continuous of order 6. 7.3. Logarithmic gradients The logarithmic gradient of a measure 1A on IR" with a smooth density f is the
mapping V f If. Although in infinite dimensional spaces neither f nor V f make sense separately, it turns out to be possible to define their ratio. Logarithmic gradients of measures were introduced in [141. We shall see that logarithmic gradients of Gaussian measures are precisely the drifts of the symmetrizable diffusions generated by linear stochastic differential equations.
Let X be a locally convex space, let H C X be a continuously and densely H be the embedding defined embedded separable Hilbert space, and let j,,: X' above in (7.2.1).
7.3.1. Definition. Let a Radon measure p on X be differentiable along all X such that
vectors from jN(X'). If there exists a Borel mapping 0.1 : X =QH(k)
VkEX',
(7.3.1)
Chapter 7. Applications
340
then this mapping J31 is called the logarithmic gradient (the vector logarithmic derivative) of the measure p, associated with H.
In other words, the logarithmic gradient is a Borel mapping 0H : X - X satisfying the integration by parts formula
f ai (k)f(x) p(dx) = - f f (x)x.(k, /H(x))X p(dx), X
'dk E X'. f E FC".
X
The reader is warned that if p is a Gaussian measure, then H in the foregoing definition may be different from its Cameron-Martin space H(p).
7.3.2. Example. Let X = H = R" and let p be a measure on R" differentiable along all vectors. Denote by p the density of p. Then QN(X) _ ©P(x) P(x)
PROOF. Identifying (R")' with R", one has j,, (k) = k for any k. It remains to note that 8kp(x) = (k,Vp(x)). 7.3.3. Example. (i) Let p be a centered Radon Gaussian measure on a locally convex space X and let H be its Cameron-Martin space. Then
QH(x) = -x. (ii) Let X = R", let H = 12, and let p be the countable product of probability measures it,, defined by smooth densities p" on the real line with p'" E L'(1111). Then R" - R°` is given by the equality
(3,(x))n = pn(xn) Pn(xn) PROOF. Claim (i) follows from Theorem 5.1.6.
(ii) In this case X" is the space Ra of all finite sequences and j (k) = k for a n y k E R o x . The logarithmic derivative of u along k = (k1, ... , k", 0.0, ...) is the function " x F-'
kip: (xi)/P1(xi).
Therefore, (p;,/p") t is the logarithmic gradient. Let us consider an example, where the space H involved in the definition of logarithmic gradient differs from the Cameron-Martin space.
7.3.4. Example. Let y be a centered Radon Gaussian measure on a Hilbert space X with the Cameron-Martin space H(y) = T(X) and let H = Q(X) C T(X), where T and Q are injective nonnegative operators with Q2(X) C T2(X). Then
3H(7) = -R', where R is the operator satisfying the equation Q2 = T2R. PROOF. First of all, note that, by condition, the operator R exists algebraically. By the closed graph theorem (see Appendix), R is continuous. One has: (k
(Rk,-X)x = (T-2T2Rk,
-Xj, = 1T2Rk(X)I which is the desired relationship, since in our case we have j, (k) = Q2k = T2Rk.
0
7.3.
Logarithmic gradients
341
In an infinite dimensional space, a Gaussian measure may fail to have the logarithmic gradients along certain densely embedded Hilbert spaces H (even if jH(X*) C H(-y)) 7.3.5. Example. Let H = 12 and let X Then
{(xn):
1x1l2
n-2xn=1
x J 3HW) = 1 (xn): E n2xn < 00}. n=1
l
Let us take for p the countable product of Gaussian measures on IR1 with densities
pn(t) = n(2n)-112 exp(- 1 n2t2J.
In this case, the space H(p) = j,, (X') is embedded into X by a Hilbert-Schmidt operator, but H generates no logarithmic gradient. PROOF. Indeed, otherwise we would have = -n2xn,
3' (x) n
since for the n-th coordinate functional In : x
xn one has jH (In) = en, where
{en } is the standard basis in 12, and
=pn(xn)/pn(xn) _ Thus, the series
E/
\2
n-21 H(x) 1 =
n=1
n2xn.
n2x n=1
diverges p-a.e., which contradicts the inclusion 3' (x) E X .
0
7.3.6. Proposition. Let y be a centered Radon Gaussian measure on a locally convex space X and let H be a separable Hilbert space, which is continuously and
densely embedded into X. Suppose that H C H('y). Then 3y exists. Moreover, there exists an operator T E C(H(y)) such that H = T(H(y)), and 3y coincides with the measurable linear extension of -TT'. PROOF. Note that, by the closed graph theorem, the natural embedding H H(y) is continuous. Denote by T the composition of an arbitrary orthogonal isom-
etry H(y) .- H with the embedding of H into H(y). Then T E C(H(y)) and H = T(H(y)). We know that every continuous linear operator A on H(y) admits a unique extension to a measurable linear mapping A: X -+ X. Let us apply this fact to the operator A = -TT'. For any k E X' and any h E H(y), one has
-X.(k,TT'h)a = -(7,-1 jH(k),T`h)H(,)
(jx(k),h)H(,),
since j, (k) E H = T(H(y)). The functional h (jH(k),h)H(-)) is the restriction to H(y) of the measurable linear functional -v, where v = jH (k) (see Chapter 3). This means that the measurable linear functionals x -. (k, TT' (x)) and v coincide
Chapter 7. Applications
342
we may put 3H(x) = -YT, (x).
7-a.e., since they coincide on H(-y). Since v
0
In the finite dimensional case, a probability measure with linear logarithmic gradient is necessarily Gaussian. The example constructed below shows that the situation is different in infinite dimensions. This phenomenon is closely related to the uniqueness problem for measures with a given logarithmic derivative. The uniqueness problem, as well as the existence problem, admits several different settings. The various possibilities existing here are discussed in [821. Note that difficulties arise in the case where one has to compare logarithmic gradients of mutually singular measures. One of the possibilities is to fix some Borel version of 3. Simple examples show that even on the real line different probability measures with smooth almost everywhere positive densities may have equal logarithmic derivatives. In-
deed, let V be a smooth function such that y,(0) = 0 and :p > 0 on R' \ {0} and p dx = 1. Put
J
s):= 2v on [0,oo),
c1p
on (- -x., 01,
where c > 0 is such that Jt'dx = 1. Then the measures p := pdx and n := Odx are different, but have one and the same logarithmic derivative
rli'/rl'. To
be more specific. let V(x) := x2p(x), x E R', where p is the standard Gaussian density on IR' .
One can verify (see 191)) that two probability measures on IR' with equal continuous (or at least locally integrable) logarithmic gradients coincide. This is no longer true in infinite dimensional spaces even for linear logarithmic derivatives.
7.3.7. Theorem. Let X be an infinite dimensional separable Hilbert space. Then there exist a separable Hilbert space H densely embedded into X by means of a Hilbert -Schmidt operator and two different nondegenerate centered Gaussian measures it, and p2 on X such that for 3 and Mill one can take one and the same continuous linear operator A on X. PROOF. Let us use Example 7.3.4. Let X = L2[0,1] and let. H = W'2[0, 11 be the Sobolev space of all functions with the absolutely continuous third derivative such that fill E L2[0,11, f(,"(0) = f(,)(1) =0, 0:5j:53 .
Define operators (A,. D(A;)), i = 1, 2, on X by the formula D(AA)
{u E l%.2.210,1] 1 u(0)
= u(l) =
0},
D(A2) := {u E 1V2.2[0,111 u(1) - u'(1) = 0 = u(0) + u'(0)}. A3
A2
= -A.
It is known that (A D(A,)), i = 1. 2, are injective nonnegative self-adjoint operators on X and that T, := A; ' are injective nonnegative self-adjoint Hilbert-Schmidt operators on X. In order to get the injectivity, note that, for any u E D(Aj), one has
7.3.
Logarithmic gradients
it
343
it
1
- f u"(t)u(t) dt = 0
u'(t)2 dt + u'uI ; = J u'(t)2dt > / u(t)2 dt, 0
0
0
and, for any u E D(A2), one has i
1
-
r u"(t)u(t) dt = / u'(t)2 dt + u'uI01 0
0 1
r
=J u'(t)2dt+u(1)2+u(0)2> fu(t)2dt. 0
0
Since the embedding H C X is a Hilbert-Schmidt mapping, there exists an injective
nonnegative self-adjoint Hilbert-Schmidt operator Q on X such that Q(X) = H and I 1H = IQ-1 - Ix. It is clear that
Q(X)=HCT,(X)nT2(X)and T12h=T22h, Vh E H. Since Q2(X) C Q(X) and T,2(X) C T,(X), i = 1,2, then our claim follows from Example 7.3.4 if we choose for the p,'s the centered Gaussian measures on X with the covariance operators T,2, i = 1, 2, and take A = R', where R = T 2 Q2 = T2 2 Q2. The theorem is proven.
7.3.8. Corollary. Let p = apI + (1 - a)µ2, where p1 and µ2 are the centered Gaussian measures from the previous theorem and a E (0,1). Then p is a nonGaussian probability measure, but f3 = ,0"H' = A is a bounded linear operator.
The situation described in Theorem 7.3.7 is not possible if H is the CameronMartin space of the two measures.
7.3.9. Proposition. Let u be a Radon probability measure on a locally convex
space X, let A be a continuous linear operator on X, and let f31 = A p-a.e. Suppose that every set from E(X) up to a set of measure zero coincides with a set from the o-field generated by the functionals k o A. k E X'. Then p is a centered Gaussian measure. In particular. this is true if the operator A is injective.
PROOF. Let k E X Put 1 = koA and h = j,, (k). Suppose that ,31(x) equals Ax p-a.e. Then i3 (x) coincides p-a.e. with -1(x). Let us show that I is a centered Gaussian random variable on (X, M). To this end, note that, by virtue of the integration by parts formula, one has dt p(tl) = i f l(x) exp(itl(x)) p(dx) = i
J
8l, exp(itl(x)) µ(dx) = -tl(h)µ(tl),
whence µ`(t1) = exp(-'-2l(h)t2). Thus, all functionals k o A. k E X', are centered Gaussian random variables on (X, p). By condition, this implies that p is the centered Gaussian measure with covariance Q(koA) = (k, Aj (k)). The last claim follows from Proposition A.3.12 in Appendix.
Let p and v be two equivalent probability measures on X constructed according to Example 7.3.8 such that 31 = 3N = A, where A is a continuous linear mapping on X. Let be continuous linear functionals on X such that jH (ft),. .. , j, are orthonormal in H and let Y = m Ker f j. Put P =
Chapter 7.
344
Applications
(Ji .... , f,) : X -. R" and L = Ker P. Denote by iry the natural projection of X to Y and put py = it o Ire. vy = v o 7t}.1. Then the measures p and v have equal Gaussian conditional measures py = vy on the subspaces L + y, y E Y. Indeed, it is known (see [821, (911) that, for almost all with respect to both py and vy points y E Y, the measures py have the logarithmic gradients, associated with the space H. spanned by iH (fl),... , jH (fn), given by By(x) = PB(z), x E y + L. This implies that pU = vy is a Gaussian measure. For other examples of this phenomenon in the theory of Gibbs measures, see 12861. See also 15651 for related examples.
7.4. Spherically symmetric measures In infinite dimensions, there is no any analogue of Lebesgue measure, hence it is more difficult to define nontrivial symmetries of measures. It is easily seen that Dirac's measure at zero is the only spherically symmetric probability measure on an infinite dimensional Hilbert space X. However, one can consider H-spherically symmetric measures on X as measure that are invariant with respect to the action of the group of orthogonal operators on H. A more precise definition is as follows.
Let a separable Hilbert space H be densely and continuously embedded into a locally convex space X as described above.
7.4.1. Definition. A probability Radon measure it on X is called H-spherically symmetric if its Fourier transform has the form
dl E X',
p(l) _ where
is a function on 1R'.
7.4.2. Theorem. Let p' be a centered Radon Gaussian measure on a locally convex space X with the infinite dimensional Cameron-6fartin space H. Then a measure p on E(X) without atom at zero is H-spherically symmetric if and only if it is the mixture of the Gaussian measures u': B '-+ p'(t-'B), i.e., xr
p(B) =
J0
pi(B)a(dt),
BE E(X),
(7.4.1)
where or is some probability measure on (0,oc).
PROOF. By condition, p`(f) = j.(Ij8 (f)( ), where cp is a function on [0,00). By the Lebesgue theorem, cp is continuous. Since i is nonnegative definite, Schonberg's theorem applies (see [800, Theorem 4.2, Ch. IV1), which yields the representation xr
J expt- I t2[hI2 ) a(dt) 0
with some probability measure a on (0, oc). This yields
x
x
µ(f) = Jexp(_t2IjH(f)I) a(dt) = pt(f)a(dt), J 0
0
whence the conclusion.
7.4.3. Corollary. Let H be a separable infinite dimensional Hilbert space continuously and linearly embedded into a locally convex space X. Suppose that there exists a Radon probability measurep on X that is H-spherically symmetric and has
7.4.
Spherically symmetric measures
345
no atom at zero. Then there exists a centered Radon Gaussian measure p' with the Cameron-Martin space H such that (7.4.1) holds. PROOF. Suppose first that X is complete. As above, we get the representation
f 0
with some probability measure a on (0, oo). Let e > 0. Let us find r > 0 such that a((r, oo)) > 1 - e. By the additional completeness assumption, there is an absolutely convex compact set K with u(K) > 1 - E. Now let C be any cylinder containing K. Denote by t' the cylindrical additive set function with the Fourier transform exp(-I jH (f )I2/2) and note that (7.4.1) holds true for the cylindrical sets B. There exists a cylindrical absolutely convex set Q with a compact base such that K C Q C C (see the proof of Lemma 2.1.6). Clearly, Fit(Q) < i '(Q) if t > r. Hence 1 - c < µ(Q) < e + µ'(Q), which gives p'(C) > 1 - 2e. Therefore, (pl)*(K/r) > 1 - 2e. It remains to apply Theorem A.3.19 in Appendix. In the general case (where X may not be complete), one can consider p on a completion Y of X and get the corresponding Radon Gaussian measure µ' on Y. Then p' (X) = 1. Indeed, letting K C X be any compact set of positive p-measure and denoting by
L its linear span, we get by the zero-one law that p'(L) = 1, since otherwise
0
pt(L) = 0 for every t > 0.
There is no similar characterization in the finite dimensional case, since such a mixture has a positive density. However, Problem 7.6.16 and the next result enable one to describe differentiable H-spherically symmetric measures both in finiteand infinite dimensional spaces as the measures possessing logarithmic gradients, is a real function. associated with H, of the form 31j (x) = c(x)x, where
7.4.4. Proposition. Let p be a probability Radon measure on X differentiable along all vectors from j,4 (X "). Suppose that
8H (x) = c(x)x,
(7.4.2)
where c is a measurable real function on X. Then u is H-spherically symmetric. If H is infinite dimensional and y is a centered Radon Gaussian measure on X with the Cameron-Martin space H, then there exists a probability measure a on (0, oc) such that
x
µ(B) = J..v(tB)a(dt).
(7.4.3)
Conversely, if an H-spherically symmetric measurep has the logarithmic gradient ,3H, then (7.4.2) holds true.
PROOF. First of all, note that it suffices to prove the first claim for all finite dimensional projections of p, which have the form P,,x = (10)'... 1.(x)), where
the functionals li E X* are such that the vectors e; = jH(li) are orthogonal in H (moreover, it suffices to consider only two dimensional projections). Let B be the conditional expectation of the mapping P"/3, with respect to the a-field,
generated by ll,... ,1,,. Then it is readily seen that. B,,(x) = 3.(P,,x), where ,l 3,, is the logarithmic gradient of the measure p o on IR" (generated by the space IR"). Clearly, this conditional expectation has the form x ' c(x)P"x,
Chapter 7. Applications
346
where c is the corresponding expectation of the function c (it suffices to note that. P"3y(x) = c(x)Px). In particular, c,,(x) = d"(Px), where d" is a function on lit". It remains to make use of Problem 7.6.16. Suppose now that an H-spherically symmetric measure p has the logarithmic gradient 31f. In the case of IR" the validity of our claim is easily seen from the fact that a spherically symmetric absolutely continuous measure has a density which
depends only on the norm of the argument. Suppose that H is infinite dimensional. By Corollary 7.4.3, there exists a centered Radon Gaussian measure y, for which (7.4.3) holds true. It follows that the measure it is concentrated on a Souslin linear subspace E and that there exists a sequence C X' separating the points in E. We may assume that the vectors j form an orthonormal basis in H. The projections p o P, 1 considered above also have the logarithmic gradients 3" (associated with the spaces P"(H)). Since these projections are spherically symmetric, one gets 3,, (y) = d (y) y. Let us fix i E IN. It is readily verified that the conditional expectations g" := IE" (1 with respect to the o-field, generated by 11, ... ,1", coincide with (d,, o P")1,, whenever n > i. On the other hand, the sequence converges in measure to (1,, 3N) . Since the set Ker li has p-measure zero, the sequence {d" o P. } converges in measure to some function c, which, thereby, does
not depend on i. Thus, (l;, 3t, (x)) = c(x)1{(x) a.e., whence one gets the desired
0
relationship (7.4.2).
7.5. Infinite dimensional diffusions Let X be a locally convex space, let H be a separable Hilbert space continuously
and densely embedded into X, and let B: X
X be a Borel mapping. Let us
consider the following stochastic differential equation:
4:= t+B(Et)dt, to=x.
(7.5.1)
By a solution we mean a random process ti = (lt)i>o (called a diffusion process) in X such that there exist a filtration F = {)Qe>0, with respect to which the process l; is adapted, anF-Wiener process (W=)t>0 in X, associated with H, such that, for all t > 0, one has a.s. a
t = x+w + 21 JB)ds. 0
In the finite dimensional case this corresponds to the concept of a weak solution. It is possible to define also a strong solution, namely, to require that the foregoing conditions be fulfilled for any a priori given Wiener process (W1)1>0 with the fil-
tration Ft = a(W, : s < t). There exist some other interpretations of a solution of equation (7.5.1). It should be noted that in infinite dimensional spaces (say, in infinite dimensional Banach spaces) equation (7.5.1) may fail to have solutions even for a bounded continuous mapping B (see examples in (79J). If X is a Banach space, then the Lipschitzness of B is sufficient for the existence of a strong solution of (7.5.1). Note the following important special case of equation (7.5.1): B(x) = -x. The solution of this equation exists and is called an Ornstein flhlenbeck process. Let us recall several analytic objects related to the concept of a Markov process in a topological space X (the concept itself will not be used below; basic information concerning Markov processes can be found in (822]). Let p be a Radon
Chapter 7.
348
Applications
coincide. Let us show that Tt/2 f (x) = E f (l f) for every f r= Bb(X) and all x E X,
t > 0. By the first equality in (7.5.5), it suffices to prove that the law v of the t
random variable Wt - I
J0
e("-t)V2W8 ds equals the image of -y under the mapping
y H \Irl - e-ty. Clearly, v is a centered Gaussian measure. Let I E X* be such that jj (1) 1,, = 1. We have to evaluate the variance of the random variable t
-
I f (e-t)"2 I(W,)ds. 2Je
0
Since 1(W,) is the standard Wiener process, we can deal with the one dimensional
case. Then by formula (2.11.8), the variance of t equals that of f(s_t).f2dw3, 0 tr
which is f e°-t ds = 1 - e-. This is exactly the variance of the image of ry under 0
the mapping y -+
0
1 --e-41(y), whence our claim.
Note that -y is a unique probability measure on E(X) invariant for the OrnsteinUhlenbeck semigroup (Tt)t>o. Indeed, for any bounded continuous cylindrical
as t - oo.
function f and every x, one has Tt f (x) -+ /
If µis an invari-
ant probability measure, then ! f d= J Tt f du converges to
f fdp=
J f dry, whence
fdry.
Let X be a locally convex space and let H C X be a continuously and densely embedded separable Hilbert space. To every mapping B: X - X, one can associate the elliptic operator L defined on FCOO by the equality
Lf = &f,f +X. (f', B),,
(7.5.7)
where 00
A. f (x) := E f (x),
(7.5.8)
n=1
and {en } is an orthonormal basis in H. Note that the sum in (7.5.8) does not depend on a concrete choice of an orthonormal basis in H. For any function f of the form f = '(I .....In), where tP E Cb (Rn), Ii E X*, one has n
-9,W01,...,ln)1.(l;,B)X. t=1
If X is a Hilbert space and H = T(X), where T is a nonnegative injective HilbertSchmidt operator with eigenvectors hn, forming an orthonormal basis in X, and eigenvalues t,,, then the vectors en = tnhn form an orthonormal basis in H and 00
00
Lf = 1 t2n8hn1+ > Bnah..f, n=1
n=1
7.5.
Infinite dimensional diffusions
349
where B = The proof of the following theorem is given in [91].
7.5.2. Theorem. Let p be a Radon probability measure on a locally convex space X and let H C X be a continuously and densely embedded separable Hilbert space.
(i) Let (1, B) E L2(µ),
Vl E X.
(7.5.9)
Suppose that the operator L given by (7.5.7) is symmetric on FC- C L2(µ). Then the logarithmic gradient 3'N exists and coincides with B p-a. e. (ii) If the logarithmic gradient 3;, exists and the mapping B = 8H satisfies condition (7.5.9), then the corresponding operator L is symmetric. In addition. for any f E .FC', one has (L f, f)L2(u) < 0-
7.5.3. Remark. According to the F iedrichs theorem, statement (ii) implies that the operator L has a nonpositive self-adjoint extension, i.e., it extends to the generator of a symmetric Markov semigroup on L2(p). If there is a Radon Gaussian measure on X with the Cameron-Martin space H, then, according to [15], there exists a diffusion process with invariant measure A. for which the generator of the transition semigroup coincides with 2L on .FC'°. Therefore, the previous theorem shows that the logarithmic gradients of measures are, up to factor 2, the drifts of the symmetrizable diffusions.
We already know from the previous chapters that stochastic differential equations are closely related to nonlinear transformations of Gaussian measures. Hence it is natural to ask about the conditions of the absolute continuity of the distributions of diffusion processes and their transition probabilities and invariant measures with respect to Gaussian measures. Under very broad assumptions, the transition probabilities and invariant measures of the diffusion processes on 1R° given by equation (7.5.1) are absolutely continuous with respect to the standard Gaussian measure. The situation is completely different in the infinite dimensional case. First of all, typically, the transition probabilities P(t. x, ) are mutually singular for different t. For instance. this happens in the case of the Wiener process W . where the transition probability P(t.0, - ) is the image of the Gaussian measure ry equal to the distribution of I under the mapping x i x. Secondly, the transition probabilities and invariant measures may be mutually singular with respect to all Gaussian measures (see [91]). We discuss here the interesting and important special
case, where B(x) = -x + v(x) with a vector field v: X -» H (which corresponds to "small perturbations" of the Ornstein-Uhlenbeck process). Let H be, as above, the Hilbert space associated with Wt, let 7 be the distribution of 1V1, and let
B(x) _ -x + v(x),
v: X
H.
We shall assume that condition (7.5.9) is satisfied. The study of invariant measures of the diffusion generated by (7.5.1) is closely related to the elliptic equation
L'p=0,
Chapter 7. Applications
350
which is understood in the following weak sense:
I Lf(x)p(dx)
= 0,
V f E .FC°`.
(7.5.10)
x Under very broad assumptions, any invariant measure of the diffusion given by equation (7.5.1) satisfies equation (7.5.10) (e.g., this is true if sup I vI H < oo). The following result is due to [694].
7.5.4. Theorem. Suppose that sup Iv(x)I,, < oc. Then there exists a process (.i )t>o,:Ex satisfying equation (7.5.1), and, in addition, this process has an invariant probability measure p equivalent to the measure y.
We turn now to the results concerning the regularity of solutions of elliptic equation (7.5.10), which gives some information about the invariant measures of diffusion processes (7.5.1). In the case where X = 1R" and B is a smooth mapping, classical Weyl's theorem states that every solution of equation (7.5.10) is an absolutely continuous measure with a smooth density with respect to Lebesgue measure. The following extension of Weyl's result was obtained in [91]. Note that unlike the classical situation, the corresponding differential operator is not defined on all distributions, since the coefficient B is only integrable with respect to a solution p. In
particular, this result applies to singular drifts B which need not be locally integrable with respect to Lebesgue measure. For example, the measure with density x2 exp(-x2/2) satisfies the corresponding equation with B(x) = 2x-' - x.
7.5.5. Proposition. Let p be a probability measure on IR" such that (7.5.10) is fulfilled, where Lcp = AV + (Vip, B) and B is a Borel vector field on 111" with I B IE L2(µ). Then p has a density p E In particular, p is differentiable along all vectors from IR". In addition, the following estimate holds true:
f
TpI dx < JIB(x)12/2(dx). p
(7.5.11)
Moreover, Vp/p is the orthogonal projection of B onto the closure of the set of the gradients of the functions from Co (1R") in the space L2(p,IR").
7.5.6. Theorem. Let X, H, and y be the same as above. Suppose that a probability measurep on X satisfies equation (7.5.10), where
B(x) = -x+v(x), v: X - H, Ivies E L2(p). Then:
(i) the measure it is absolutely continuous with respect to the measure -y and its Radon-Nikodym derivative p has the form p = F2, where the function F is in the Sobolev class W2.1 (y);
(ii) The measurep is differentiable along all vectors h E H and j3H (x) = -x + H is the orthogonal projection of v onto the closure of u(x), where u: X the set {DH f I f E .FC°°} in the Hilbert space L2(p, H). PROOF. (i) We may assume that u = v, since p satisfies equation (7.5.10) with B1 (x) = -x + u(x). This follows from the equality
f x.(f',u-v).
du= f (DHf,u-v)udp=0, b'f E FC'.
7.5.
Infinite dimensional diffusions
351
Let {en } be an orthonormal basis in H such that e = j (1 ), 1 E X'. We shall temporarily consider both measures y and p on the a-field generated by the functionals {1,} (replacing v by the corresponding conditional expectation). For any F E L2(p, H), the sequence Fn = E,, [Flan] of the conditional expectations of F with respect to the a-fields o generated by ii i ... , ln, converges to F in L2(p,H) (see [800, Ch. II, Theorem 4.1]). Let H. be the linear span of e 1 . . . . . en and Pn : X - Hn, Pnx = 11(x)el + ... + In (x)en. The space Hn is equipped with the inner product from H. Note that 1, (ej) = (e;, a j ),, for all i, j. Hence P. I,,
is the orthogonal projection in H onto Hn and IPnh - hl,, -. 0, as n - no, for F in all h E H. Therefore, for the mappings Fn defined above, one has L2(p, H). Indeed,
f IF(x) -
p(dx)
< f I F(x) -
f
,1i(dx)
N p(dx) + f I P3F(x) -
IF(x) - PnF(x)I2 Ez(dx) +
f
IF(x) - FF(x)IH p(dx) -. 0,
since the first term on the right-hand side( tends to 0 by the Lebesgue theorem. Put
Vn .- IE,,[Pnvlan] = PnEµ[vlan], Therefore,
v
bn := E [PnBlan]
in L2(p,H)as n-+oo.
Note that bn(x) = -Pnx + vn(x). Let An :=poP,,1. There exist Borel mappings b,: H Hn such that bn = 'b o P p-a.e. It is easily verified (this verification is found in (91, Proposition 3.3J), that the measure An on H satisfies the equation L , p = 0, where n
Lnu=Ea,y
Vu ECb (Hn).
u+(V,, Zi,bn)H.,.
i=l
According to Proposition 7.5.5, the measure pn has a density fn with respect to the
standard Gaussian measure yn on Hn. Let qn be the standard Gaussian density
on H (recall that H is equipped with the inner product from H). By virtue In y = of Proposition 7.5.5 one has pn := fngn E W"1(Hn) and a"" addition, I AH I H,, E L2 (pn ). Therefore, QXn(Z)
z+ V,,f fn(z)(z)
for pn -a . e. 2 E H
.
(7. 5.12)
On the other hand, according to Proposition 7.5.5, one has
(z)+dn(z)
forpn-a.e. zEJIn,
(7.5.13)
where the mappings do : Hn -+ Hn are such that
f (V.,f(z),dn(z)) H.,
V f E Cb' (H.).
(7.5.14)
Chapter 7. Applications
352
By virtue of relationships (7.5.12) and (7.5.13), we get Pnx
vn(x) = bn(x) + Pnx =
_
')H
By (7.5.14) one has
f
p-a.e.
fn(Pnx)
(VHffl(Pflx)(p)")
p(dx) =0.
(7.5.15)
Indeed, (7.5.15) is derived from (7.5.12) as follows. By 191, Theorem 2.8], there exist functions q; E Cb (HH), i E IN, such that
qi - Q N, The mapping S: z
in
2(pn, H.)
-z on Hn coincides with QXnQ, where Q(z) _ -2(z,z)H,,.
It is easily verified that the mapping S is also in the closure of {V,,, f I f E C6 (Hn)} (the latter follows from in LZ(pn, Hn), making use of the fact that S E L2(pn, v in L2(p, H), the square integrability of bn and v,, with respect to p). Since v,, then, by virtue of (7.5.15), there exist two mappings d and G from L2(p, H) such that do P. Pn -y d and o Pn / f n o P. - Gin L2 (p. H). It is easily seen from
(7.5.14) that the mapping d is orthogonal to {D f : f E FCG }, hence also to C. Since we assume that t, = u, we get that d is orthogonal to v. whence d = 0. Thus,
fn0Pn
v inL2fig. H), n- oc.
(7.5.16)
We shall now use the logarithmic Sobolev inequality. Since ff E
by
virtue of Proposition 7.5.5, we may put Vn
fn o& E
11'2.1
(7)
and apply the logarithmic Sobolev inequality top. Moreover,
JX fn(Pnx)-t(dx) = NJ 4 J (Da
= / I VN" fn(x) I2yn ^rn(dx)
n(x)I H - Y
f
X
H
r Vj fn(x) 12 Hn
fn(X)7n(dX) = 1.
MX)
Hn
A (x)
f,, (x) 1'n(dx) = J I n
H,
V, fn (x) 12
fn(x)
f .l
X
H
pn(dx)
2
fn(Pnx)
p(dy),
H
where the use of the chain rule is easily justified by replacing fn by fn+E and letting f f) o Pn If,, o Pn e tend to 0. By virtue of (7.5.16), the norms of the mappings in L2(p,H) are uniformly bounded by some constant. C. Therefore, sup f pn(x)2log I;Fn(x)I -t(dx) < Ci2. n
X
7.6.
Complements and problems
353
This estimate implies the uniform integrability of the functions f, o P. _ ,pn on (X,ry). Since (fn o Pn)nEIN is a martingale with respect to and the measure y, we conclude that this martingale converges to some function p E L' (y).
Put A := p 7. Then, for
WnA and all 1 = clli +... + cln,. we get
J exp(il) dAn -. x fexP(il)dA.
X
On the other hand, as n >r m, one has
r
f exp(il)dAn=1
x
x
r
H
r
= f exp(il) dµ = J x H
dµ = J exp(il) dµ. x Therefore, u = \ = py on the u-field E({1,}) introduced above. It follows that ,u = py on E(X), since the sequence {l;} was arbitrary and, for the measure y (hence also for py), the sets from E(X) coincide with sets from E({l,}) up to sets of measure zero. Both measures are Radon, which yields that p = py on 8(X). Since W - yr =: y in measure y and one has the estimate C2/4.
II DHVnII
it follows that V E 14"2whence claim (i). (ii) Let h E H. By Lemma 5.1.12 we get ph E L2(-y). This yields the equality
3h = -h + (h,W -'D.W)H. Taking vectors of the form h = jH k, k E X', we arrive at the equality 3H (x) = -x + 2+p-' DH w(x) k-ax.
The equality W-IDH:p = u follows from relationship (7.5.16) taking into account that the sequence V. converges to V in measure y and is bounded in W2,1 (-Y) (which implies that the arithmetic means of its subsequence converge to Win W2-1(7)). O
7.6. Complements and problems
Gaussian comparisons In the study of the trajectories of Gaussian processes several different types of comparison of their covariances have proved to be efficient. We encountered one of
these types in Chapter 3, where the covariances were compared by means of the usual ordering W < V for quadratic forms. Another type of comparison suggested by Slepian [7131 makes use of the process itself and is transparent in the well-known Slepian inequality (7131.
7.6.1. Theorem. Let l; and n be two centered separable Gaussian processes on a set T with covariance functions KK and K,,. Suppose that KE(t, t) = K,,(t, t) and KE(s, t) < K,,(s, t) for all s, t E T. Then one has
P(suplt > M) > P(supnr > M), C
T
VAf E Et'.
Chapter 7.
354
Applications
The Slepian inequality was generalized by Gordon (306] - [308). Slepian and Gordon inequalities are special cases of the following comparison theorem proved in (391); related results were obtained in [610], [611]; see also the proof in [471, Theorem 3.11].
7.6.2. Theorem. Let t = (t , ... , ") and il = (rla.... , r)") be centered Gaussian vectors in iR" and let
A = {(i,j): Efifj < E?.nj}, B = {(i,j): E{,t7 > Erliq,}. Suppose that a function f E 4V (1R") is such that 82,82, f > 0 if (i, j) E A and 8=,a--, f < 0 if (i, j) E B (more generally, these inequalities can be interpreted in the sense of the generalized functions; then we need not require that f be locally Sobolev). Then E f (l;) < E f (rl). One more natural way of comparing the covariances of centered Gaussian processes t; and q was used by Sudakov, Fernique, Markus, and Shepp (see [732], [243],
(526]), who considered the following condition: EIt;, - &12 < E]p, - ?It]' for all s, t E T. According to [526], if the process q on [0, 1] has continuous trajectories, then the process t does also. An analogous statement is true for the processes on separable metric spaces. Proofs can be found in [119, § 91. Interesting connections between Gaussian measures, the path properties of Gaussian processes, and the geometry of Banach spaces, in particular, applications of the various inequalities discussed above, are found in [306), [307]. [308], (547], and (605].
Logarithmic gradients and linear stochastic equations Let us mention several additional results concerning linear logarithmic gradients and linear stochastic differential equations (see [82]). As it was mentioned above, a probability measure p with linear logarithmic gradient A is invariant for some diffusion process t with drift A/2 (and this process is Gaussian). Since not every process generated by a linear stochastic differential equation has an invariant measure, the question arises concerning a characterization of the linear mappings which are logarithmic gradients of measures. The next result is proved in [82].
7.6.3. Proposition. Let y be a centered nondegenerate Radon Gaussian mea-
sure on a locally convex space X and let A: X - X be a y-measurable linear mapping. Suppose that X is sequentially complete. Then the following conditions are equivalent: (i) There exists a separable Hilbert space H densely embedded into X such that
j (X') C H(') and A = ON; (ii) The function (f, g) '-+ -(f, A'g) on X' is an inner product. where
A': X' - H(y), (A'k,h)H(. _ (k, Ah), i.e., (f, g) '-' (g, Aj5 (f )) is an inner product on X
The next result can be proved along the same lines as Proposition 7.3.9.
7.6.4. Proposition. Let y be a Gaussian measure on a locally convex space X and let p be a probability measure on E(X) differentiable along some linear subspace D C H(y) such that, for all h E D, the functions 3r', and 3; admit equal
7.6.
Complements and problems
355
modifications which are continuous linear functionals. Assume, in addition, that such functionals separate the points in X. Then y = µ. This result can be used to get an infinite dimensional analogue of Proposition 1.10.2 characterizing Gaussian measures.
7.6.5. Proposition. Let y be a centered Radon Gaussian measure on a locally convex space X with the Cameron-Martin space H and let {t:,,} C X' be an orthonormal basis in X. Suppose that it is a Radon probability measure on X such that X' C L2(µ), (tt,t;j)L2(,,) = b(;J, and
U(p):=sup(
1
ff2dµ
dp,
fE.F}=1,
f JD.fJ2 where F is the collection of all functions f E L2(µ) of the form f = cp(II,... ,1.), 1JJ
cp E C"(IR"), li E X. Then µ = -y. PROOF. Put en = Ry(an). The same reasoning as in Proposition 1.10.2 shows
that
Jn9diL=JO9dti, V9 E.F. X
X
Since (e,, } is a basis in H and e" = j (t n ), we conclude that
f 9dP. =
f
x
x
b'l; E X','dg E .F.
This yields the equality /3K (x) = -x. Therefore, by Proposition 7.6.4 (or Proposi0 tion 7.3.9), µ = y. An important for applications class of Gaussian diffusion processes on infinite dimensional spaces X is connected with the equations of the form
dX1 = d1l't + AX,dt,
X() = x,
(7.6.1)
where W1 is a Wiener process, associated with a Hilbert space H C X, and A is the generator of a strongly continuous semigroup (Ti)t>o on H. One of the first problems arising in this connection is the interpretation of (7.6.1), since H has measure zero with respect to the distributions of K',. This problems arises even in the case, where the semigroup (T1)t>o is defined on the whole space X, since the domain of definition of the generator may be very narrow. The following result from [92) enables one to overcome this difficulty.
7.6.6. Theorem. Let (Ti)t>() be a strongly continuous semigroup with generator (A, D(A)) on a separable Hilbert space H. Then H can be embedded linearly and continuously into some Hilbert space E in such a way that H is dense in E, (Tt)t>0 extends to a strongly continuous semigroup (T,E)t>o on E, and H turns out to be embedded into domain D(AE) of the generator of the extended semigroup (equipped with the Hilbert norm IIAExIIE + IIxIIE) by means of a Hilbert-Schmidt operator. Moreover, it is possible to choose E in such a way that the natural embedding of H to D((AE)2) is a Hilbert-Schmidt operator.
7.6.7. Corollary. If the conditions in Theorem 7.6.6 are satisfied, then there exists a continuous Gaussian process (Xl )t>o with values in E such that, for all
Chapter 7.
356
Applications
x E D(AE), one has XT E D(AE) and equation (7.6.1) is satisfied with A = AE. where (Ht)j>0 is a 14'Fiener process in E associated with H. In addition.
Xf = TEx +
Wt
+
I AETTE,W. ds.
t > 0.
0
Applications to partial differential equations We have already encountered probabilistic representations of solutions of various partial differential equations by means of Gaussian functional integrals. Another example of this type is the so called Feynman-Kac formula- Let us consider the Cauchy problem Su((ttt,x)
= 2Au(t,x)+V(x)u(t.x).
u(O,x)= f(x).
The Feynman-Kac formula is the following path integral representation of the solution of this Cauchy problem (valid under certain conditions on V, of course):
f (w(t) + x) exp (r V(w(s) + x) ds) Pt (dw).
u(t.x)
0
For related information, see [263]. [386], [387]. [406], [639]. [704].
Limit theorems Gaussian measures play an important role in the limit theorems. Let us make several remarks about one of the most important of them - the central limit theorem (the abbreviation: CLT). Let X be a locally convex space and let {Xn ) be a sequence of X-valued independent centered random vectors with one and the same Radon distribution p. Put Sn-X,+...+X,.
vn Note that the distribution of S coincides with the measure p". defined by the equality p" (A) = (p +... * p)(n'12A). where the convolution is n-fold. The central limit theorem concerns the following two problems: 1) convergence of the sequence of random vectors S. (in a suitable sense); 2) if this sequence converges to some random element Y, then what is the rate of convergence on a certain class of sets? More precisely, let M be a fixed class of subsets of X (say, a certain class of balls in a Banach space). Then the problem is to estimate the quantities
An(M) = sup ,P(S E Af) - P(Y E XI)I. M EM
For example, a typical problem of this sort is to estimate
On (j, r) = I P(f (S.) < r) - P(f(Y) < r) where j is some function on X (usually a norm or a smooth function). We shall consider only !Radon probability measures p such that
f Y
t(x)2p(dx) < oc,
V1 E X'.
7.6.
Complements and problems
357
In this case we say that the measure p has the weak second moment. A measure p on X has the strong second moment if f q(x)`p(dx) < oc
x
for every continuous seminorm q on X.
7.6.8. Definition. Let X be a locally convex space. (i) A probability measure p with mean m. on X is called pre-Gaussian if it has the weak second moment and there exists a Gaussian measure y with mean in. on X such that
f fgdp= f x
fgdy,
`df.gEX'.
x
(ii) A probability measure p with zero mean on X is said to satisfy the central limit theorem (CLT) if the sequence {p'") is uniformly tight. A probability measure p with mean m is said to satisfy the CLT if the measure p_,,, with zero mean satisfies the CLT. (iii) The space X is called a space with the CLT property if every probability measure p on X with zero mean and the strong second moment satisfies the CLT, X is called a space with the strict CLT property if the CLT is fulfilled for every probability measure p on X with zero mean and the weak second moment.
Note that if X is a separable Frechet space, then, as the next lemma shows, the definition of the CLT given in (ii) becomes equivalent to the classical one requiring the weak convergence of the sequence {p""} to a Gaussian measure.
7.6.9. Lemma. Let p be a probability measure with zero mean on a locally convex space X. If the sequence (p" I is uniformly tight, then it converges weakly to some centered Radon Gaussian measure y. In addition, p is a pre-Gaussian measure.
Proof is left as Problem 7.6.21. On the space X = lR", every probability measure with the weak second moment satisfies the CLT. Certainly, such a measure has also the strong second moment. The situation is different in the infinite dimensional
case. For instance, the space C[0,1] does not have the CLT property. Moreover, there exists a pre-Gaussian measure with compact support in C[0,1], which does not satisfy the CLT. On the other hand, there exists a probability measure with compact support in C[0,1], which is not pre-Gaussian. Finally, there exists a measure on C[0,1], which satisfies the CLT, but has no strong second moment (see [592] concerning such examples). It is known that any Hilbert space has the CLT property. Since in a Hilbert space the covariance operator of a probability measure p is nuclear precisely when p has the strong second moment, we see that in this case the class of all pre-Gaussian measures coincides with the class of all measures satisfying the CLT (and also with the class of all probability measures having the strong second moment). As the space C[0,1] shows, these three classes of measures may be all different for general Banach spaces. The equality of all the three classes characterizes Hilbert spaces (more precisely, Banach spaces linearly homeomorphic to Hilbert spaces). In other words, a Banach space is linearly homeomorphic to a
358
Chapter 7.
Applications
Hilbert space if and only if the existence of the strong second moment of a probability measure is equivalent to the validity of the CLT for this measure. It is known that every probability measure with the strong second moment on a Banach space X satisfies the CLT precisely when X is a space of type 2. Therefore, on any non-Hilbert space of type 2 there exists a measure, which satisfies the CLT, but has no strong second moment. If every measure on X satisfying the CLT has the strong second moment, then X is known to be a space of cotype 2; moreover, this property is a full characterization of the spaces of cotype 2. Note also that X has cotype 2 precisely when every pre-Gaussian measure on X satisfies the CLT. Proofs of these assertions and the corresponding references can be found in (592, Ch. 31, [472, Ch. 10]. Let us mention several properties of locally convex spaces with the strict CLT property. This property was introduced in (73], where the proofs can be found.
7.6.10. Theorem. A Banach space X has the strict CLT property precisely when dim X < oc. The strict CLT property is inherited by the closed subspaces and is retained by the strict inductive limits of increasing sequences of closed subspaces, by countable products, arbitrary direct sums, and the countable projective limits.
7.6.11. Example. Let X be the dual to a complete nuclear barrelled locally convex space Y. Then X with the strong topology has the strict CLT property. For example, this is true if X is the dual to a nuclear Freshet space. The following spaces have the strict CLT property: Co [a, b], S(IRk), S(IRk)' IR" A detailed survey of the results connected with the problem of estimating the rate of convergence in the central limit theorem can be found in [592], [58]. An Cn_,/2 important achievement in this area was the proof of the estimate in the case, where U is a ball in a Hilbert space and the vector X, has the strong third moment (assuming the existence of the sixth moment this estimate was first obtained by F. Gotze, and the improvement involving only the third moment is due to V. Yurinsky). It has been recently shown by V. Bentkus and F. Gotze [67] that if X, takes values in a Hilbert space and has the strong fourth moment, then Cn'I, provided the topological support of the limit Gaussian measure has dimension at least 9. V. Bentkus discovered that, for general Banach spaces, no
moment restrictions enable one to get an estimate better than Cn-1/6 even if the distribution of the norm with respect to the limit Gaussian measure has a bounded density. If no assumptions are made concerning the distribution of the norm, then the rate of convergence on balls can be arbitrarily slow (see [641]). It remains an open problem what is the rate of convergence on balls in the case, where the limit Gaussian measure is the Wiener measure on C[O,1]. The central limit theorem is just but one problem in the growing area of limit theorems for infinite dimensional random elements. An extensive and interesting material, including the study of convergence of sums of independent random vectors, is presented in the works cited in Bibliographical Comments. One of the related questions is the law of the iterated logarithm (see [191], [305), [435], [442], [472], [476), [487], [726]). In its simplest formulation it states that if {t;,,} is a sequence of independent random vectors in a separable Banach space X with one and the same centered Gaussian distribution y, then with probability 1 the sequence E"=, 1;j/2n log ogn has as a cluster point (in the topology of X) every element of the unit ball U of the Hilbert space H(y). The same is true for a locally convex space X if y is a Radon measure (see [96], [753]).
7.6.
359
Complements and problems
A random vector with values in a locally convex space X is called (see [494],
[771]) stable of order a E (0.2] if, for every n, there exists a vector a E X such that, for any independent copies y1, ... , f,, of the vector , the random vector n-"({1 + +Sn) - a" has the same distribution as . A measure is called stable if it is the distribution of a stable random vector. Stable of order 2 random vectors are precisely Gaussian vectors. The distribution of any stable vector is a mixture of Gaussian measures (see [749]).
Problems 7.6.12. Let S. _ E {,,,k, where for every n E V.
are independent
k=1
centered Gaussian random variables with variance n-'. Show that S" - 1 in the square mean and deduce (7.1.3). Hint: show that IE(S" - 1)2 = nE(f!.1 -1/n)2 = 2/n; to prove (7.1.3) use the martingale convergence theorem (see [337, §2.2, Theorem 2.3]).
7.6.13. Let f be a continuous function on 11' such that the function t'- f (w,(w)) has bounded variation on [0.1] for a.e. w. where (w1)r>o is a standard Wiener process. Show that f is constant. Hint: see [257]. 7.6.14. Let 7 be a centered Radon Gaussian measure on a locally convex space, let
H be the Cameron-Martin space of 1, and let f E IV2'1(')). Put p = f µ. Show that
3 (x) = -x+DHf(-)/f(x). 7.6.15. Show that in the situation of Proposition 7.3.9 the measure µ is a unique probability measure with the logarithmic gradient A generated by H.
7.6.16. Show that a measure p on IR" with the logarithmic gradient 3" (generated by IR") is spherically symmetric if and only if there exists a real function c(.) on IR" such that 01'(x) = c(x)x p-a.e. In addition, every such function c admits a spherically symmetric modification. Hint: show that, for every orthogonal operator S, one has a.e. the equality p(Sx) = p(x), where p is a density of p. To this end, letting Te be the group of rotations in angle tin a fixed two dimensional plane, verify that etp(T,x) = 0 for a.e. t and a.e. x, choosing a modification of p such that the function t - p(Ttx) is absolutely continuous for a.e. x.
7.6.17. Prove that the measure µ defined by equality (7.4.3) has the logarithmic gradient associated with H precisely when f t ' o(dt) < oc. Hint: the necessity of this 0
condition (noted in [568]) reduces to the one dimensional case by taking the measure u o I - ', where I E X' is not zero. The sufficiency part is trivial. 7.6.18. Is the standard Ornstein-Uhlenbeck process a martingale? 7.6.19. Let (C. (t)) be a sequence of martingales on a common probability space with a given filtration. Show that the process sup, {,(t) is a submartingale with respect to the same filtration.
7.6.20. Prove that if a probability measure µ on a locally convex space X is stable of some order and convex (i.e., satisfies (4.2.2)), then it is Gaussian. Hint: reduce the statement to the one dimensional case and use the fact that any convex measure has the second moment, whereas among the stable measures only Gaussian measures have this property (see (697, Ch. III, §5]). 7.6.21. Prove Lemma 7.6.9. Hint: make use of the relative weak compactness of the sequence of measures p" and the uniqueness of its possible weak limit, which follows from the central limit theorem for the one dimensional projections.
Chapter 7. Applications
360
7.6.22. Let X be a separable Banach space, which contains a closed linear subspace linearly homeomorphic to the space co. Show that there exists a probability measure p on X with compact support such that p is mutually singular with every pre-Gaussian measure on X. In particular, this is true for the space X = C[0,1]. Hint: use the method of Example 2.12.10; see also [93].
7.0.23. Let 7 be the countable product of the standard Gaussian measures on the real line considered on the Hilbert space X of all sequences (x,,) with the finite norm n-2x,2,) 1". Define a probability measure it on X by n=1
=
JA/tt)dt, 0
where p is any positive probability density on (0, oo) such that f t2p(t) dt = oo. Show that for any finite collection of p-measurable linear functionals mapping V: R" -+ X, one has III x
0
and any Borel
- (0(11(x), ... ,1,,(x)) 112 p(dx) = oo.
X
Prove that, more generally, the same is true if 1; (x) are replaced by measurable functions of the f o r m 1 j (x) = f j (x, 11(x), ..., 1 _ ( z ) ) , where f, =1 1 is a measurable linear functional,
f: X x Rj -'
R, (x, y) -, f j (x, y) is measurable linear in x and Borel in y. Hint: use Anderson's inequality and Proposition 6.11.4.
APPENDIX A
Locally Convex Spaces, Operators, and Measures We do not understand many matters not because our concepts are weak, but because these matters do not belong to the circle of our concepts. Ko.sma Prutkov
A.1. Locally convex spaces
Basic definitions Proofs of the facts presented below and an additional information concerning locally convex spaces can be found in (670], (220]. A nonnegative function p on a real linear space X is a called a seminorm if p(Ax) = (AIp(x) and p(x + y) < p(x) + p(y) for all reals A and all vectors x, y E X. A real linear space X is called a locally convex space if it is equipped with a family of seminorms P = (p,).EA on X separating the points (i.e., for every nonzero element x E X there exists an index a E A such that po(x) > 0). The topology on X generated by the family P consists of the open sets which are arbitrary unions of the basis neighborhoods of the form
{x: p,,(x-a)<e,, i=1,...,n}, o;EA, aEX,nEN. Clearly, different families of seminorms can define one and the same topology. A normed space is a special case of a locally convex space. The topological dual to a locally convex space X (the space of all continuous linear functionals on X) is denoted by X'. Sometimes we use also the algebraic dual X' which is the space of all (not necessarily continuous) linear functionals on X. However, the term dual is reserved for the topological dual throughout this book. Every locally convex space X has a Hamel basis, i.e., a collection of linearly independent vectors {v0} such that every element in X is a finite linear combination of the vectors v,. A mapping Y A between linear spaces X and Y is called affine if A = L + a, where L: X is a linear mapping and a E Y is a fixed vector. A typical example of a locally convex space arising in the theory of random processes is the space IRT of all real functions on a nonempty set T equipped with the topology of pointwise convergence, or, in other words, the topology generated by the family of seminorms
pt(x) = jx(t)(,
t E T.
The space IRT is called the product of T copies of IRI. In particular, if T is the set
N of all natural numbers, then the corresponding space is denoted by IR". The dual to IRT coincides with the linear span of the functionals x - x(t), t E T (see [670, p. 137, Theorem IV.4.3]); this is clear from the fact that a linear functional 361
Appendix
362
f bounded on the neighborhood {x: Ix(t;)I < c, i < n} is a linear combination of the functionals 61, : x - x(t; ), i < n, since it is zero on n i Ker b,, . The linear span of a set A in a linear space is denoted by span A. For any sets A and B in a linear space X and any scalar A, we put
AA:={AaIaEA}, A+B:={a+blaEA,bEB}. A set A in a locally convex space X is called bounded if, for every neighborhood
of zero V in X, there exists A > 0 such that. A C AV. This is equivalent to the boundedness on A of every continuous linear functional.
A set A in a locally convex space is called symmetric if A = -A. A set A in a locally convex space is called convex if Aa + (1 - A)b E A for all A E [0, 1] and a, b E A. A convex set. A is called absolutely convex (or convex balanced) if AA C A
for every scalar A with JAI < 1. Clearly, this is equivalent to the convexity and symmetry of A. The convex hull of a set A is the minimal convex set (denoted by cony A) containing A. The absolutely convex hull absconv A of a set A is defined analogously. The closed absolutely convex hull of a set A is the minimal absolutely convex closed
set containing A. We say that a locally convex space (X, r,,) is continuously embedded into a locally convex space (Y, T,.) if X is a linear subspace in Y and the natural embedding (X, r,) --. (Y, r,.) is continuous. If, in addition, X is dense in Y, then we say that X is densely embedded. Let E be a linear space and let F be a linear subspace in the space of all linear functionals on E, separating the points in E (i.e., for every two different elements in E, there is a functional from F taking on these elements different values). Denote by a(E. F) the locally convex topology on E generated by the family of seminorms
pt(x) = If
f E F.
This is the topology of pointwise convergence on F if the elements of E are considered as functionals on F. Two typical examples: the weak topology a(X, X') on the locally convex space X and the *-weak topology a(X',X) on its dual. An important property of the topology a(E, F) is that the dual to (E.a(E, F)) coincides (as a linear space) with F, i.e., every linear functional that is continuous in the topology a(E, F) has the form .r -+ f (x), f E F. In particular, any continuous in the topology a(X',X) linear functional F on the space X' has the form F(f) = f (a) for some a E X. The Mackey topology rst(X',X) on X` is defined by means of the serninorms
KE1C. p,K(f)=supif(x)I, zEK where K: is the family of all absolutely convex a(X,X')-compact subsets of X. A proof of the following Mackey theorem can be found in [670, p. 131, Ch. IV, 3.2, Corollary 1].
A.1.1. Theorem. Every linear functional F on X' continuous in the Mackey topology Tkj (X'. X) has the form F(f) = f (a) for some a E X X. A topological space T is called metrizable if the topology of T is generated by a metric. A locally convex space is metrizable precisely when its topology is generated by a countable family of seminorms. A complete metrizable locally convex space is called a Pr6chet space. For example, the countable product of the real lines fx is a
A.1.
Locally convex spaces
363
Frechet space. Any Banach space (i.e.. complete normed space) is a Frechet space. The most typical examples of Banach spaces encountered in the theory of Gaussian measures are: the space I" of all bounded sequences x = (x,,) with IIxii = sup Ixnl, n
its closed subspace co consisting of all sequences converging to zero, the spaces Lo(o), where p E [1, oc], and the space C[a, b] of all continuous functions on [a, b] with the sup-norm. Every locally convex space X is completely regular, i.e., for every point x E X and every neighborhood U of x, there exists a continuous function f : X -. [0, 1] such that f (x) = 1 and f = 0 outside U (it suffices to be able to construct such
a function for x = 0 and any neighborhood of the form U = (p < 1}, where p is a continuous seminorm; in this case one can put f (z) = 1 - p(z) if z E U and
f(z)=0ifz
U).
A.1.2. Lemma. Let K be a compact set in a completely regular topological space X and let U be an open set containing K. Then: (i) There exists a continuous function f : X -. [0,1] equal I on K and 0 outside U;
(ii) Every continuous function y on K extends to a continuous function i' on X such that sup_v ktI = suph Jpi and >L' = 0 outside U. PROOF. A proof of (i) can be found in [220, p. 19]. For the proof of (ii) it suffices to find a continuous extension of to X with preservation of maximum and multiply it by the function from (i). The Stone-Weierstrass theorem implies the existence of a bounded continuous function g on X equal p on K. Now we can replace g by the function 0(g), where 0(t) = t if Iti < sup I'i, 0(t) = sup Icpl if ItI > sup l0I.
0
Recall that a mapping F between topological spaces is called sequentially continuous if F(x,,) -y F(x) whenever x, -+ x.
A partially ordered set A is called directed if, for every a and 0 from A, there
is 7 E A such that a < -y and 3 < y. A net of elements of the set X is a subset {xa}aEA C X indexed by a directed set A. The concept of a net generalizes that of a sequence. A net {xa})EA in a locally convex space X is called fundamental (or Cauchy) if it is fundamental with respect to every seminorm q from some family of seminorms
generating the topology of X (i.e., for every e > 0, there exists L E A such that
q(x,, -x3)<eforalla>A,3>\F). A.1.3. Definition.
(i) A locally convex space X is called sequentially complete if every Cauchy sequence in X converges. (ii) A locally convex space X is called complete if every fundamental net in X converges.
(iii) A subset A of a locally convex space X is called sequentially closed if it contains the limit of every convergent sequence of its elements. In a similar manner one defines the completeness and the sequential completeness for subsets of X. It is clear that every complete locally convex space is sequentially complete. An infinite dimensional Hilbert space with the weak topology gives an example of a sequentially complete locally convex space which is not complete (Problem A.3.25). Similarly to metric spaces, locally convex spaces possess completions.
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364
A.1.4. Theorem. Every locally convex space X has a completion X. i.e., there exist a complete locally convex space 9, a linear subspace Xo everywhere dense in k and a linear homeomorphism h: X - X0. The product X x Y of locally convex spaces X and Y possesses the natural structure of a locally convex space: the corresponding family of seminorms is defined by (x, y) +--+ p(x) + q(y), where p and q are representatives of the families of seminorms defining the topologies of X and Y, respectively.
Convex sets and compact sets Let us describe a construction connected with convex sets which finds numerous applications in measure theory on linear spaces. Let A be an absolutely convex set
in a locally convex space X. Denote by E the linear span of A. Put PA (x) = inf {r > 0: x E rA}, r E EA . The function p,, on E,, is called the Minkowski functional (or the gauge function) of the set A. A.1.5. Theorem. Let A be an absolutely convex sequentially closed bounded set in a locally convex space X. Then the function p, is a norm on E . whose closed unit ball is A. In addition. the natural embedding (EA , p,,) into X is continuousIf A is sequentially complete, then (E4, p4) is a Banach space. The proof can be found in [220. p. 444, Lemma 6.5.21.
Let us formulate a number of results about compact sets in locally convex spaces that we use in the main text. A.1.6. Proposition. In any complete locally convex space, the closed absolutely convex hull of a compact set is compact.
The previous statement may fail for not necessarily complete spaces. Part (ii) of the next proposition is due to [6441. The proof below is borrowed from [901.
A.1.7. Proposition.
(i) The metrizability of a compact space K is equivalent to the existence of a sequence of continuous functions separating the points in K. A compact set K in a locally convex space X is metrizable if and only if there exists a sequence {ln} C X' separating the points of K. (ii) The closed absolutely convex hull k of any metrizable compact set K in a locally convex space X is metrizable: if X is sequentially complete. then k is a metrizable compact space. PROOF. (i) Clearly, on any metrizable compact set there is a sequence of continuous functions separating the points. Recall that if on a set K one has two Hausdorff topologies r3 and r2 with respect to which K is compact and the natural embedding (K. rl) - (K, r2) is continuous, then this mapping is a homeomorphism. Therefore, if continuous functions fn separate the points of a compact set K, the metric
B(x, l/) = E 2-n
Ifn(x) - .fn(y)I
1 + Ifn(x) - fn(y)I n=1 generates the initial topology of K. This simple observation implies also that on a compact set K in a locally convex space X the weak topology coincides with the initial one and. in addition, that the weak topology coincides with every topology
A.2.
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Linear operators
on K generated by any family of continuous linear functionals separating the points in K. Therefore, in the case where there is a countable family with this property, the corresponding topology is defined by the aforementioned metric. Conversely, if a compact set K in a locally convex space X is metrizable, then the weak topology on K has a countable base of the form
{x: Il' (x - a)I < k-l, j = 1..... n(a)}, a E A, P E X', k E IN,
where A C K is an at most countable set. Therefore, there exists an at most countable family of continuous linear functionals separating the points in K. (ii) Let K be a metrizable compact set in a locally convex space X. Assume
first that X is complete. According to the Riesz theorem, the dual to C(K) is identified with the space of signed Borel measures on K. By the Banach-Alaoglu theorem, the closed unit ball U in C(K)' is compact in the *-weak topology. Since
the space C(K) is separable (see Problem A.3.23), then, by virtue of (i), U is compact metrizable in the weak topology. Let us consider the mapping
I: U -+ X, I(m) = rxrn(dx), K
where the integral is understood in the sense of Pettis (see Section A.3 below), and its existence follows from the completeness of X (see the proof of Lemma A.3.20 below). It is easy to see that this mapping is continuous if U is equipped with
the *-weak topology and X is given the weak topology. Therefore, the absolutely convex set 1(U) is weakly compact in X. Moreover, by the metrizability of U, this set is metrizable (see [231, Theorem 4.4.15]). Clearly, I(U) contains the closed absolutely convex hull of K, since K C 1(U) by virtue of the equality k = I(bk), where bk is the probability measure at the point k. Therefore, the closure of the absolutely convex hull of K is a metrizable compact set as a closed subset of a metrizable compact space (in fact, as can be easily shown, 1(U) coincides with the closed absolutely convex hull of K). It remains to note that the first claim from (ii) follows now from the existence of a completion of X. 0 A.2.
Linear operators
Bounded operators Recall some well-known facts from the theory of linear operators. We consider below only real spaces.
The range of a linear operator A on a space X is denoted by A(X). Ker A stands for the kernel of the operator A (the preimage of zero). Denote by £(X, Y) the space of all continuous linear operators from a locally convex space X to a locally convex space Y. Let £(X) := L(X, X ). If X and Y are normed spaces, then C(X, Y) is equipped with the operator norm II - IIc(x.r) An operator A on a normed space is called compact if it takes the unit ball to a precompact set. The space of all compact operators from X to Y is denoted by IC(X,Y). Put K(X) :=1C(X,X). The following useful result is called the closed graph theorem (see [670, p. 78, Theorem III.2.3]).
A.2.1. Theorem. Let X and Y be two &3 chet spaces (e.g., Banach spaces). A linear mapping A: X -. Y is continuous if and only if its graph {(x, Ax), X E X}
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366
is closed in XxY. In particular, if Banach spaces X and Y are continuously linearly embedded into a locally convex space Z and X C Y. then the natural embedding
X -. Y is continuous. Let H be a Hilbert space. In the definitions and statements below for the sake of simplicity of formulations we use the notation which means implicitly that the spaces in question are infinite dimensional; clearly, in the finite dimensional case we have in mind finite bases, etc.
A.2.2. Definition. An operator A E C(H) is called symmetric if (Ax.y) _ (x, Ay) for all x, y E H. A symmetric operator A E C(H) is called nonnegative if
(Ax,x)>OforallxEH.
Note that in real spaces (unlike the complex ones) the positivity of the quadratic form (Ax, x) does not imply the symmetry of A. For every nonnegative operator B E C(H), there exists a unique nonnegative operator C E C(H) denoted by v such that C2 = B. For A E C(H) we put IAI:=
Note that for any h E H one has (A.
Ax, x) = (Ax, Ax). An operator K E C(H) is compact precisely when so is the operator IKI. According to the Hilbert-Schmidt theorem, for any compact symmetric linear operator (I AI x, I AIx) =
A on a separable Hilbert space, there exists an orthonormal basis {e } such that
Ae =
a tend to zero.
A.2.3. Definition. An operator on a Hilbert space which preserves the inner product is called isometric (or an isometry). A linear operator is called orthogonal if it is invertible and preserves the inner product. For example, the operator x
(0, xl , x2, ...) on 12 is isometric, but not or-
thogonal. The polar decomposition of an operator A is the representation
A=UTAI, where U E C(H) is a linear isometry on the closure of I AI (H) given by U(I AI x) = Ax
and zero on the orthogonal complement of IAI(H). Note that U is well-defined., which follows from the fact that if I AI v = 0, then Av = 0. The operator U is called a partial isometry. If an operator A is injective (i.e., has zero kernel) and has the dense range, then U is an orthogonal operator. The polar decomposition can be written also in the form
A = AA' V, where V is the operator adjoint to the partial isometry from the polar decomposition for A. This representation yields the following simple, but useful fact: for every operator A E C(H) with dense range. one can find an injective nonnegative symmetric operator B such that B(H) = A(H). Indeed, by the factorization by the kernel of A this claim reduces to the case where A is injective. The range of the symmetric nonnegative operator B = AA' is dense as well, hence, as one can easily verify, this operator is injective. In addition, B(H) = A(H), which follows from the formula above, since V is orthogonal.
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Linear operators
A.2.4. Proposition. Let E be also a Hilbert space. Then K(H, E) coincides with the closure of the class of the finite dimensional continuous operators with respect to the operator norm.
A.2.5. Definition. Let H and E be two Hilbert spaces. An operator A E £(H, E) is called a Hilbert-Schmidt operator if the series (A.2.1)
IIAeaIIE a
converges for some orthonormal basis {ea} in H. If the space H is nonseparable, then the membership of A in the class of Hilbert-
Schmidt operators means that A is zero on the orthogonal complement of some separable subspace Ho C H and C'
,IIAen IIE
for some orthonormal basis {en } in Ho.
A.M. Proposition.
(i) If the series in (A.2.1) converges for some orthonormal basis in H, then it also does for every orthonormal basis in H and its sum does not depend on a basis. (ii) A symmetric operator A E £(H) is a Hilbert-Schmidt operator precisely when there exists an orthonormal basis {ea} in H consisting of the eigenvectors corresponding to the eigenvalues as among which there exist at most
countably many nonzero values an such that E an < 00. n=1
(iii) Let A = UTAI be the polar decomposition of an operator A E 1(H). Then A is a Hilbert-Schmidt operator precisely when so is IAl. PROOF. It suffices to prove these statements for separable H. Let {en}, {gyp;} and {i4,} be three orthonormal bases in H. Then IIAenII2 =
,
F(Aen. vi)2 = E E(en, A",Pi)2 = F IIXiGiII2. i=1
n=1
x x Since (A')* = A, we get E II Aen II2 = E IIA'j II2. Sufficiency of the condition in n=1
j=1 oc
x
(ii) is clear. Note that A is compact by the estimate II E x,Ae,112 <_ E IIAe,l1211XI12, i=n ion n
which yields the convergence of the finite dimensional operators E x,Aei in the i=l
operator norm. By the Hilbert-Schmidt theorem, A has an eigenbasis, whence the necessity of the condition in (ii). Statement (iii) is obvious. 0 The class of all Hilbert-Schmidt operators from H to E is denoted by ?l(H, E). We put
?t := ?t(H) := 7{(H, H).
A.2.7. Definition. Let H and E be separable Hilbert spaces. A continuous linear mapping A: Hk - i E is called a k-linear Hilbert-Schmidt mapping on H if
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for some orthonormal basis {en} in H one has
x IIA(e,,,....e,,)IIE < 00.
In this case the corresponding sum is finite for every orthonormal basis in H and is independent of bases. Denote by flk(H, E) the space of all k-linear Hilbert-Schmidt mappings from
H to E. Put Nk := 7ik(H,lR1). The space RA, is naturally isomorphic with the space f(H,Wk_1). Indeed, for every 41 E fik, the operator
T: H -» fk-1, T(h)(al,... ,ak-1) = 41(al.... ak_j,h), is Hilbert-Schmidt; conversely, for any T E 11(H,1lk-1), the k-linear form
(hl,-.. ,hk)'-' T(hk)(hl,.-- ,hk-1) is Hilbert-Schmidt. Note that the space N &.(H, E) with the inner product (A,B),.
_
elk)) il.....i}=1
is separable Hilbert. In particular, the set of all Hilbert-Schmidt operators on H equipped with the inner product (A,B)% = E(Aen,Ben)H n=1
is a separable Hilbert space.
A.M. Definition. Let H be a separable Hilbert space. An operator A E £(H) is called nuclear (or a trace class operator) if I AI has an orthonormal basis formed by its eigenvectors corresponding to the eigenvalues an such that Qn < 00.
n=l For a proof of the next result, see, e.g., {296, Ch. 111, §81.
A.2.9. Proposition.
(i) For any nuclear operator A. the sum of the series En I(Aen,en)H does not depend on an orthonormal basis {en}. (ii) A symmetric operator A E 1(H) is nuclear precisely when for every orthonormal basis (en) in H the series
x E(Aen,en)H n=1
converges.
(iii) A symmetric operator A E C(H) is nuclear if and only if there exists an orthonormal basis {en} in H consisting of the eigenvectors A corresponding to the eigenvalues on such that IanI < oo. n=1
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369
Linear operators
Denote by C(,) (H) the class of all nuclear operators on H. Clearly, C(i) (H) C N(H) C IC(H). For every A E L(j) (H), the sum
x trace A n=1
(which is independent of an orthonormal basis {en}) is called the trace of the operator A. The function IIAII(1) := trace JAI
is a norm on C(1)(H), with respect to which this space is Banach.
A.2.10. Proposition.
(i) The composition of two continuous operators between Hilbert spaces is a Hilbert-Schmidt operator if so is at least one of these two operators. In addition, if A E 1{(H) and B E £(H), then IIABIIN and IIBAIIN are majorized by IIAIIHIIBIIc(H)
(ii) The composition of two Hilbert-Schmidt operators on H as well as the composition of a nuclear operator and a continuous operator on H is a nuclear operator. (iii) If A E f(H), then A' E fl(H) and IIAIIx = IIA*IIht. PROOF. Note that (iii) has been shown in the proof of Proposition A.M. It suffices to prove (i) for operators A and B acting on one and the same space H. Since IIBAenII <_ IIBIIG(H)IIAefhl, we have BA E 1{(H) and IIBAII, : IIBIIL(H)IIAII,t
Using that IIB'IIc(H) = IIBIIc(H), we get from (iii) that AB = (B'A')' E R(H) and IIABII, <_ IIBIIc(H)IIAIIN. Finally, the composition of two Hilbert-Schmidt operators is nuclear by the inequality I(ABe;,e;)I = (Be,,A*e;)I < (IBe;II The second claim in (ii) follows from the first one together with the obvious observation that any nuclear operator A can be written as A = A, A2, where Al and A2 are Hilbert-Schmidt operators (e.g., using the polar decomposition A = UTAI, one A,
The following theorem describes an interesting connection between nuclear operators and functionals on the space of operators. Its proof can be found in [296, Ch. III, Theorem 12.3].
A.2.11. Theorem. Let T E L(1)(H). The functional K -+ trace(TK) on X (H) is continuous and its norm equals IITII(,) In addition, every continuous linear functional on 1C(H) admits such a representation, i.e., the space )C(H)' is naturally isomorphic to 4(1)(H). A.2.12. Proposition. Let H1, H2, E be Hilbert spaces, let A; E L(H;, E), i = 1, 2, and let A,(H,) C A2(H2). If A2 is compact or Hilbert-Schmidt, then so is the operator A1. If H, = H2 = E and A2 is a nuclear operator, then A, is nuclear as well.
PROOF. If A2 is injective, then the claim follows from Proposition A.2.10. Indeed, the operator A21 A, is well-defined by virtue of the inclusion A1(H,) C A2(H2) and, moreover, is continuous by the closed graph theorem, since the relationships xn x, A2'A,xn -+ y imply that A1xn -+ A2y, whence A,x = A2y, hence A2'A,x = Y. In addition, A, = A2A2'A,. The general case reduces to the case above by replacing A2 by the operator A2: H2/Ker A2 -+ E.
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370
A.2.13. Lemma.
(i) Let (T, µ) be a measurable space and let K be in L2(T xT, pop). Then the operator
Tx(t) = fK(ts)x(s)J4ds) T
on L2(µ) is Hilbert-Schmidt. Conversely, every Hilbert-Schmidt operator on L2(µ) admits such a representation.
(ii) Let H be a Hilbert space and let A: H - C[a, b] be a continuous linear mapping. Then the composition of this mapping with the natural embedding C[a, b] -* L2[a, b] is a Hilbert-Schmidt operator. The same is true when C[a,b] and L2 [a, b] are replaced, respectively, by L'° (St, µ) and L2(St, µ), where (Q, µ) is any probability space. (iii) Let W2.1[0,1) be the Sobolev space. Then its natural embedding to L2[0,1) is a Hilbert-Schmidt operator.
PROOF. The first claim is an exercise in functional analysis (see, e.g., [296, Ch. III, §9]). The proof of (ii) can be found in [603, Theorem 19.2.6] (the simpler case where 12 is a compact space is considered in [603, Proposition 17.3.7]). Claim "72.1 [0, 1] (iii) follows from (ii), since C C[0,1].
A.2.14. Definition. A continuous linear operator D on a Hilbert space H is called diagonal if there exists an orthonormal basis in H consisting of the eigenvectors of D.
Let us recall a theorem due to von Neumann (see 1326, Theorem 14.13]).
A.2.15. Theorem. For every symmetric bounded linear operator A on a separable Hilbert space H and every e > 0, there exist a diagonal operator DE and a symmetric Hilbert-Schmidt operator SE on H such that A = D, +S, and JIS,, 11%(H) :5 c.
A.2.16. Lemma. Let E, H be two Hilbert spaces, A E G(E, H). Suppose that H is separable and the operator A is injective. Then E is separable as well.
PROOF. The set A'(H') is dense in E' by the injectivity of A. Hence the space E' is separable, which implies the separability of E.
Semigroups and unbounded operators A linear mapping A defined on a dense linear subspace D(A) in a Hilbert space
H and taking values in H is called a densely defined linear operator. A densely defined operator is called closed if its graph is a closed set in HxH. If A is a densely
defined linear operator, then the domain D(A') of the operator A' is defined as the set of all vectors y E H such that the functional x H (Ax, y) is continuous on D(A) with the norm from H. By the Riesz theorem, there exists z E H such that (Ax, y) = (x, z) for all x E D(A). By definition, A'y = z. Note that the set D(A') may coincide with {0}.
We say that A is a symmetric operator in a (real) Hilbert space H if A is a linear mapping from a dense linear subspace D(A) C H (called the domain of A) to H such that (Ax, y) = (x, Ay) for all x, y E D(A). For any symmetric operator, the adjoint operator is defined at least on D(A), hence is also densely defined. A.2.17. Definition. A symmetric linear operator is called self-adjoint if it coincides with its adjoint (i.e., D(A*) = D(A) and A' = A on this domain).
A.3.
Measures and measurability
371
Unlike the case of bounded operators, a symmetric operator may fail to be self-adjoint.
A.2.18. Definition. Let X be a Banach space. A family (Te)e>o C £(X) is called a strongly continuous semigroup on X if To = I. Tt+, = TtT, for all t, s > 0, and, for every x E X, the mapping t Ttx from [0, oc) to X is continuous. One of the fundamental results in the theory of operator semigroups states that the linear subspace
D(L) :_ {h E X :
limTtht-h
exists in the norm of X }
is dense in X (see 1214, p. 620, Lemma VIII.1.8)). In addition, the linear operator L defined on D(L) by the equality
Lh=iim
Th - h t
is closed. This operator is called the generator of the semigroup (Tt)t>o.
A.3. Measures and measurability Measures and integrals Concerning the facts from the Lebesgue integration theory mentioned below, see [697, Ch. II]. The term "measure" means a countably additive bounded nonnegative
measure on a o-field of sets M. For two measures p and v on M, the absolute continuity of p with respect to v is denoted by p 4Z v. If p << v and v K it, then p and v are called equivalent (notation: p - v). The mutual singularity of two measures is denoted by p 1 v. For every measure p on M, the symbol Mm denotes the Lebesgue completion of M with respect to p. The sets from M are called p-measurable. The functions measurable with respect to M are called pmeasurable. A mapping that coincides with a given mapping F p-a.e. is called a modification or version of F. For p > 1 by LP(p) we denote the Banach spaces of p-measurable functions whose absolute values are integrable in power p. The norm in LP(p) is denoted by II . IILD(,.) or by II IIp. The integral of a function f over
a set A with respect to a measure p is denoted by J f (x) p(dx) or by A
In A
the case of integrating over the whole space the limits of integration are sometimes
omitted. The indicator function of a set A is denoted by IA (IA(x) = 1 if x E A, IA (X) = 0 if x j9 A). If p is a measure and A is a p-measurable set, then the measure PIA := IA p (i.e., pI A(B) = p(A fl B)) is called the restriction of p to the set A. It is known that every signed measure m (a countably additive real function on a o-field B in a space 0) can be written as m = m+ - m-, where m+ and mare mutually singular nonnegative measures on B called the positive and negative parts of m, respectively. The quantity Ilmil := m"(1?) + m' (1) is called the total variation of m (or shortly the variation of m). The variation is a norm on the linear space of all signed measures on B making it into a Banach space. The variation distance IIp - vII between two nonnegative measures p and v on o-field B can be written as
IIp - VII = sup{l (B) - v(B)I + Ip(1l\B) - v(S2\B)I, B E B}.
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372
X
Let pn be probability measures defined on o'-fields Bn in spaces X. Put X,,. Let B = ®- 1Bn be the o-field generated by all sets of the form
B = Bi x B2 x
x B. X Xn+1 X Xn.t2
, where Bi E Bi. Recall that the countable
product 000pn is a probability measure p on B (called a product-measure) defined n=1
by p(B) = µ1(B1) IA. (Bn) for the sets B of the form above. It is readily seen that this set function is well-defined. A well-known theorem in measure theory states p is countably additive (which is not obvious) on the algebra generated by such sets,
x
hence it uniquely extends to a measure on B denoted also by ®pn and called the n=1
product of the pn's. The construction of countable products enables one to define arbitrary products (8) A,, of probability measures on or-fields S. in spaces Xa. To
'
a
this end, it suffices to note that the v-field (&,B., generated by the sets 11a Ca, where Ca E Ba and only finitely many of the C,,'s differ from Xa, consists of the m
sets of the form E = C x Y, where C E ® Ban and Y = its#a X0. Hence we n=1
may put ®pa (E) = ® pa.. (C) a
n=1
Let p be a measure on a measurable space (X, B), let Y be a space with a v-field E, and let f : X - (Y, E) be a p-measurable mapping (i.e., f -1(E) C Br,; such mappings are also called (B,,, E)-measurable). Then the measure
pof-': A~p(f-1(A)) on E is called the image of the measure p under the mapping f . A function cp on Y is integrable with respect to pof -1 precisely when the function po f is p-integrable on X. In this case the following identity called the change of variables formula holds true:
f
w(y) IAof-1(dy) = I
v(f(x)),u(dz).
(A.3.1)
X
Y
A.3.1. Definition. A family .F C L1 (p) is called uniformly integrable if
lim sup f
C-. ofET If1>>C
if (x) I p(dx) = 0.
A sufficient condition for the uniform integrability is the estimate sup fey
f
I1(x)I log If (x)I p(dx) < oo.
Concerning the uniform integrability we refer the reader to Meyer's book [541, Chapter II, §2] and Shiryaev's book (697, Chapter II). The next classical result is called the Vitali-Lebesgue theorem.
A.3.2. Theorem. Let If, } C L' (p) be a sequence convergent almost everywhere (or in measure) to a function f. If the sequence {fn} is uniformly integrable, then it converges to f in the norm of L1(p).
Let (X, M, p) be a space with measure and let A C M be a one more or-field. Recall that for every integrable function f there exists a function EA f measurable
Measures and measurability
A.3.
373
with respect to A such that
r E-4f (x)g(x)p(dx) = ff(x)9(x)P() for every bounded function g measurable with respect to A. The function IEAf is called the conditional expectation of f with respect to A.
A.M. Definition. Let (X,M. p) be a probability space, T C IR' and let At, t E T, be an increasing family of a-fields contained in JAI. The family {ft}tEr of µ-integrable functions is called a martingale with respect to {At} if
IEA, f, = f V t. s E T. s < t. If in the relationship above we replace the sign "=" by ">". then we get the definition of a submartingale.
A.3.4. Example. Let f E L' (P), where (52,.x', P) is a probability space, and let {A, }tET C .P be an increasing family of a-fields. Then the family {IE't` f }tET is a martingale with respect to {A, },E ,..
The following two results obtained by Doob (the last statement in the next theorem is due to P. Levy) play an important role in probability theory. Their proofs can be found in 1541, Ch. V. §3J or [697, Ch. VII, §3J.
A.3.5. Theorem. Let { fn } be a martingale on a probability space (X. M, P) with respect to an increasing sequence of a-fields {An} in M. If the family {fn} is uniformly integrable. then there exists a function f E L'(P). measurable with respect to the a-field A, generated by {An}. such that IE'4^ f = fn for all n. In f almost everywhere and in L'(P). If f E L'(P), where r > 1. then addition. fn there is the convergence in L'(P) as well. Conversely. if f E L'(P) is measurable with respect to A. then {EA,, f } is a uniformly integrable martingale convergent
to f a.e. and in L'(P). The next result is Doob's inequality.
A.M. Theorem. Let (f,, } be a submartingale with respect to an increasing .sequence of a-fields such that the functions fn are nonnegative and Supllfn1IL2(P)
Then sup fn E L2(P) and II sup fnI[LJ(P) <_ 2 sup, IlfnilL1(P) n
a-fields in locally convex spaces Recall that for every family F of functions on a set X, there exists the smallest a-field E(X, F) (denoted also by E({F})), with respect to which all functions from F are measurable. This a-field is generated by the family of all sets of the form
{xEX: f
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374
80(X) is the Baire or-field generated by C(X ): B(X) is the Borel o-Held of X generated by all open sets.
Clearly. E(X) C Bo(X) C B(X). It is readily verified that 80(X) = B(X) for every metric space X. The proof of this fact and the following theorem can be found in [800, Ch. I].
A.3.7. Theorem. Suppose that X is a separable Frechet space. Then, for every family r C X' separating the points in X, the following equalities hold true:
cm I')=E(X)=Bo(X)=B(X). In addition, there exists a countable family r with such a property. However, in some important cases B(X) is strictly larger than .6(X).
A.3.8. Example. Let T be an uncountable set and X = IRr . Then E(X) _ B0(X) is strictly smaller than B(X). PROOF. The equality E(IRT) = B0(Il1T) follows from Theorem A.3.9. It follows from Lemma 2.1.2 that E(1RT) is not equal to 13(]R-').
The following deep result was proved in 1219].
A.3.9. Theorem. Let X be a locally convex space with the topology a(X, X'). Then E(X) = B0(X,a(X,X')). Moreover, for every function f continuous in the topology a(X,X'). there exist a continuous function g on U1 and functionals l; E X' such that if > 0} = {g o rr > 0), where ir(x) = (ll (x),12{x).... ). A Wrapping F from a topological space X to a topological space Y is called Borel
if F-'(B) E B(X) for every B E 8(Y). If Y = 1R1 with the standard topology, then F is called a Borel function.
Radon measures Let us recall some basic notions and results from measure theory on topological
spaces. The proofs of the results mentioned below and further references can be found in 1674] and 18001.
A.3.10. Definition. Let X be a topological space. (i) A countably additive measure on 8(X) is called a Borel measure. (ii) A countably additive measure on 80(X) is called a Baire measure. (iii) A Borel measure p on X is called a Radon measure if, for every B E B(X ) and every e > 0, there exists a compact set K C B such that p(B\K) < e. A measure p is called tight if condition (iii) is satisfied for B = X.
A.3.11. Theorem. Every Borel measure on any complete separable metric space is Radon.
If p is a Borel (e.g., Radon) measure on a topological space X. then by pmeasurable sets we always mean the elements of B(X ),, (the Lebesgue completion of B(X) with respect to p). Any Radon measure on a locally convex space E is uniquely determined by its values on 6(X ).
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Measures and measurability
375
A.3.12. Proposition. Suppose that it is a Radon measure on a locally convex space X. Then, for every p-measurable set A. there is a set B E E(X) such that
p(ADB)=0. Moreover, if G C X' is an arbitrary linear subspace separating the points in X. then such a set B can be chosen in E(X,G).
PRooF. Let e > 0. Let us find compact sets K C A and S C X such that
p(K) > p(A) - e and p(S) > p(X) - e. We may assume that K C S. There exists an open set U D K with u(U) < p(K) + e. Since on the compact set S the initial topology coincides with the weak topology, there is a set V open in the weak
topology such that V n S = U n S. By the compactness of K one can find a set IV which is a finite union of open cylindrical sets such that K C IV C V. Then we have IV E E(X) and p(IV ZD A) < p(W L K) + e < p(V\K) + e < 3e,
whence our claim. The same proof works for C replacing X', since the topology a(X,G) coincides with a(X, X') on S.
A.3.13. Corollary. Let p be a Radon measure on a locally convex space X. Then the collection of all bounded cylindrical functions on X is dense in LP(p) for every p E [1, co). In addition, the linear space T generated by the functions of the form exp(i f ), f E X. is dense in the complex spaces LP(p). Moreover, both claims remain valid if X' is replaced by any linear subspace C C X' separating the points in the space X.
A.3.14. Definition. Let p be a Borel measure on. a topological space X. A closed set S,, C X is said to be the topological support of p if p(X 0 and there is no smaller closed set with this property. Every Radon measure has the topological support (see Problem A.3.35). In measure theory an important role is played by Souslin sets defined as the images of complete separable metric spaces under continuous mappings to Hausdorff topological spaces. Hausdorff topological spaces that are continuous images of complete separable metric spaces are called Souslin spaces. Non-Borel sets of this kind were discovered by M. Ya. Souslin. For example. the orthogonal projection of a Borel set in 1R.2 to lR' may fail to be Borel, but it is a Souslin set. It is known (see [185)) that there exist an infinitely differentiable function f : IR' IR' and a Borel set B C 1111 such that f (B) is not Borel. N. N. Lusin established the measurability of Souslin sets. In the next theorem we have collected the most important properties of Souslin sets frequently used in measure theory. Their proofs can be found in [674, p. 124, Chapter 11, Corollary 1: p. 95, Theorem 2: p. 103, Corollary 3; p. 107, Corollary 16] or in [344].
A.3.15. Theorem. Suppose that X and Y are Hausdorff topological spaces and f : X -. Y is a mapping. Then: (i) Every Souslin set in Y is measurable with respect to every Radon measure on Y. (ii) If X is a complete separable metric space and f is continuous, then f(B) is a Souslin set in Y for every Borel set B C X. If, in addition, f is injective, then f (B) is Borel in Y.
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376
(iii) If X and Y are Souslin spaces and f is a Bore! mapping, then the images and preimages of Souslin sets are Souslin. If f is injective. then f (B) is Bore! in Y for every Bore! set B in X. It is known that every Borel measure on any Souslin space is Radon (see [674,
p. 122]). On every Souslin space X, there exists a countable set of continuous functions separating the points. Therefore, all compact sets in Souslin spaces are metrizable. Hence every Borel measure on any Souslin space is concentrated on a countable union of metrizable compact sets. Note also that all Borel subsets of Souslin spaces are Souslin. It is worth mentioning that a space which is a countable union of its Souslin subspaces is Souslin itself. However, the complement of a Souslin set may fail to be Souslin: moreover, if the complement of a Souslin set is Souslin, then this set is Borel (see 1674, Corollary 1, p. 101]). A Souslin space may be nonmetrizable (for example, the space 12 with the weak topology). However, sequentially closed sets in Souslin spaces are Borel (see [674, Corollary 1. p. 109]). In particular, any sequentially continuous function on a Souslin space is Borel. A proof of the following result can be found in [674, Lemma 18, p. 108].
A.3.16. Proposition. Let X be a Souslin space. Then B(X) is generated by some countable family of sets. In addition, B(X) = E(X, If,)) for every sequence of Borel functions f separating the points in X. Finally, if X is a Souslin locally convex space, then such a sequence can be chosen in any set G C X* separating the points in X. A very important object connected with a measure on locally convex space is its Fourier transform.
A.3.17. Definition. Let X be a locally convex space and let p be a measure on E(X ). The Fourier transform µ of the measure p is defined by the formula
µ: X' - C. µ(f) = JexP(if(x)) p(dx).
(A.3.2)
x A.3.18. Proposition. Any two measures on E(X) with equal Fourier transforms coincide.
According to Corollary A.3.13, any two Radon measures with equal Fourier transforms are equal. The following theorem may be useful for constructing Radon measures. A.3.19. Theorem. Let X be a locally convex space, let G be a linear subspace in X' separating the points in X. and let p be an additive nonnegative function on the algebra 1c of all cylindrical sets generated by G. Suppose that p has the following property: for any e. there exists a compact set KK C X such that p(C) < e for every cylindrical set C E IZc which is disjoint with K, Then p uniquely extends to a Radon measure on X. Let X be a locally convex space and let p and v be two measures on 6(X) - Then the measure p8v is defined on E(XxX) (note that E(XxX) = e(X)6(X), which can be easily deduced from the equality (X xX )' = X'xX' ). The image of this measure
under the mapping X x X - X, (x, y) '-- x + y, is called the convolution of the measures p and v and is denoted by p * v. It is easily verified that, letting A = p * v, one has A = µv. With the aid of Theorem A.3.19 one readily proves that the product
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Measures and measurability
377
of Radon measures µ;, i = 1.... , n, on a locally convex spaces X; is uniquely extended to a Radon measure p on X 1 x . x X,,. More generally, if µ are Radon probability measures on locally convex spaces X,,, then the product-measure ®µ n=1
extends uniquely to a Radon measure on X = 11n 1 X,,. By the product of Radon measures we always mean the result of this extension. Certainly, for reasonable spaces (e.g., separable metric or Souslin), there is no need to consider extensions. since the product-measure is defined on the Borel a-field of the product space from the very beginning. It is known (see [800, p. 60. Ch. I. Theorem 4.11) that if p is the aforementioned product of two Radon measures µ1 and p2, then for every B E 13(XIxX2), thefunction x2 µ1(B?,), where Bx, = {x1 E XI: (x1,x2) E B}, is Borel and p(B) = it I(Bx,)l12(dx2)
J
XZ
In particular, if p and v are Radon measures on a locally convex space X, then their convolution p * v extends uniquely to a Radon measure (again denoted by µ * v). By the convolution of Radon measures we always mean the result of this extension. In this case, according to [800, p. 64, Ch. I, Proposition 4.4]), for every B E 8(X), the function x - µ(B - x) is Borel and the following equality holds true:
µ * i'(B) = f µ(B - x) v(dx). X
Pettis integral Let f : (X, B(X )},) (X, t'(X )) be a measurable mapping on a locally convex space X with a Radon measure p. The element. m E X is called the Pettis integral of the mapping f if for every I from X' the function l(f) is integrable with respect to p and its integral equals 1(m). Put f f(x)p(dx):= m.
A.3.20. Lemma. Let µ be a Radon probability measure on a sequentially complete locally convex space X concentrated on a metrizable compact set K. Then any
sequentially continuous linear mapping A: X - X has Pettis integral which is an element of the closed convex hull of the compact set A(K). PROOF. According to Problem A.3.36, there exists a sequence of probability measures µ with finite supports in K that converges weakly to the measure is. For the measures p,,, the Pettis integrals In := fk obviously exist and are elements of the convex hull Q of the compact set A(K). Since for any I E X' the function 1 o A is continuous on the metrizable compact set K (being sequentially continuous), by construction, the sequence 1(I,,) converges to fh l(Ax) µ(dx). Hence the sequence {I,,} is a Cauchy sequence in the weak topology. Note that if X is complete, then the closure of Q is compact, and the initial topology coincides with the weak one on this closure. Hence converges to some point m E X. Clearly, m. is the Pettis integral of A.
Suppose now that X is only sequentially complete. The compact set A(K) is metrizable (as a continuous image of a metrizable compact space, see [231.
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378
Theorem 4.4.15]). By virtue of Proposition A.1.7, the closure of Q is a metrizable compact set as well. Therefore, we conclude again that the sequence {In } converges to some point m., which is the Pettis integral of A. 0
If X is a separable Banach space and (S2,µ) is a space with measure, then for measurable mappings f : 1 -. X satisfying the condition 11f11, E L1(µ), the notion of the Bochner integral is defined by the analogy with the Lebesgue integral for scalar functions, see [214, Ch. 1111). In this case the Pettis integral exists as well
and coincides with the Bochner one. Let us denote by U'(µ, X) the Banach space of all p-measurable X-valued mappings f such that IIfIILI(P.X,
{ff(x)II x µ(d2)}
1/p
< oc.
S1
This notation is also used in the case when X is a normed space, but then it is additionally required that f be Bochner integrable.
Random vectors Let (52,.x', P) be a probability space, X a locally convex space. The symbol is used to denote the expectation of a random variable t: on fl (i.e., lE,t; is the Lebesgue integral of the measurable function C). A measurable mapping l; : S2 lE
(X,E(X)) is called a random vector in X. The measure Pt(C) = P(t:-(C)) is called the distribution (or the law) of C. Clearly, every probability measure on E(X) can be obtained in such a form (with the identity mapping C(x) = x). If we have a family of probability measures µn on X, then there is a family of independent random vectors f,, on one and the same probability space S2 such that
x
x
n=I
n=1
Pt, = FIn (take Si = fi X,,, Xn = X, P = ® p,,, Cn(w) = wn) A random process (tt)tET is, by definition, a collection of random variables on a probability space
(f),.P, P). In this case Ct...... t,,.B = {w:
(w)) E B} E,F
(A.3.3)
for every Borel set B E B(IRn) and any t I , ... , to E T. Therefore, we can define a
measure on the algebra R(R') of the cylindrical sets of the form (A.3.3) by the formula
µ'(Ct,... J_B)=P((tt,,...,EtjEB). This measure is automatically countably additive, hence it uniquely extends to a countably additive measure on E(RT) denoted by pt and called the distribution of the process t in the functional space (or the measure generated by f). Conversely, any probability measure p on E(lRT) is the distribution of the random process
t't(w)=w(t)ifwetake S2=lRrandP=p. Note that f o r any finite collection t1, ... , t,, E T, the formula above defines a probability measure Pt,..... t on R" called the finite dimensional distribution of {-
It is clear that if {sl,....sk} C {ti.... ,t"}, i.e., s, = t7i, i = 1,... ,k, then the image of Pt,,.... t under the mapping n
k
coincides with Pa,...,,,,, (i.e., the projections are consistent). The following result is a celebrated theorem due to Kolmogorov (421) (see its proof also in 1822, Ch. 51) .
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Measures and measurability
379
A.3.21. Theorem. Suppose that for every finite set of points tt, ... , to E T. a probability measure P......t on JR" is given such that the aforementioned consistency property is satisfied. Then there exists a probability measure P whose finite dimensional projections are exactly P,,.
Another Kolmogorov's result enables us to construct measures on the space C[a,b] (its proof can be found in [822, Ch. 5]). A.3.22. Theorem. Let fit. t E [a. b]. be a random process such that for some a > 1, C > 0, and e > 0. one has IE{fr - l;. JO < Cat - slh+`,
dt, s E [a, b].
Then there exists a random process 77, t E [a, b]. with continuous trajectories such that for each t one has nr = tt a.s. In particular. the process 171 has the same finite
dimensional distributions as r (hence Ee" = µ£). In addition. (µ')*(C[a.b]) = 1. Moreover, (pt )'(W [a, b]) = 1 for every b E (0, a/a), where H5[a, b] is the set of all functions satisfying the Holder condition of order 6.
Note that the same is true for the processes with values in separable Banach spaces (see [289. Ch. 3, §5] ).
Problems A.3.23. (i) Let K be a metrizable compact space. Show that the space C(K) with the sup-norm is separable. (ii) Show that the product I' of the continuum of segments is separable, but the Banach space C(I') is not.
A.3.24. Prove that if X a separable normed space, then there exists a countable family of continuous linear functionals separating the points in X, hence X' is separable in the s-weak topology. The converse is not true. A.3.25. (i) Show that any reflexive Banach space is sequentially complete in the weak topology. (ii) Let X be an infinite dimensional normed space. Show that X is not complete in the weak topology (i.e., there exists a net which is Cauchy in the weak topology, but has no limit in the weak topology). In addition, the space X' is not complete in the s-weak topology.
A.3.26. Construct an example of a locally convex space X such that there exists a sequentially continuous linear functional on X that is not continuous.
A.3.27. Let K be a convex compact set in a separable metrizable locally convex space X. Show that K is the intersection of a sequence of closed half-spaces: if K absolutely
convex, then p,, has the form p,; (a) = supi l,(x) for some sequence (1,) C X'.
A.3.28. Let X be a Banach space. Prove that if a linear mapping A : X -- X is continuous from the weak topology to the norm topology, then the range of A is finite dimensional.
A.3.29. Show that there exists no continuous norm on the space 1R" with its natural topology.
A.3.30. Let X be a Hilbert space, let Y be a locally convex space, and let L E C(X,Y). (i) Show that L takes the closed unit ball to a closed set. (ii) Show that if K E K(X,Y). then K maps the closed unit ball to a compact set. A.3.31. (i) Give an example of two nonnegative nuclear operators on a Hilbert space which have dense ranges intersecting only at zero. (ii) Let K be a compact operator on an infinite dimensional separable Hilbert space H. Prove that there exists a compact
Appendix
380
operator S such that its range is dense in H, but intersects the range of K only at zero. (iii) Let X be a Banach space and A E L(X). Show that if the set A(X) is dense and does not coincide with X, then there exists an operator B E £(X) such that the set B(X) is dense and intersects B(X) only at zero. Hint: see [6821.
A.3.32. Let K be a compact set in a Hilbert space H. Show that K is contained in a compact ellipsoid of the form A(U), where A is a symmetric compact operator on H and U is the unit ball of H. Hint: it suffices to consider separable H with an orthonormal basis {en }: construct an increasing sequence of natural numbers k(n), n E IN. such that k(n)
(r,e))2 < 2-" for all z E K and n > 1. Let Ae. = aye,, where a. = 1 if
j
fn(W) < x}. A = {W E f2: 3 lim fn(W)I, E _ {W E n: sup n
nx
A.3.35. Show that every Radon measure has the topological support.
A.3.36. Let K be a compact metric space and let Q C K be a countable set dense in K. Prove that for every Borel measure p on K, there exists a sequence of linear combinations of Dirac's measures 6q, q E Q, convergent weakly to p. IR 1 be a bounded Borel A.3.37. Let X and Y be Souslin spaces and let f : X xY function. Show that the function g(x) = sup f (r. y) is measurable with respect to every 'EY
Borel measure on X.
A.3.38. Let {{n} be a martingale with values in a separable normed space E and let q be a continuous norm on E. Prove that {q(4,)} is a submartingale. Hint: using the Hahn-Banach theorem represent q as the supremum of a sequence of continuous linear functionals.
A.3.39. Let E be a separable Hilbert space, let (S1, P) be a probability space, and let S C L2(P. E) be a compact set. Show that there exists a nonnegative compact operator K in E with dense range such that F(.) E K(E) a.e. for every F E S and
sup f IIK-'F(W)IIE P(d r) < ac. FES
A.3.40. Let p be the Radon extension of the product of the continuum of Lebesgue measures A on [0, 11 (p is defined on the compact space X = [0,1]`). Prove that p(K) = 0 for every metrizable compact set K C [0,1]`. Hint: consider the base U of the topology in [0, 1]` formed by the products Ur = fl, J. where, for some finite set T, J, = (0,11 if t 1% T and J, = (a, b) with rational a, b such that lb - al < 1/2 if t E T; show that there is a set U E U such that U n K belongs to an uncountable number of the sets Ur.
A.3.41. Let A be an absolutely convex set in a locally convex space X and let E,, be its linear span. (a) Show that if A is a Borel set, then its Minkowski functional PA is a Borel function on E,,. (b) Show that if all the sets tA. t E IRI, are measurable with respect to some measure p on E(X) (or with respect to a Borel measure p), then p5 is p-measurable. Hint: write the set {p,, < c} as the countable union of the sets r,,A over all rational numbers rn < c.
Bibliographical Comments 0, non enreauine JHCT14 B CTenax ae4epHHx 6H6nHoTeK. Kor;ta paauy.Nlba TaK 411CTW. A nbint, nbsxee. 4exi HapKOTHK! H. Yy.%mnea.
13 6H6nSOTeKe
You will see there a lot of what I am telling you (although, sometimes this is said there in an indirect wav. as an alternative. etc.), because the authors are professionals and most of the real information they have incorporated in their texts.
D. V. Dcopik. Biblical archaeology
To Chapter 1 Normal density was first used by A. Moivre [551] in the central limit theorem. and was later considered by P. Laplace. Two dimensional normal distributions appeared in the works of Adrain [7] (1809), Laplace [463] (1810), Plana [617] (1813), Gauss [280] (1823) (initially random vectors were assumed to have independent coordinates) and were further studied by Bravais [110] and other researchers. Some historical information and references can be found in [4], [533]. [591], [593]. [722], [723]. [730]. The term "normal distribution" was suggested in the second half of the 19th century (F. Galton, K. Pearson, C. S. Peirce). In particular, in [594] the term "normal" applies to the curve of errors. See historical notes in [591]. For quite a long time normal distributions were also called "Laplace distributions", but for the past 40 years this name became obsolete (cf. the polemical remarks in [282]). Now the terms "normal random variable", "normal distribution", "Gaussian random variable". "Gaussian distribution". "Gaussian measure" are generally accepted. Some additional information about Gaussian measures on finite dimensional spaces can be found in the books [18], [413]. [545], [591], [769]. [770]. See also the bibliography [321]. Calculation of normal distribution density is discussed in [528]. The polynomials Hk are often called the Chebyshev -Hermite polynomials (see, e.g., [591], [7371), since they were investigated independently by P. Chebyshev and Ch. Hermite. We
use a shorten name in order to avoid confusion with other Chebyshev polynomials that are encountered more often in the approximation theory. Identity (1.2.4) was noted by many authors, see, e.g., [616]. Mehler's formula goes back to Mehler's work [540]. Logarithmic Sobolev inequality was obtained by Gross [317]. The hypercontractivity of the Ornstein-Uhlenbeck semigroup was established by Nelson
[561]. A probabilistic proof can be found in [564]. An important contribution to this direction is due to Bakry, Emery. and Ledoux [33], [34], [35], [467], [469], [471]. Theorem 1.7.1 is essentially due to B. Maurey and G. Pisier (see [605]). Inequality (1.8.1) was proved in [628] (see also [475]) and generalized in [109], whence our proof is borrowed. Anderson's inequality was obtained in [17] for a considerably larger class of
381
Bibliographical Comments
382
measures (see [119]). Lemma 1.7.7 is a classical result in the theory of information (see [443, Ch. 2] and the references therein).
To Chapter 2 Bachelier [24] was the first to consider the Brownian motion from the mathematical
point of view (in the framework of the analytical machinery of that time). From the modern point of view, Bachelier used the Brownian process to describe stock prices in some options of that epoch in France. The Gaussian property of the increments of the Brownian motion was noted in the physical works by A. Einstein and M. Smoluchowski in 1905-1906. Note also Langevin's work [462], where the equation bearing now his name was introduced. Historical comments can be found in [1131. The first rigorous construction of the integral over the Wiener measure (in the form of the Daniell integral on the space of continuous functions) was realized by Wiener (823], [824], (8251, [826]. There are several different ways of proving the existence of the Wiener measure. One of them is based on the Fourier series with random coefficients (see, e.g., [3901, [430]). One can also use the
Haar functions on this way (see [170]). A second proof (given in the text) is based on two Kolmogorov's theorems. A third possibility comes with Gross's theory of abstract Wiener spaces. There exist proofs (see [603]) using the concept of a radonifying operator.
In addition, there are modifications of those approaches (see, e.g.. [800]). Instead of Kolmogorov's theorem one can use approximations of the Brownian motion by discrete random walks (see, e.g., [64]) and then the arguments based on the weak compactness. Certainly, not all possibilities to prove the existence of the Wiener measure are mentioned here. A large series of papers on the classical Wiener measure was published in the 40-50s
by Cameron, Martin, and their collaborators (see (133] - [144] and the survey [4251). Uhlenbeck and Ornstein [782) soon after Wiener's works suggested another model of a diffusion corresponding to a Gaussian random process bearing now their names; this type of diffusion was investigated further in [204] and (8161. The first general definition of a Gaussian measure on an infinite dimensional space is due to Kolmogorov 14221. Intensive investigations of Gaussian measures on Banach spaces and the distributions of Gaussian processes in functional spaces started in the 50s (see, in particular, [261J, (262), [2671, (311), 1335]), [480), [556), [630). In the 60s the general theory of Gaussian measures on linear spaces was developed in [159], [240], [278), [2851, (3121 - [3151, [3711, [3961, 1432], [563], [638]. [658], [6961, (6621, [6881, [8101 - (8131. The books by Shilov and Fan Dyk Tin' [696], Neveu [563], and Rozanov (6581 were the first monographs devoted to the theory of Gaussian measures. An extensive bibliography is contained in 1563). Most of the results of the linear theory presented in this book were already known that time (although, not always in the general form known now). In subsequent years, a significant progress was achieved in the unification of the theory; connections between Gaussian distributions in the function spaces and Gaussian measures on Banach and locally convex spaces were clarified (see also comments to Chapter 3. Finally, note that large parts of the theory of Gaussian measures are presented in the books (1781, [189], 12891, 1338]. (3591, [4451, [491], [563), [658), [6961, [832), [835), and surveys [27]. [28]. (81J. Kuo's book [445] has been one of the most widely cited references in the theory of Gaussian measures over the past 20 years. Theorem 2.2.12 is taken from [753]. Additional results concerning characterizations of Gaussian measures on infinite dimensional spaces can be found in [3301, [4051, [631].
The fact that the integral of a Gaussian process over the parametric set in 1R' is a Gaussian random variable was noted by Pitcher [6091 (this fact follows directly from [206, Ch. II, Theorem 2.8]). In the general case the assertion in Example 2.3.16 was proved by Rajput [634]. Another proof is found in [504]. The simplified argument given in the text was suggested by Vakhania [7971. The Cameron-Martin formula was proved originally in [137] for the classical Wiener space and the absolutely continuous functions h E C[0,1) such that h' has bounded variation. Later, in the works [5291, (136], and [745), it was noticed independently that instead
Bibliographical Comments
383
of the boundedness of variation of the derivative it suffices to require that h' E L2(0,1] (the necessity of this condition was noted in [6771). The conditions for the absolute continuity
and formulas for the density of a shift of a general Gaussian measure were obtained in [311]. The Cameron-Martin space is often called the reproducing kernel Hilbert space of a Gaussian measure: this term is used also for the Hilbert space of all measurable linear functionals by analogy with the Hilbert space obtained as the completion of the pre-Hilbert space of functions on the set T with respect to the inner product generated by a kernel K(., ) on T2 (such a construction was considered by Aronszajn (21]). The zero-one law for Gaussian measures follows directly from the classical zeroone law of Kolmogorov [421] (and in this sense can be regarded as Kolmogorov's result); Cameron and Graves [136] proved the zero-one law for the Wiener measure. More general formulations were found in [397]. The zero-one law for subgroups was proved in [380],
[397], [372), [31], (477]. An important event in the theory of Gaussian measures was the discovery by Hajek [324], [325] and Feldman (238] of the dichotomy "equivalence or singularity". Related results were obtained in the works in mathematical physics [677], [683). Further progress was achieved in Rozanov's papers [653] - [658). In the 60s
and 70s those results were generalized and specified for various special classes of Gaussian processes by many authors (see [288], (342]. [400), [590], [658], [687], [688], [805]. (808], [815), [830]). It was noticed in [478) that the dichotomy "equivalence or singularity" can be deduced from the zero-one law. The proof given in the text is due to Talagrand [753]. Concerning zero-one laws and equivalence/singularity, see also (132], [218], [244], [363), [664]; (212] and [242] give short proofs of the zero-one law for stable measures.
For alternate proofs of the Ito-Nisio theorem, see [243], (491]. Related results are obtained in (581], [582]. Natural modifications were studied by Tsirelson [774], [775], who obtained Theorem 2.6.4. Results related to Fernique's theorem were obtained in [709], [461]; (364] and [128] study the integrability of seminorms of Gaussian vectors in non locally convex topological vector spaces.
Measurable linear functionals on the classical Wiener space are described in [136], [310]. More general situations were considered in [810]. [696). See (720] concerning
generalizations of the large numbers law for measurable linear functionals. The stochastic integral of a non-random function against a Brownian path (the Paley-Wiener-Zygmund stochastic integral) was introduced in [586] (see also [5851). Infinite dimensional Gaussian distributions are actively used in the financial mathematics (see [698)).
To Chapter 3 Many results in the theory of Radon Gaussian measures were obtained originally in more special cases (for example, for the Wiener measure or for the product-measures). For this reason, it is sometimes embarrassing to attribute the priorities. One of the first detailed expositions of the theory of Radon Gaussian measures was Borell's article [96], which influenced considerably this area. The main tool of transferring the classical results to the general locally spaces setting are theorems 3.4.1, 3.4.4, and 3.5.1 which are essentially due to Tsirelson [774], [775). Modifications of the original Tsirelson proofs were suggested in [863]. [752]. In addition to the works already cited, Gaussian measures on locally convex spaces were investigated in [196], [240], [577], [633], [635], [666). [772]. The first correct proof of the separability of the spaces L2 (y) for Radon Gaussian measures was published in [667] (independently, but later, this fact was proved in [774], [96]); the proof given in the text follows [667] (Lemma 1.8.8 used in this proof was noted in [667] and later in [4871). Convergence of series and sequences of Gaussian vectors was investigated in [97], [117], [127], [149]. [151], [371), [374], [434], [460], [583], (774], [775], [796], [800], [815], and in the papers cited therein. A detailed exposition of the related problems has
384
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been given by Yurinsky [835]. Expansions of Gaussian processes in the series with respect to the eigenfunctions of integral operators go back to Karhunen's and Loeve's works [407]. [506]. The works of Its and Nisio (371] and Tsirelson (774], (775] were of particular importance for this direction. Representation (3.5.2) for general separable Banach spaces was obtained in [374], [434] and for separable Frechet spaces in [633]. Concerning Gaussian function series, see [55], (117], [390], (523], [774]. [775]. and historical notes in (392]. Theorem 3.5.7 was obtained in (149]; Theorem 3.5.10. answering a question posed in [445], is due to [457]. Supports of Gaussian measures were studied in [367]. [583], [833]. Theorem 3.6.5
for Banach spaces is due to [118]; for Frlchet spaces it was proved (by the argument presented in the text) in [73]. The first example of a Gaussian measure on a separable Banach space without Hilbert support was constructed by Dudley [209]. It was shown in [319] and (716] that the classical Wiener measure on C[0.1] has no Hilbert support ([319] mentions also an unpublished proof given earlier by S. Kwapien). The short proof of a more general fact given in the text is borrowed from [93].
Measurable linear functionals and operators were investigated (in addition to the works mentioned in the comments to Chapter 2) in [98], [254], [578], [665]. Theorem 3.7.6 and Proposition 3.7.10(i) go back to Gross's works [312], [314]: Assertion (i) in Proposition 3.7.10 is generalization of [445, Corollary 1.4.41; assertion (ii) in this proposition extends a theorem of Goodman (see (445, Theorem 1.4.6]). The weak convergence of Gaussian measures was considered in [41], (151]. [161]. [162], (163]. (247]. [303]. [801].
Abstract Wiener spaces were introduced by Gross [314] and investigated by many authors in the 60s and 70s (see (25], [29], [30], [43], [211]. (213]. [316]. [398]. [432] (435], [445], [662]). The fact that any nondegenerate Gaussian measure on a separable Banach space can be represented as an abstract Wiener measure was proved by Sato (662],
Jain and Kallianpur (374], and Kuelbs (433]. An analogue of the filtration a(w,. s < t) on the classical Wiener space can be introduced for abstract Wiener spaces (see [793]). See (265], [753] for interesting examples exhibiting various set-theoretical pathologies arising for non Radon Gaussian measures on Banach spaces (such as I" ).
There are two directions, which link the material of Chapter 3 with the theory of locally convex and Banach spaces, but are not discussed in this book. The first of them is concerned with various notions of a radonifying operator. Let X and Y be two locally convex spaces. A continuous linear operator T : X -. Y is called 1-radonifying if, for every cylindrical Gaussian measure v on X with the continuous Fourier transform, the measure voT-1 is tight. In the case, where X and Y are Hilbert spaces, this class is precisely the class of all Hilbert-Schmidt operators. An additional information can be found in (557]. (603], [800]. The second direction deals with the characterization of the quadratic forms on X' which are Gaussian covariances. In the case of a Hilbert space this class coincides with the family of all nonnegative forms generated by nuclear operators, hence, coincides with the class of covariances of all probability measures p with f Iixl[2 p(dx) < --<,. The situation is different in general Banach spaces; see Remark 3.11.24 and remarks in section 7.5. Further references can be found in [164], [499], [557], [750], [796], (799], [800]. Various estimates connected with Gaussian measures can be found in [39], (293]. [302], [304], [587], [760], [837]. Functions separating Gaussian measures are discussed in [482], [495]. Gaussian measures on non locally convex linear spaces are considered in [124], [125]. [126], [127], [243]. and in the references given therein. Gaussian measures on projective spaces are discussed in [652]. There exist analogues of Gaussian measures on more general spaces (for example, on groups, on the spaces over p-adic numbers, in the noncommutative analysis); see [232]. [328], [334], [418], [419], [814]. Evaluation of Gaussian functional integrals is discussed in [221]. Applications of Gaussian measures in the quantum field theory can be found in [60], [266], [291], [520], [703], (704]. Concerning applications
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in statistics, see, e.g., [122], [359], [360], [389), [399]1 [443), (504), (521), (6071, [637], (777). Applications of Gaussian measures in the theory of complexity of algorithms are discussed in [773). Concerning Kolmogorov's widths and related objects on Gaussian spaces, see (510]. [773], and the references therein.
To Chapter 4 The logarithmic concavity of a broad class of measures including all Gaussian measures was proved in [628] (for convex sets) and in 1941. The arguments based on the Brunn-Minkowski inequality were applied in [840] for the proof of certain special cases of the logarithmic inequality and the logarithmic concavity of the function x i-. y (A + x) for the absolutely convex sets A C R'. The isoperimetric inequality for Gaussian measures on infinite dimensional spaces was proved by Sudakov and Tsirelson [736] and Borell [95]. The exponential integrability of H-Lipschitzian functions was first proved in [172). Ehrhard's works [222] -- [225] became a considerable contribution to this direction; we follow these works in our exposition. Additional results connected with the isoperimetric inequalities and Brunn -Minkowski type inequalities can be found in [38], [102], [471], [472]. Corollary 4.4.2 was obtained (with a different proof) in [776]; for more details about the distribution of the maximum of a Gaussian process, see [189], (197], [198], [490], [491], [734], [754], [756]. Proposition 4.4.3 is taken from [172]. The proof of Theorem 4.6.1 given in the text is borrowed from (671]. Clearly, this result follows immediately from gidak's inequality (Corollary 4.6.2) proved by §idak [699], (700], and Khatri (416] (see comments given in [701] concerning the reasoning in (675] and several other papers). This inequality was conjectured by Dunn (215], who also proved some special cases.
Onsager and Machlup [579] investigated the behavior on balls of the measure p generated by a diffusion process , (with a constant diffusion coefficient). Their work became a starting point of intensive investigations (see (152], (361]). The first general results on the existence of the Onsager-Machlup functions were obtained in [579], [747], [264] for the Wiener measure on C[0, 1] with the sup-norm and for Gaussian measures on Hilbert spaces with the usual norm. A stronger result - the existence of the limit in (4.7.1) for any bounded absolutely convex set V in a locally convex space and t' = eV -was obtained by Borell [98). Later Dudley, Hoffmann-Jorgensen, and Shepp (345] proved
that the Onsager-Machlup function exists for the product y of the standard Gaussian measures on the real line, provided that the -(-measurable norm q has the form q(x) = sup q(xl, ... , x,,, 0, 0, ... ). In [689], Shepp and Zeitouni considered the case, where y is the Wiener measure on C[O,1] and q is a norm satisfying certain special conditions; they proved that under their conditions the limit in (4.7.1) exists for all h E H(y). All these results were improved in [80]. Various results related to the Onsager-Machlup functions are obtained in [98], [302] - 1304], [438], [496), (500], [609]. An important problem of the theory of random processes concerns estimates and asymptotics of the distributions of small values of Gaussian processes. A closely related to that is the investigation of Gaussian measures of small balls in Banacb spaces. The distribution of the absolute value of the Wiener process and the asymptotics of the Wiener measure on small centered balls are special cases of more general results obtained by Kolmogorov and Leontovich [423] and Petrovskii (597]. The asymptotic behavior of Gaussian measures of small centered balls in Hilbert spaces was described by Zolotarev (842]. His result was generalized in [343) and precised in different directions by many authors. Concerning Gaussian measures of small balls and related problems as well as additional references, see [37], (61), [166], [193), [356), (437], [441], [442), (484], (486], (488], [492], [493], (498], (550], [553], [559], [569), (684), (746], (748], [762], (783), [817], [843]. The large deviations principle for Gaussian measures appeared with Schilder's theorem [672] for the Wiener measure. Its extension to general separable Banach spaces is due to Donsker and Varadhan (see [203]). Borell (96] considered the case of a general locally convex space. Ben Arous and Ledoux [54) gave formulations in non-topological
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terms. Our exposition follows [54] (although the space X was assumed to be separable Banach in [54], exactly the same proofs are applicable to general locally convex spaces). Closely related to large deviations, Laplace's method of estimating Gaussian integrals is discussed in detail in [53], [612], [643]. In connection with the material of this section, see also [111], 11121, [264], [402], ]439], [489], 15341, [558], [8361.
To Chapter 5 Differentiability of measures on infinite dimensional spaces was considered first by Pitcher [608] and later in a series of Fomin's papers (see [260] and the surveys (23), [82]. [93]). At present the theory of differentiable measures is a large area of infinite dimensional analysis with interesting applications in stochastic analysis and mathematical physics (see [82], [93], [178]). Useful integration by parts formulas for Gaussian measures were found in [180] -- [182]. More general integration by parts formulas were obtained in (414] (see also (76], [82], (931). The directional derivatives of measures were introduced by Fomin, who considered also the differentiability of Gaussian measures. Directional logarithmic derivatives of measures appeared in 1231. Logarithmic gradients were introduced in [14]. The analyticity of Gaussian measures was established in 1141, 156], (645]. Sobolev classes over infinite dimensional spaces were defined first by N. N. Frolov (see [2681 --- [2711), who studied also their embeddings. Later such classes were considered by many other authors; see [26], 1251], [361], [426] -- (428), (464), [5421 - [5441, [596], [695], (738), [7391, (819], [821]. The principal results about the equivalence of the different definitions of Sobolev classes belong to Meyer and Sugita. A proof of Meyer's equivalence shorter than the original argument of Meyer is due to Pisier [606]. Riesz transforms of Gaussian measures on R" are considered in 1322]. Sobolev classes of mappings with values in certain special Banach spaces are considered in (518] in connection with the study of stochastic flows. Non Gaussian Sobolev classes are discussed, e.g., in [59), [821, [87], 188). A different approach to the Sobolev classes over Gaussian measures is given in the book 15701. Gaussian analysis in terms of the Fock spaces (Xk in our notation) is presented in [3811. Divergences of vector fields and derivatives of measures along vector fields were studied
in [180], [1811, [4261- [428], 1512], [6361, [711], and in many works, connected with the Malliavin calculus (see the surveys [82], 193]. [1771). In the general case (for non Gaussian measures on manifolds) these concepts were introduced in 11791. Relations with extended stochastic integrals are investigated in [283], [567]. Divergence is one of the central objects in the Malliavin calculus discussed in Chapter 6 (see, e.g., [517]). Extension of the logarithmic Sobolev and hypercontractivity to infinite dimensions is straightforward due to their dimensionless character. A more detailed information about hypercontractive semigroups can be found in [3181. In addition to the works already cited, the Ornstein-Uhlenbeck semigroups were discussed in (474], (600], [784]. In our exposition, we used the papers [469], (471]. The development of the Malliavin calculus motivated a considerable interest in Gaussian capacities. Important results are obtained in 1273). [2741. (275]. 12941. 13411. [4041, (448], [513], [740], [7511 (see also [1471). Lemma 5.9.8 was proved in [121. Tightness of Gaussian capacities (giving, in particular, their topological invariance) was established in [4481. [740]. [12] for Banach spaces and in 12521, (891, [90) for locally convex spaces. The most general result presented in the text is due to (90]. The question about the topological invariance of Gaussian capacities was raised by Ito and Malliavin. More general Gaussian capacities associated with the operator semigroups (71e)e>o are discussed in [82). (4041,
[412]. Many classical results which are concerned with "almost everywhere properties" can be obtained in a sharper form as "quasi-everywhere properties" (see (3971. [515]). Large deviations estimates in terms of capacities can be found in [102], [8331. The exponential integrability of Sobolev functions, the logarithmic Sobolev inequality. and the Poincard inequality are studied in 19], 110], [47]. [2721, where one can find also non-Gaussian generalizations.
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A discussion of nonlinear Wiener functionals, which are determined by their restrictions to the Cameron-Martin space, can be found in [1521, [575], [741], (742), [743], [838]. Various problems connected with functions from Sobolev classes and approximations by such functions are discussed in (75], (2071, [277], (300], (601]. In [571], [573], [785], [786], one is concerned with the characterization of independence of random variables from W-(-y) by means of their derivatives. In particular, [786, Theorem 6.4] gives the converse to Problem 5.12.40 (Craig's theorem; see [413, 15.13]). Distributions on spaces with Gaussian measures, in particular, Sobolev classes with negative orders of differentiability are considered in [339], (446], [576], (819], (821]. Concerning holomorphic functions on abstract Wiener spaces, see [235], [236], [517], (519], [743], [744]. Measurable polynomials (in the form of multiple stochastic integrals or in an abstract form) appeared in the works of Wiener [827], Cameron and Martin [143], Ito [366], Segal [676], and were further studied in [810], [715[1 [673]. and many other works; see, e.g., [99]. [178], [187], [201], (207], (289], [337], (351], (352], [353], (382], [399], [460], [466], [511], (531], [534], [572], [710], [728]. Abstract measurable polylinear mappings were introduced by Smolyanov [715], where, in particular, their relation to measurable polynomials was investigated. Various problems related to multiple stochastic integrals (limit theorems, estimates of distributions) are investigated in [199], [200], [305], [509], [584], [763], (764]. Second order measurable polynomials and double stochastic integrals were considered by many authors; see, e.g., [336], (527], (604], [619], [620], [649], [651], [681], [696]. [807], [809], [810]. For quadratic forms, Proposition 5.10.9 was obtained
in [2]. Proposition 5.10.10 extends the results from [329] and [650]. Part of the proof of Proposition 5.10.7 and Corollary 5.12.15 are due to Borell [99]. Closely related results are found in [103], (460], [466]. Zero-one laws for polynomials presented in Section 5.10 generalize earlier results obtained in [2], [329], [650]. An extension of the zero-one law
to certain quasi-analytic functions is given in [449, Theorem 6.11; note that Proposition 5.10.10 for real functions follows from the theorem cited, but can be extended to quasi-analytic mappings in the spirit of [449]. The Wick products of Gaussian variables and polynomials in Gaussian random functions are discussed in [201], [381], [520], (703).
It was suggested in [315] and [383] to take for universally null those sets that are zero with respect to all homotetic images of a fixed Gaussian measure. Clearly, the corresponding class of sets is larger than that of all Gaussian null sets.
To Chapter 6 The principal results presented in this chapter concerning linear transformations of Gaussian measures have been known for a long time (sometimes in a slightly less general form, but with very similar proofs); see, e.g., Segal's work [677], and later works [312], [563], [566], [658], [679], [680], [696], [810], [812], [828]. Related problems were discussed in detail in the books [696], [563], [658], [320]. Linear transformations of Gaussian measures, in particular, of the Wiener measure, were investigated also in [28], (589], [806]. Another aspect of linear transformations is considered in [702]. There is a lot of works devoted to the equivalence conditions for the measures generated by Gaussian processes and fields from various special classes (clearly, the straightforward application of the general theorem may face insuperable difficulties of computational character). For a related discussion, see (16], [123], [153], [167], (289], [298), (299], (337], (338]1 [342], [359], (393], [394], [401], [548], (658], [661], (719], [802) -- [804], [808], [829], [831], and the references therein. The description of Gaussian measures equivalent to the Wiener measure (see Example 6.5.1) is due to Shepp (687] (see also (2981) who gave a different proof (a partial result was obtained in [806]). The result in Example 6.5.2 is also borrowed from [687].
Bibliographical Comments
388
The results presented in Section 6.6 go back to the pioneering works [137], [139], [145], [135], where shifts, linear transformations, and then more general nonlinear transformations Tw = w + F(w) were studied consecutively in the case of the classical Wiener
space. One of the basic assumptions in this circle of problems is that F takes values in the Cameron-Martin space. Further progress was due to Maruyama, Girsanov, and Skorohod. The main result in Maruyama's paper [530] is the proof of the existence of the transition density of any nondegenerate one dimensional diffusion. In order to get that statement, it was proved in [530] that the distribution of a one dimensional diffusion with the unit diffusion coefficient and drift f is equivalent to the Wiener measure (under mild integrability conditions on f, e.g., for bounded f) and its Radon-Nikodym density is the function A given by the equality A(w.) = exp (f. f (w,) dw, - s fa f(w, )z ds) . However, for quite a long time Maruyama's paper was unknown to most of the researchers in the field. The aforementioned absolute continuity result was obtained independently in [630]. It was proved By Girsanov [290] and Skorohod [706] that the distributions of two multidimensional diffusion processes (note that the multidimensional case is more complicated and cannot be handled by Maruyama's arguments) with one and the same diffusion coefficient A are equivalent under very broad assumptions on drift f (in par-
ticular, if A = 1, they are equivalent to the Wiener measure). In addition, Girsanov's paper contains a more general result which in the case A = I states that the process t
l;e = w: - f f (s, w) ds is Wiener with respect to the measure Q = A A. P provided f (t, w) 0
is an adapted square-integrable process and lEA = 1. As shown in [290], the condition lEA = 1 is satisfied in many important cases. In the framework of abstract Wiener spaces (or Hilbert spaces with Gaussian measures), related results were obtained in [312], [32), [660], [708], [444], [712], and other papers. In this setting, the progressive measurability condition is replaced by certain smoothness of F. Ramer's result [636) became a decisive step in this direction. Further improvements are due to Kusuoka [447], whose method was used in our exposition. Theorem 6.6.7, which generalizes Kusuoka's result, is obtained by Ustanel and Zakai [789]; their proof (reproduced in the text) is a modification of an argument from [447]. Buckdahn's works (see 11161) gave an impetus to active investigations of Girsanov's type transformations without nonanticipativity conditions. The distributions of the sequences C. + n", where the g"'s are independent standard Gaussian random variables and the N's are some random variables on the same probability space, are investigated in [388]; this is related to the transformations of R" of the form 1 + F, where F; R°° -+ 1 satisfies the nonanticipativity condition of Example 6.7.4 considered in that paper. Interesting additional results can be found in [27], (229], (284], [287], [452], (453], [570], [787] - [791], [794], [839], [841]. Analytic transformations of R" preserving the standard Gaussian measure are considered in [226], [503]. Applications of the results about equivalent transformations of Gaussian measures to quasiinvariant measures and diffusions on infinite dimensional manifolds are described in [44), (208], [228], (415], [562], [685], [729). The Malliavin calculus [512] appeared as a new method of proving the smoothness of the transition probabilities of multidimensional diffusion processes with possibly degen-
erate diffusion coefficients. One of its first impressive applications was the proof of an important special case of the celebrated Hormander theorem about hypoelliptic second order differential operators. It was soon realized that Malliavin's method is a beautiful and efficient tool in the study of nonlinear transformations of measures (not necessarily Gaussian) on infinite dimensional spaces. The Malliavin calculus has a lot of interesting links with functional analysis, stochastic analysis, and the topology of manifolds. The introduction into this calculus presented in this book has only aimed at describing the basic ideas conformably to the Gaussian case. Various interpretations of Malliavin's method can be found in the works of many authors following [512]; see, e.g., [51], (65], [82], [93],
Bibliographical Conwwnts
389
[177]. [189]. [361]. [516]. [517], [570]. [574]. [693]. [727]. [819]. and the extensive bibliography in [82]. [93), [517], [570]. In all such modifications the essence of the method remains invariable, though; they differ rather by the terminology (for example, derivatives along vector fields are called "differential operators". "operators carr6 du champ". etc.) However. the concrete situations to which the general method applies are most diverse. Additional results concerning smoothness of the distributions of various functionals can be found in the works cited and in [77], [281]. [514]. [555], [621]. Applications of the Malliavin calculus to the study of asymptotics are given in [454]. [820]. See [362] for applications to occupation densities. The first general results about the absolute continuity of the distributions of Wiener functionals were obtained by Shigekawa [692], [693] with the aid of methods of the Malliavin calculus. However. in many cases for this purpose it is more efficient to apply the standard facts from the geometric measure theory to the conditional measures on the finite dimensional subspaces. The idea of using conditional measures for the study of nonlinear (typically, infinite dimensional) images of measures was employed systematically in (289] and [710]. For mappings to IR°. the same idea was used later. e.g.. in [74]. [93]. [106]. [107]. [187]. (188]. Certain refinements of this method are discussed in [189]. The absolute continuity of the finite dimensional images of Gaussian measures was investigated also in [449]. [450]. (690]. In particular, Corollary 6.8.6 was proved in [449] and [690] for the expansions in Hermite polynomials: note that [449. Proposition 7.1] extends this result to quasi--analytic functions discussed in [449]. See [537] concerning the absolute continuity of infinite dimensional images. Lemma 6.9.7 was suggested by Uglanov [780]. The construction of the surface measures described in this chapter follows Airault and Malliavin [11]. Different approaches were suggested earlier in [445], [710]. [778] [780]. As a principal difference (cf. [778]) note that in the case of a Banach space. when defining the neighborhood of a surface, instead of the Cameron-Martin space norm one
can use the norm of the space itself (e.g., in the case of a Hilbert space X to use the unit normal vectors with respect to the norm of X). Clearly, this leads to a different. although close. theory. However. as shown in [75]. Malliavin's method is efficient on this way as well. Green's and Stokes formulas are discussed in [721], [301], and in the works cited above. See [253] concerning surface measures and Hausdorff measures. Gaussian spherical measures are investigated in [331]. [332]. [333]. Supports of measures induced by sufficiently regular mappings are discussed in [8]; in particular, it is interesting to know when does the support of the measure 7 of-' coincide
with the closure of F(H(-y)). The connectivity of the support of the measure 1 o F ' for any F E 14 (-,,1Rd) was shown in (233] by the aid of the Malliavin calculus (for the functions from this result was given later in [207]). The boundedness and smoothness of the distribution density of the norm of a Gaussian
vector have been studied by many authors: see [42]. [189]. [436]. [438]. (440). [491], [497]. [592], (780], and the references therein. The case of Hilbert spaces has been investigated especially well: see [355]. [356]. (500). [535]. [588). [640], [807]. [842], where. in particular. some asymptotic expansions can be found. As we have seen, the smoothness of the finite dimensional images of measures can be verified by means of appropriate estimates of their Fourier transforms. This leads naturally to the investigation of the infinite dimensional oscillatory integrals: see [53]. [74]. [77], [82]. [93]. [514]. [517], [519], and the references therein.
To Chapter 7 Many results and constructions of the abstract theory of Gaussian measures originated in the study of the trajectories of Gaussian random processes. Kolmogorov's sufficient condition of continuity of the sample paths (Theorem A.3.22) was first published in Slutsky's work [714]. Important early results in this direction are due to Hunt [354] and Belyaev [48]. (49]. [50]. In the 60s the two principal directions in the study of Gaussian processes were the equivalence problem (commented above) and the continuity and boundedness of
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sample paths. The latter problem was investigated in [131], [192], [239], [279], [376], (424], [523) - (525], [629]. Various geometric characteristics generated by the covariance functions of Gaussian processes (such as the metric entropy) have deep connections with supports of Gaussian measures. An outstanding contribution to this area is due to Dudley [209), [210), Sudakov [731] - [733], Fernique [243], and Talagrand [755], (758), [759], [761]. Various aspects of the sample path theory (including additional references) are presented in the books [5], [6], [359], [471], [472], (481], (491], [611), (829]. Conditions for a Gaussian process to have sample paths of bounded p-variation are found in [379]; analogous problems for the usual variation and decompositions of Gaussian martingales are considered in (378]. Numerous interesting properties of the trajectories of the classical Brownian motion and some related processes and fields are discussed in [40], [104], [115], [369], [390], [406], [481], [560], [639). The papers [115], [171], [357], [358] give some information about Sobolev and Besov classes containing Brownian and other Gaussian
processes sample paths. Properties of the fractional Brownian motion are discussed in (522). Various problems of the theory of Gaussian processes are treated in [216], [610]. For applications of the wavelet decompositions to the study of Gaussian processes, see [55], [817]. A survey of subgaussian processes is given in [377]. Markov properties of Gaussian
random processes and fields and connections with Gibbs distributions are discussed in [202], [286], [346], [552], [613], [647]. [648], [657], [659]. There exists an extensive literature devoted to infinite dimensional Wiener processes and more general diffusions; concerning the problems discussed in this chapter and further references, see [14], [15), [19], (79], [90], [92), [176], [178], [248], [249], [250], [255], [259], [365], [368], [445], (546]. An analogue of the Ornstein-Uhlenbeck process corre-
sponding to the so called Levy Laplacian (defined as a certain limit of n-E"., 0,',) is discussed in [1]; although this process has compact state space, it preserves many Gaussian features. Some additional information about logarithmic gradients of measures is found in [82]. The fact that a probability measure with the logarithmic derivative ,Q(x) = -x is Gaussian (see Proposition 7.3.9) follows from a result in [178] as noticed in (76]; this simple fact
was proved in another way in [647) in the case of the Wiener measure; see also [568], [648], [91) (the latter paper contains a more general result). Expression (7.4.2) for the logarithmic gradient of an H-spherically symmetric measure was derived in [568]; the fact that this relationship implies that u is H-spherically symmetric is due to [82). A characterization of H-spherically symmetric probability measures as the measures that are ergodic with respect to the group of rotations of H is given in [330]. In relation to the symmetry properties of Gaussian measures note that the standard Gaussian productmeasure on 1R" can be represented as a certain limit of normalized surface measures on finite dimensional spheres (see, e.g., [340], 15381).
Our exposition in Section 7.5 follows [78], [91]. Analogous results for nonconstant diffusion coefficients have been obtained in (85]. The differentiability of the transition probabilities of infinite dimensional diffusions is investigated in [52), [176], [282], [445], [554], [602]. Boundary value problems in abstract Wiener spaces are studied in (178], [417), [445], [505]. See (258] and [646] concerning Martin boundaries on abstract Wiener spaces. A much more detailed discussion of the limit theorems for infinite dimensional random elements related to Gaussian measures can be found in [20], [58), (117), (429]. (472), (592),
(669], (800]. A good account of various results connected with normal approximations and asymptotic expansions in finite dimensions is given in [03].
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Index
absconv A, 25, 39, 362 A. 124
E(X. F). 40. 373
a 44, 100
lAf.373
Bb(X ). 347
IE , :378
BAL, 195 B(X), 374 8)(X), 374 B(X)I. 97, 98, 374 conv A, 25, 39. 362 cov (fs, ti). 53
F'. 245 F'"I. 206
CT, 243
H (T, d. 6). 333
C,.,, 248
H(7). 44. 100
C[a, bl, 363 270 Cti (32), 12
H(µ, v). 92 HP-'(0). 12
(({F}). 373
.FC". 207 GP-"(-7. E). 215
GPI (?, E), 237 Hk. 7
HP-'(-)). 217 HP-'(?. E). 217 HIP"( y). 237
Cb (1R"). 12 co, 363
DF, 205 DE F, 206 DE"F, 206
Hi (?, E). 237
Df, 236
H"(?). 217
DH"F, 212, 213 D(L), 371
H x (?. E). 217 N. 367
DP." (Y). 213
?{(H, E), 367 ?{k. 368 ?{k(H. E). 368 NC'(-I, H). 304 'HCt.uw (?. H). 304 h. 60. 100
217
HP.x(?. E). 217
DP -I(?,E), 213 DP."(?, E). 213
D"(7), 213 D'° (?, E), 213 det 2, 289
dH, 223 dhµ, 207
/.A. 371
!k, 8 Ker A. 365
d"µ, 207 dh l ... dh" µ. 207
K:(X ). 365 ? (X, }").:365 L. 11 LP(µ, X). 378
EA, 364 E(X), 39. 373
C(X)I 40 427
428
Index
L(X), 365 L(X,Y), 365
tr(E, F), 362 o(X,X'), 362
LOO), 369 1-,363
o(f), 45, 100
M,, 371
WY,, 229
PW554 (Pt)e>o, 270
(h,k)H(,), 60
Pd(-y), 255
II-IIp,371
Pd(y,Y), 255
6,2 IILIixa, 13 II
-
Ilp.r, 217
PA , 364
213 215
p(.,a,o2), 1 RKHS, 44
llfllxp-r, 217
Rw, 56
llfJJwp.', 211 llµ - t.II, 371 JAI, 366 Ihltr(7), 44
R,, 44, 100 RT, 361 R°°, 361 span A, 362 S(L, e), 157 Sd('Y), 168
S(R'), 12
V H , 236 Oh F, 206
8vf, 238 00
® µn, 372
(Tt)t>o, 9, 78, 215
n=1
UN , 44, 100
lim inf f (s), 68
ug, 247
lim sup f(8), 68
V,(°)f, 216 Vrf, 215 W(',' 1 [0, 11, 56
WP.r(Q), 12
WP"(y), 211 WP,'(-y, E), 211
Wp" (7n), 13 Wp'r(7n, E), 16 W?c (Rn), 12 WP'(-y), 237 W)oe (y, E), 237
W°°(1), 212 W°°(y,E), 212 X', 361
X',361 X7, 44, 100
Xk, 8, 78 Xk(E), 9, 79 py, 340 Ph , 207 pu , 238 yh , 100
AH, 270, 348 6v, 238 Ap, 301 AK, 290 µ', 3, 43, 376 µ s v, 376, 377
µt, 378 µh, 40
µo f-1, 40, 372
µw,371 µv,371 v, 371 µIA, 371 A
s-t
absolutely convex hull, 25, 362 set, 25, 362 abstract Wiener space, 136 affine
function, 79 measurable, 80 mapping, 42, 361 subspace, 66 Anderson inequality, 28, 77, 165 automorphism measurable, 284 measurable linear, 284 Baire measure, 374 Banach-Sales property, 220 Ben Arous-Ledoux class, 195 Bochner integral, 378 Borel function, 374 mapping, 374 measure, 374 Brownian bridge, 58 motion, 54 path, 335, 336 sheet, 58 Brunn-Minkowski inequality, 27
Cameron-Martin formula, 61 classical, 85 Cameron-Martin space, 44, 100 capacity, 243 Carleman inequality, 289
Index
centered Gaussian measure, 1, 42 central limit theorem, 3, 356 change of variables formula, 372 chaos decomposition. 78 Chebyshev-Hermite polynomial, 7 Chentsov-Wiener field, 58 closed absolutely convex hull, 362 compact linear operator, 365 compactness in Sobolev classes, 267 complete space, 363 completely regular space, 363 completion of a locally convex space, 364 concave function. 171 conditional expectation, 114, 220, 373 of Gaussian vector, 140 conditional measure, 140, 326 continuous polynomial, 250 convex
function, 171 hull, 25, 362 set, 25, 362 topological support, 166 convolution of Gaussian measures, 44 of measures, 376
of Radon Gaussian measures, 98 of Radon measures, 377 correlation inequality. 177 covariance, 44
429
of a norm, 327 of a polynomial, 319, 330 of a process, 378 of a quadratic form, 321, 330 of a second order polynomial, 328 divergence, 238 Doob inequality. 373 theorem, 373 dual, 361 algebraic, 361 topological, 361 Dudley integral. 334
Ehrhard's inequality. 162 ellipsoid of concentration, 5 entropy. 23 equilibrium potential, 247 equimeasurable rearrangement, 198 exceptional set. 269 expectation, 378
extended stochastic integral, 242 extension Lipschitzian, 265 measurable linear, 125
function, 53
Fernique theorem, 74 Feynman integral, 94 Feynman-Kac formula, 356 Fisher's information, 36
operator, 4, 44
flow generated by a vector field, 324
factorization, 108
cylindrical measure, 136 set. 39
degree of a polynomial, 250 derivative generalized, 215 generalized partial, 214 logarithmic, 207 Sobolev, 12, 212 stochastic. 212 vector logarithmic, 340 determinant Ftedholm-Carleman, 288 differentiability along subspace, 205 Fomin, 207 Fr4chet, 205 Gateaux, 205 Hadamard, 205 stochastic Gateaux, 213 differentiable mapping, 205 measure, 207 diffusion process, 86, 346 distribution of a nonlinear functional, 327
formula
Cameron-Martin, 61, 85 change of variables, 372 Feynman-Kac. 356 integration by parts, 207 Ito, 86 Mehler, 9 Stokes, 323 Fourier transform, 3, 43, 376 Fourier-Wiener transform, 94 fractional Brownian movement, 57 Ornstein-Uhlenbeck process, 58 Frechet space. 362 Fredholm-Carleman determinant, 288 function H-Lipschitzian, 174, 223, 261 p-measurable, 371 affine, 79 Borel, 374 concave, 171 convex, 171 gauge. 364 log-concave, 27 measurable convex, 171 nonnegative definite, 53 Onsager-Machlup, 182 quasicontinuous, 244
Index
430
smooth cylindrical, 207 functional measurable linear, 80 proper linear, 80 gauge function, 364 Gaussian k-symmetrization, 157 capacity, 243 conditional measure, 326 diffusion, 331 measure, 3, 42 measure on LP, 58, 148 measure on IRT, 52, 150 null set, 269 orthogonal measure, 88
Hellinger, 92
Pettis, 377 stochastic iterated. 89 Ito, 85 multiple, 89 Paley-Wiener-Zygmund, 83 integration by parts formula, 207 invariant measure, 347 isometric operator, 366 isoperimetric inequality, 167 iterated stochastic integral, 89 Ito formula, 86 stochastic integral, 85 Ito-Nisio theorem, 68
process, 52
Radon measure, 97 random variable, I vector, 42 Gaussian measure Radon-Nikodym density, 291 generalized derivative, 215 partial derivative, 214 generator, 371 Girsanov's theorem, 309 Gross measurable seminorm, 137 Hamel basis, 361 Hellinger integral, 92 Hermits polynomial, 7 Hilbert transform, 231
Hilbert-Schmidt mapping, 367 operator, 367 theorem, 366 homogeneous polynomial, 249
Langevin equation, 87 Laplacian AN, 270 large deviations, 196 large numbers law, 72 Lebesgue completion, 40 Levy theorem, 335 Lipschitzian extension, 265 locally convex space, 361 nuclear, 155 log-concave function, 27 measure, 28 log-concavity, 27 log-log law, 336, 358 logarithmic derivative, 207 derivative along a field, 238 gradient, 340 Sobolev inequality, 16, 226 Lusin's condition (N), 281, 330 Lusin's property (N), 281
hypercontractivity, 17, 227
image of measure, 40 Inequality Anderson, 28, 77, 165 Blachman-Stam, 36 Brunn-Minkowski, 27 Carleman, 289 correlation, 177 Doob, 373 Ehrhard, 162 generalized Poincar6, 36, 266 isoperimetric, 167 logarithmic Sobolev, 16, 226 Poincar6, 18, 226, 228
8idak,178 Slepian, 353 integral Bochner, 378 Dudley, 334 Feynman, 94
Mackey topology, 362 majorizing measures condition, 334 Malliavin calculus, 316 mapping H-Lipschitzian, 174
k-linear Hilbert-Schmidt, 368 7-measurable polynomial, 255 p,measurable, 372 affine, 42, 361 Borel, 374 differentiable, 205
Hilbert-Schmidt, 367 measurable linear, 122 polynomial, 250 preserving Gaussian measure, 284, 325 proper linear, 122 ray absolutely continuous, 212 sequentially continuous, 363 stochastically differentiable, 212 Markov semigroup, 347
Index Martin's axiom, 90 martingale, 373 mean, 1, 4, 44 mean-square deviation, 1 measurable, 122 affine function, 80 in the sense of Gross, 137 linear automorphism, 284 linear extension, 125 linear functional, 80 linear mapping, 122 linear operator, 122 linear space, 90 polynomial, 250, 274 process, 58 quadratic form, 257 seminorm, 74 measure H-spherically symmetric, 344 r-additive, 143 Baire, 374 Borel, 374
canonical cylindrical Gaussian, 136 conditional, 140, 326 cylindrical, 136 differentiable, 207 differentiable along a field, 238 Gaussian, 1, 42 nondegenerate, 119 Gaussian on LP, 58, 148
Gaussian on B", 3 Gaussian on BT, 52, 150 Gaussian orthogonal, 88 invariant, 347 log-concave. 28 nondegenerate Gaussian, 119 of finite Cp,,.-energy, 271 pre-Gaussian, 357 product, 372 quasiinvariant, 312
Radon, 374 Radon Gaussian, 97 stable, 359 standard Gaussian, 1 surface, 321 tight, 374 Wiener, 54 median, 176 of a convex function, 204 Mehler formula, 9 metric entropy, 333 metrizable space, 362 Meyer equivalence, 231 theorem, 231 Minkowski functional, 364 mixture of Gaussian measures, 344, 359 modification
431
continuous, 335 natural of a Gaussian process, 69 of a mapping, 371 quasicontinuous, 245 separable of a process, 70 modulus of continuity of we, 336 modulus of convexity, 328 multiple stochastic integral, 89 multipliers theorem, 230
natural modification of a Gaussian process, 69 negligible set, 269 net, 363 nondegenerate Gaussian measure, 119 nonlinear equivalent transformation, 305 nonnegative operator, 366 normal distribution. 2 distribution function, 2 nuclear operator, 152, 368 space, 155 Onsager-Machlup function, 182 operator compact, 365 covariance, 44 diagonal, 370 Hilbert-Schmidt, 367 isometric, 366 measurable linear, 122 nonnegative, 366 nuclear, 152, 368 Ornstein-Uhlenbeck, 215 orthogonal, 366 quasinilpotent, 290, 325 self-adjoint, 370 symmetric, 366 trace class, 368 Ornstein-Uhlenbeck operator, 12, 215 process, 87, 346, 347 fractional, 58 stationary, 57 semigroup, 9, 78, 215 orthogonal operator, 366 oscillation, 67 oscillation constant, 93
partition of unity, 225 Pettis integral, 377 Poincar4 inequality, 18, 226, 228 polar decomposition, 366 polarization identity, 250 polynomial, 249 -y-measurable, 250
Chebyshev-Hermite, 7 distribution, 319, 330
432
Hermite, 7 homogeneous, 249 mapping, 250 pre-Gaussian measure, 357 process diffusion, 86, 346 diffusion Gaussian, 331 Gaussian, 52 measurable, 58 Ornstein-Uhlenbeck, 87, 346, 347 random, 378 separable, 67 symmetrizable diffusion, 347 Wiener, 54, 337 product-measure, 92, 372 Gaussian, 52, 99 Radon,377 progressive measurability, 85 proper linear, 80 proper linear version, 122 property (E), 285
quadratic form distribution, 321, 330 measurable, 257 sequentially continuous, 259 quadratic variation, 335 quasi-everywhere, 244 quasicontinuity, 244 quasicontinuous modification, 245 quasiinvariant measure, 312 quasinilpotent, 290 Radon Gaussian measure, 97 measure, 97, 374 Radon-Nikodym density of Gaussian measure, 291 property, 209 random process, 378 variable Gaussian, 1 vector, 378
reproducing kernel Hilbert space, 44 restriction of a measure, 371 second quantization, 153 self-adjoint operator, 370 semigroup Markov, 347 Ornatein-Uhlenbeck, 9, 78 strongly continuous, 371 seminorm, 361 Gross measurable, 137 measurable, 74 separable modification of a process, 70 process, 67
Index
separant, 67 sequentially complete apace, 363 continuous mapping, 363 set absolutely convex, 25, 362 bounded,362 convex, 25, 362 cylindrical, 39 exceptional, 269 Gaussian null, 269 negligible, 269 Souslin, 375 symmetric, 25, 362 universally zero, 269 shift of measure, 40 $idak's inequality, 178 signed measure, 371 Skorohod theorem, 130 Slepian inequality, 353 Sobolev class, 12, 211, 214 derivative, 12, 212
norm, 13, 211 space, 13, 211
Sobolev embedding theorem, 13 Souslin set, 375 space, 375 space
abstract Wiener, 136 complete, 363 completely regular, 363 continuously embedded, 362 Fr4chet, 362 locally convex, 361 metrizable, 362
of cotype 2, 152, 358 of type 2, 152, 358 sequentially complete, 363 Sobolev, 13, 211 Souslin, 375
with the Radon-Nikodym property, 209 spherically symmetric measure, 344 stable measure, 359 random vector, 359 standard distribution function, 2 Gaussian measure, 1, 4 stochastic derivative, 212 stochastic differential equation, 86 stochastic integral, 83, 88 extended, 242
iterated, 89 itb's, 85 multiple, 89 Paley-Wiener-Zygmund, 83
433
Index
strong second moment, 357 submartingale, 373 support
uniform integrability, 372 uniformly tight family, 130 universally zero set, 269
Banach, 121 connected, 313
Hilbert. 121 of measure, 375 topological, 119, 166, 375 convex,166
surface measure, 321 symmetric Gaussian measure, 1, 42 operator. 366 set, 362
symmetrizable diffusion process, 347
tensor product, 154 theorem Alexandrov. 129 Bernstein, 30 central limit, 356 closed graph. 365 Cramer, 32 Darmois-Skitovich, 33 Doob, 373 Fernique, 74 Girsanov, 309 Gnedenko, 30
Hilbert-Schmidt, 366 lto-Nisio, 68 Kakutani. 92 Kolmogorov, 378 Lebesgue-Vitali, 372 Levy, 335 Mackey, 362 Meyer. 231
multipliers, 230 normal correlation, 7, 140
Polya, 32 Prohorov, 130 Rademacher, 261 Rademacher-Ellis, 330 Seidenberg-Tarski, 317 Skorohod, 130 Sobolev embedding, 13 Tsirelson, 109 tight measure. 97, 374 tightness of capacity, 247 topological support convex, 166
of a Gaussian measure, 119 of a measure, 375 trace, 369
trace class operator, 368 transformation linear equivalent, 286 nonlinear equivalent. 305 Tsirelson theorem, 109
variance, 1
vector
Gaussian, 42 random, 378 stable, 359 vector logarithmic derivative, 340 version
C.-quasicontinuous, 249 of a mapping, 371 proper linear. 122 weak
compactness of measures, 129 convergence of measures, 129 second moment. 357 sequential compactness, 129 Wiener chaos, 78
field. 58 measure, 54 process, 54, 337
infinite dimensional, 337 modulus of continuity, 336 zero-one law, 64 for polynomials, 256
Selected Titles in This Series (Continued from the front of this publication)
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20 Kenneth R. Goodearl, Partially ordered abelian groups with interpolation. 1986 19 Gregory V. Chudnovsky, Contributions to the theory of transcendental numbers. 1984 18 Frank B. Knight, Essentials of Brownian motion and diffusion. 1981 17 Le Baron O. Ferguson, Approximation by polynomials with integral coefficients. 1980 16 O. Timothy O'Meara, Symplectic groups. 1978 15 J. Diestel and J. J. Uhl, Jr., Vector measures. 1977 14 V. Guillemin and S. Sternberg, Geometric asymptotics. 1977 13 C. Pearcy, Editor, Topics in operator theory. 1974 12 J. R. Isbell, Uniform spaces. 1964 11 J. Cronin, Fixed points and topological degree in nonlinear analysis. 1964 10 R. Ayoub. An introduction to the analytic theory of numbers. 1963 9 Arthur Said, Linear approximation. 1963 8 J. Lehner, Discontinuous groups and automorphic functions. 1964 7.2 A. H. Clifford and G. B. Preston, The algebraic theory of semigroupe. Volume II. 1961 7.1 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Volume 1. 1961 6 C. C. Chevalley, Introduction to the theory of algebraic functions of one variable. 1951 5 S. Bergman, The kernel function and conformal mapping. 1950 4 O. F. G. Schilling, The theory of valuations. 1950 3 M. Marden, Geometry of polynomials. 1949 2 N. Jacobson, The theory of rings. 1943 1 J. A. Shohat and J. D. Tamarkin, The problem of moments. 1943
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