Lecture Notes in Mathematics Editors: A. Dold, Heidelberg E Takens, Groningen B. Teissier, Paris
1720
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
K. D. Elworthy Y. Le Jan Xue-Mei Li
On the Geometry of Diffusion Operators and Stochastic Flows
Springer
Authors K. David Elworthy Mathematics Institute University of Warwick Coventry CV4 7AL, United Kingdom E-mail: kde @ maths.warwick.ac.uk
Xue-Mei Li Department of Mathematics University of Connecticut 196 Auditorium Road Storrs, CT 06269, USA E-mail:
[email protected]
Yves Le Jan Ddpartement de Mathdmatique Universit6 Paris Sud 91405 Orsay, France E-mail: Yves.LeJan @math.u-psud.fr Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Elworthy, David: On the geometry of diffusion operators and stochastic flows / D. Elworthy ; Y. Le Jan ; X.-M. Li. - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer, 1999 (Lecture notes in mathematics ; 1720) ISBN 3-540-66708-3 Mathematics Subject Classification (1991 ): 58G32, 53B05, 60H 10, 60H07, 58B20, 58G30, 53C05, 53C21, 93E15 ISSN 0075- 8434 ISBN 3-540-66708-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission tbr use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1999 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author Printed on acid-free paper SPIN: 10700319 41/3143du-543210
Contents Introduction
3
C o n s t r u c t i o n of connections 1.1 1.2 1.3 1.4
C o n s t r u c t i o n of c o n n e c t i o n s . . . . . . . . . . . . . . . . . . . . . Basic Classes of E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . A d j o i n t connections, t o r s i o n skew s y m m e t r y , b a s i c f o r m u l a e . . . E x a m p l e : H o m o g e n e o u s spaces c o n t i n u e d . . . . . . . . . . . . .
The 2.1 2.2 2.3 2.4 2.5
infinitesimal generators and associated operators T h e irrelevance of drift in d i m e n s i o n g r e a t e r t h a n 1 . . . . . . . . Torsion Skew S y m m e t r y . . . . . . . . . . . . . . . . . . . . . . . The 'divergence operator' J ..................... H S r m a n d e r form g e n e r a t o r s on differential forms . . . . . . . . . O n t h e infinitesimal g e n e r a t o r . . . . . . . . . . . . . . . . . . . . 2.5.1 E x a m p l e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 S y m m e t r i c i t y of t h e g e n e r a t o r .,4 q . . . . . . . . . . . . . .
D e c o m p o s i t i o n of noise and f i l t e r i n g 3.1 3.2 3.3
5
7 7 14 18 26 30 30 35 37 42 47 47 49 57
A c a n o n i c a l d e c o m p o s i t i o n of t h e noise d r i v i n g a s t o c h a s t i c differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . C a n o n i c a l d e c o m p o s i t i o n of t h e G a u s s i a n field W t . . . . . . . . F i l t e r i n g o u t r e d u n d a n t noise . . . . . . . . . . . . . . . . . . . . 3.3.1 W h e n A does n o t b e l o n g to t h e i m a g e of X . . . . . . . . 3.3.2 T h e inverse d e r i v a t i v e flow . . . . . . . . . . . . . . . . . 3.3.3 I n t e g r a b i l i t y of c e r t a i n C r n o r m s for c o m p a c t M . . . . . 3.3.4 T h e s e m i g r o u p on forms: B o c h n e r t y p e vanishing t h e o r e m s 3.3.5 B i s m u t f o r m u l a e . . . . . . . . . . . . . . . . . . . . . . .
57 60 63 69 72 72 73 75
Application: Analysis on spaces of paths
76
4.1 4.2 4.3
78 82 83
Integration by parts and Clark-Ocone formulae .......... Logarithmic Sobolev Inequality ................... A n a l y s i s on C i d ( D i f f M ) . . . . . . . . . . . . . . . . . . . .
Stability of stochastic dynamical s y s t e m s
. . .
87
6
Appendices A B C D
U n i v e r s a l C o n n e c t i o n s as L - W c o n n e c t i o n . . . . . . . . . . . . . C r e a t i o n a n d A n n i h i l a t i o n o p e r a t o r s ( n o t a t i o n for s e c t i o n 2.4) . . Basic formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . L i s t of n o t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95 95 101 103 109
Introduction The concepts of second order semi-elliptic operator, Markov semi-group, diffusion process, diffusion measures on path spaces essentially give different pictures of the same fundamental objects, with related Riemannian or sub-Riemannian geometry. Here we consider a different layer of structure centred around the concepts of sums of squares of vector fields, stochastic differential equations, stochastic flows and Gaussian vector fields; again essentially equivalent, and this time with associated metric linear connections on tangent bundles and subbundles of tangent bundles. The difference between these two levels of structure can be seen from the fact that if a semi-elliptic differential operator on functions on a manifold M is given a representation as a sum of square of vector fields ("HSrmander form") it automatically gets an extension to an operator on differential forms. In exactly the same way representing a diffusion process as the one point motion of a stochastic flow determines a semi-group acting on differential forms (by pulling the form back by the flow and taking expectation.) Given a regularity condition there is an associated linear connection and adjoint 'semiconnection' in terms of which these operators can be simply described (e.g. by a Weitzenbock formula) as can m a n y other important quantities (e.g. existence of m o m e n t exponents for stochastic flows). Moreover in the stochastic picture the connections remain relevant in the collapse from this level to the simpler one giving new results and new proofs of results e.g. on p a t h space measures. In more detail: Chapter i is connected with the construction of linear connections of vector bundles as push forwards of connections on trivial bundles. This is a direct analogue of the classical and elementary construction of the covariant derivative of a vector field on a submanifold of Euclidean space, leading to the Levi-Civita connections (Example 1B). Narasimhan & R a m a d a n ' s theorem of universal connections is evoked to assure us that all metric connections can be obtained this way (Theorem 1.1.2). We then go on to consider the various forms in which this construction will appear in situations described above. (E.g. how certain Gaussian fields of sections determine a connection.) Homogeneous spaces give a good class of examples described in some detail in w B. The notion of adjoint connection or semi-connection on a subbundle E of the tangent bundle T M to our underlying manifold M is described in w A semi-connection allows us to differentiate vector fields on M in E-directions. They play an important role in the theory. One difficulty is that the adjoint of a metric connection may
4
Introduction
not be metric for any metric (Corollary 1.3.7). In general HSrmander type hypoellipticity conditions on our generator A, or equivalently on E, play little role in this article. However in Theorem 1.3.9 we show how they are related to the behaviour of parallel translations with respect to associated semi-connections. In chapter 2 we concentrate on a generator ,4 given in HSrmander form, and its associated stochastic differential equation (s.d.e.). A first result is T h e o r e m 2.1.1 which shows in particular t h a t (for d i m M > 1) any elliptic diffusion operator can be written as a sum of squares with no first order term, or equivalently t h a t any elliptic diffusion is given by a Stratonovich equation with no drift term. The extension A q of A to q-forms is shown to have the form A q = - ( d ~ § 5d) for a certain operator ~ from q-forms to q - 1 forms (Proposition 2.3.1) and also a Weitzenbock form .A q -- 89 2 - 89 q (if there is no drift t e r m A) (Theorem 2.4.2). Driver's notion of torsion skew s y m m e t r y is investigated in w in order to discuss the operators ~, and when they are L 2 adjoints of the exterior derivative d, in w Later, w the semigroups associated to these operators are used to obtain BSchner type vanishing theorems under positivity conditions on
R q.
The question of the symmetricity of ~4q with respect to some measure on M is discussed in w Theorem 2.5.1 gives a fairly definitive result for J[q with the zero order terms removed. However conditions under which R q is symmetric seem not so easy to find if q > 1. For q = 1 this reduces to symmetricity of the Ricci curvature R i c which is shown in Proposition C.6 of the Appendix to hold in the torsion skew symmetric case if and only if the torsion tensor T determines a coclosed differential 3-form, c.f. [Dri92]. These sections are not used later in this article. The main applications in stochastic analysis start with Chapter 3. The basic idea is t h a t the diffusion coefficient of an s.d.e often has a kernel: so t h a t there is "redundant noise" from the point of view of the one point motion. We extend the results from the gradient case in [EY93] to our more general, possibly degenerate, situation giving a canonical decomposition of the noise into its redundant and non-redundant parts. We then show how this can be used to filter out the redundant noise in general situations. (This filtering out corresponds to the collapse in levels of structure mentioned above.) On the way we have to discuss conditional expectations of vector fields along the sample paths of our process, Definition 3.3.2. All this is done in some generality, e.g. allowing for the possibility of explosion. The main application is to the derivative process T~t of a stochastic flow: Theorem 3.3.7 and Theorem 3.3.8. When the redundant noise is filtered out the process becomes a "damped' or Dohrn-Guerra type parallel translation using the associated semi-connection. This procedure works equally for the derivative of the It5 m a p w ~-~ ~t(Xo)(W) in the sense of Malliavin Calculus from which follow integration by parts theorems for possibly degenerate diffusion measures, Theorem 4.1.1. For gradient systems, using [EY93], this method was used by [EL96] and was suggested by [AE95]. The Levi-Civita connection
appears in that case (which is why gradient systems behave so nicely), but in the degenerate case which is allowed here the connections are on arbitrary subbundles of T M and there is no unique particularly well behaved connection to use. Hypoellipticity is not assumed. The "admissible" vector fields are those which satisfy a natural "horizontality" condition, w B and w C. Closely related is a Clark-Ocone formula (Theorem 4.1.2) expressing suitably smooth functions on path space as stochastic integrals with respect to the predictable projection of their gradient. From this we use the method given in [CHL97] to obtain a Logarithmic Sobolev inequality for our diffusion measures, Theorem 4.2.1. Our "damping" of the parallel translation means that no curvature constants appear: indeed since in general we have no Riemannian metric given on M it would be unnatural to have such constants. Logarithmic Sobolev inequalities automatically imply spectral gap inequalities and the constancy of functionals with vanishing gradient (or equivalently whose derivatives vanish on admissible vector fields), Corollary 4.1.3: a non-trivial result even for Frechet smooth functions on path space for the case of degenerate diffusions. In Theorem 4.1.1 the corresponding results are proved for the measures on paths on the diffeomorphism group DiffM of M coming from stochastic flows, or equivalently from Wiener processes on DiffM [Bax84]. Chapter 5 is concerned with applications to stability properties of stochastic flows. In particular upper and lower bounds for moment exponents are obtained in terms of the Weitzenbock curvatures of the associated connection and a generalization of the second fundamental form to our situations: Theorem 5.0.5. This gives a criterion for moment stability in terms of 'stochastic positivity' of a certain expression in the quantities with consequent topological implications: Corollary 5.0.6. A weakness of these results is that we usually require the adjoint semiconnection to be metric for some metric. Theorem 5.0.7 shows that the lack of this condition really is reflected in the behaviour of the flow. Chapter 6 consists of technical appendices. The first gives a detailed description of how the push forward construction of connections we use relates to Narasimhan & Ramanan's pull back of the universal connections. This is needed in the proof of Theorem 1.1.2. The other appendices give the notation of annihilation and creation operators used in the discussion of the Weitzenbock curvatures in section 2.4 and some basic formulae and curvature calculations for connections given in the L-W form. The connection determined by a non-degenerate stochastic flow first appeared in [LJW84]: for this reason we have called it the LeJan-Watanabe or L-W connection. It was also discovered in the context of quantum flows in [AA96] and for sums of squares of vector fields in [PVB96]. It is used for analysis on loop spaces in laid96]. For the non-degenerate case many of the results given here were described in [ELJL97] with announcements for degenerate situations in [ELJL96].
6
Introduction
They were stimulated by [EY93]. The Chentsov-Amari a-connections in statistics are rather different. They are in general non-metric if a ~ 0 and torsion free, see [Ama85], pp42, 46. Acknowledgement: The authors would like to thank MSRI, Institut Henri Poincar~ and Universitaet Bochum (as well as their home institutes) for hospitality during the completion of this project. Support is acknowledged from EU grant E R B F MRX CT 960075 A, NSF grant DMS-9626142, DMS-9803574, Alexander von Humboldt Stiftung, and the British Council. K. D. ELWORTHY, MATHEMATICS INSTITUTE, UNIVERSITY OF WARWICK, COVENTRY CV4 7AL, UK Y. LE JAN, DI~PARTMENT DE MATHI~MATIQUE, UNIVERSITt~ PARIS SUD, 91405 ORSAY, FRANCE XUE-MEI LI, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CONNECTICUT, 196 AUDITORIUM ROAD, STORRS, CT 06269 USA. e-mail address: xmli~math.uconn.edu
Chapter 1
C o n s t r u c t i o n of c o n n e c t i o n s W e c o n s i d e r c o n n e c t i o n s on a C a
vector b u n d l e E over a s m o o t h m a n i f o l d Y M d e t e r m i n e d by a split surjection of vector b u n d l e s H ~ E ~ 0 where H -- M • H is t h e t r i v i a l b u n d l e with fibre a H i l b e r t a b l e space H . T h e c h a r a c t e r i z a t i o n of such a c o n n e c t i o n ~7 is t h a t for each x E M
~Yv ( X ( . ) e ) - 0
all v E T z M a n d e E I m a g e Y ( x ) .
W h e n M a n d E are finite d i m e n s i o n a l a n d E has a R i e m a n n i a n m e t r i c all m e t r i c c o n n e c t i o n on E can be o b t a i n e d this way for some finite d i m e n s i o n a l H . T h e s e c o n n e c t i o n s can also be considered to b e i n d u c e d by G a u s s i a n m e a s u r e s on t h e space of C ~ sections of E . In w and w some b a s i c e x a m p l e s a r e given. T h e y d e s c r i b e t h e c o n n e c t i o n s arising from c e r t a i n G a u s s i a n fields, o p e r a t o r s in H h r m a n d e r form, s t o c h a s t i c differential e q u a t i o n s , a n d h o m o g e n e o u s space structures. For E a s u b b u n d l e of T M t h e r e is also an a d j o i n t ' s e m i - c o n n e c t i o n ' ~7, inv e s t i g a t e d in w In p a r t i c u l a r we show V is m e t r i c w i t h r e s p e c t to s o m e R i e m a n n i a n m e t r i c on M if a n d only if for one set of xo, Yo E M a n d T > 0 t h e p a r a l l e l t r a n s l a t i o n ~It using V a l o n g {~T'Y~ the s o l u t i o n ~t(xo) t o t h e s t o c h a s t i c differential e q u a t i o n dxt = X ( x t ) o dBt c o n d i t i o n e d to satisfy ~T(Xo) = YO, is a b o u n d e d L(TxoM, Tu0M)-valued process.
1.1
C o n s t r u c t i o n of c o n n e c t i o n s
A . C o n s i d e r a C ~ m a n i f o l d M , a C ~ v e c t o r b u n d l e 7r : E --~ M over M a n d a C a v e c t o r b u n d l e h o m o r p h i s m X : H ~ E of a t r i v i a l b u n d l e H = M x H , w h e r e H is a H i l b e r t a b l e space. We will consider only real b u n d l e s ( a n d manifolds) here. At this s t a g e M , E , H could be infinite d i m e n s i o n a l ( b u t s e p a r a b l e , w i t h M m e t r i z a b l e ) ; however o u r m a i n focus will be on cases w i t h M a n d E finite d i m e n s i o n a l . In this s i t u a t i o n we shall w r i t e n = d i m M , p = fibre d i m e n s i o n of E , with m = d i m H if d i m H < c~.
8
Construction of connections
Suppose X is surjective and Y a chosen right inverse to X Y
H--~E
~0.
Our situation is very similar to a special case of that of Harvey and Lawson in [HL93]. Let F(E) denote the space of smooth sections of E, and set E~ = 7 r - l ( x ) , x E M. Write X ( x ) = X(x,-) : H ~ Ex. For u i n E l e t Z u be the section given by ZU(x) = X(x)Y(~r(u))u.
(1.1.1)
P r o p o s i t i o n 1.1.1 There is a unique linear connection (7 on E such that for all uo E Ezo, xo E M the covariant derivative of Z u~ vanishes at Xo. It is the 'push forward' connection defined by ~7voz = X ( x o ) d ( Y ( . ) Z ( . ) ) (Vo),
vo E Txo M, Z E r ( E )
where d refers to the usual derivative d ( Y ( . ) Z ( . ) ) : T M Y ( . ) Z ( . ) : M ~ H.
(1.1.2)
--+ H of the map
P r o o f . Certainly (1.1.2) defines a covariant differentiation. Let V be any linear connection on E. We have Z(.) = X ( . ) Y ( . ) Z ( . ) whence, for v E T, oM, (TvZ
= X ( x o ) d ( Y ( . ) Z ( ' ) ) (v) + (Tv [X(.) ( V ( x 0 ) Z ( x 0 ) ) ] = V v Z + fT~Z z(~~
(1.1.3)
Since V is assumed to be a genuine connection (not just a covariant differentiation: a point only relevant if E is infinite dimensional) and since also the map TMxE (v,u)
~
E
~
~TvZ ~'
~ives a smooth section of the bundle of bilinear maps L ( T M , E ; E) we see that ~7 is a smooth connection on E, (e.g. [Eli67]). Taking ~7 = ~7 in (1.1.3) we see that V has the property required. Uniqueness also follows from (1.1.3). // B . We shall be mainly interested in metric connections. These will arise in two, essentially equivalent, forms which we will call the metric form and the Gaussian form respectively. However the examples coming from homogeneous spaces are more easily understood in the more general non-metric framework and these will also be described below, in w
Construction of connections
9
In the "metric" form H is now a Hilbert space, inner product (,) -= (,)H and so the surjective homomorphism X induces a Riemannian metric {(, )~ : x E M} on E. The right inverse Y is chosen to be the adjoint of X, Y = X*. T h e o r e m 1.1.2 Let H be a Hilbert space, and Y the adjoint of X with respect to the induced metric on E by X . Then the connection f7 is adapted to the Riemannian metric. Moreover if M and E are finite dimensional any metric connection for any Riemannian metric on E can be obtained this way from some such X with H some finite dimensional Hilbert space.
P r o o f . Take a vector field U and a vector v E T~oM. Then
d ( U , U ) (v)
=
2(d(Y(.)U(.)) (v),Y(x0)U(x0))
=
2 ( X ( x o ) ( d (Y(-)U(.)) (v), U(xo)) = 2((TvU, U).
This shows that (7 is metric. The fact that any metric connection arises this way in the finite dimensional situation comes from Narasimhan and R a m a n a n ' s theorem [NR61] on universal connections. In the finite dimensional case the connection V is precisely the pull back of the universal connection over the G r a s s m a n i a n G ( m , p ) of p-planes in H by the map x ~-* [ image of Y ( x ) : Ex --+ H]; for details see w in the Appendix. Narasimhan and R a m a n a n show that any metric connection can be obtained as such a pull back. 9 In this situation we shall call (7 the LeJan-Watanabe or L-W, connection determined by X, or by (X, (,)), for reasons described at the end of w below. E x a m p l e 1B (Gradient systems). Let j : M ~ II~m be an immersion. Define X ( x ) : 1~m -~ T x M to be the orthogonal projection of ll{m on T~M, identified with its image under the differential dj of j, so that X ( x ) e = grad(X(.), e)•,, using the induced metric on M. Then Y ( x ) : T~M --+ ]~m is the inclusion, T j, and we have the classical construction of the Levi-Civita connection for this metric. ( T h a t it has no torsion can also be seen from the formula (2.2.3) below.) C. For the 'Gaussian form' suppose we have a mean zero Gaussian field W of sections of E. In its most general form W would be a section of the pull back p*E of F ( E ) over the projection p : l) x M ~ M where (f~, 5c, P ) is a probability space. Thus W,(co) := W ( w , x ) E E , for each x E M, w E f t . We will assume that W ( w , .) is C a for each co E ft. The more concrete manifestation comes from a Gaussian measure ~/on some subspace of the C ~ sections of E. Then (ft, if, P ) = (F(E), ~', -y), the canonical space, for 5c the a-algebra of cylindrical subsets of F(E), and we identify Wz with the evaluation map Px : F ( E ) --+ Ex, given by p~(U) = U(x). See [Bax76]. For any suitable function f on F(E) we write E f or E l ( W ) for f~ f(W(co, .))P(dw) (equivalently fr(E) f ( U ) d T ( U ) in the canonical picture). Let 7x be the law of W~, a Gaussian measure on E~. We
10
Construction of connections
make the nondegeneracy assumption that each "Yz is non-degenerate and so in the finite dimensional case has the form 7~ ( e ) =
s
e-(U,y) ~_dy
for some (,)x on Ez. P r o p o s i t i o n 1.1.3 There is a unique connection V "y on E such that the random variable V ~ W is independent of W ( x o ) for any v E T~oM, Xo E M . It is given in terms of the conditional expectation by d V~Z = ~ E { W ( x o ) I W ( a ( t ) ) = Z(a(t)) } It=o;
(1.1.4)
Vv~Z = d E W ( x o ) ( W ( a ( t ) ) , Z(a(t)))~(t ) t=o
(1.1.5)
or equivalently
for any C 1 curve a with (:r(O) = v, v E T~oM, and is adapted to the metric { (, )~, x E M }. Moreover (i) Let H.~ be the reproducing kernel Hilbert space of 7, then V "r is the L - W connection for (X, (,)H~) where X ( x , h) = p~ (h) = h(x). (ii) If E is a finite dimensional vector bundle over a finite dimensional M every metric connection can be considered as V ~ , given by (1.1.4), for some Gaussian measure ~/ on F(E) with finite dimensional support. P r o o f . Recall that the reproducing kernel Hilbert space H~ of 3, is the same as the Cameron-Martin space H of 7 and is a Hilbert space, here necessarily consisting of C a functions. Among its standard properties are: (i) The restriction of p~ to H maps onto E~, each x E M and induces the inner product (,)x. It will also be denoted by p~. (ii) The reproducing kernel k, a section of the vector bundle UxeM,yEMn(Ex, Ey) --~ M x M, defined by the reproducing property that k(x, .)(v) belongs to H each v E E~ and for all h E H , (k(x, .)v, h)H, = (h(x), v)~,
(1.1.6)
is also the eovariance of "7: k(x,y)v = E(W(x),v)~W(y),
v E E~, x , y E M.
(1.1.7)
See [Bax76]. From (1.1.7) we see k(x,y)W(x) = E{ W(y)lW(x)} E TyM
(1.1.8)
Construction of connections
11
and so
k(x,.)v -- E { W ( . ) IW(x) = v }
E H C r(E)
(1.1.9)
for all v E Ex, and x , y E M. From this we see that the defining equation (1.1.4) or (1.1.5) for Vv~Z can be written
v~z
=
dk(o(t),~o)Z(o(t))l~=o
(1.1.10)
=
~EW(~o) (W(~(t)), Z(o(t))L(~) ~=0"
(1.1.11)
If we set X ( x ) = p~ so X : H__--~ E we see from (1.1.6) that the adjoint map Y to X, using the induced Riemannian metric is just k:
Y(x) k(x, y)(.)
= p~ = k(x, .) : E~ --+ H = X ( y ) Y ( x ) : E~ ~ E~.
We are therefore in the 'metric' form discussed in Theorem 1.1.2 and V is just the L-W connection ~ of p, (,)Hr" For the characterization in terms of the independence of ( T z W and W observe that for u E E~ o the reproducing vector field Z ~~ or Z ~ as in (1.1.1) in the metric form, is given by
Z~(y)
=
X ( y ) Y ( x o ) u = k(xo,y)u
(1.1.12)
=
E { W ( y ) I W(xo) = u}
(1.1.13)
so that for any linear connection V on E we have
(YvZ ~' = E { ~ v W [ W(xo) = U} . Thus if v E Txo M we have V~Z ~ = 0 for all u if and only if
E ( ~ w, W(~o))~o = o since V . W and W(xo) are Ezo-valued Gaussian random variables (this is exactly the condition for the independence of W(xo) and VvW). By Proposition 1.1.1 this proves uniqueness, i.e. that V . W and W(xo) are independent for all Xo and v E Tzo M implies ~ = V 7. It also shows that E(V~ W, W(xo))zo = 0,
for all v E TxoM, all xo e M,
(1.1.14)
which, again because they are Gaussian vectors, implies that V~W is independent of W ( x o ) for all v E TxoM, xo e M. (The fact that the processes W ( x o )
12
Construction of connections
and ~7~vW(xo) both take values in fibres Exo of a bundle causes no difficulty in using the standard results we used: to reduce to the standard situation where the process takes values in a fixed Hilbert space/4o, say, either observe that we can find a measurable trivialization 0 : E -+ M • H0 some H0, 00 with each 0x : Ex --~ W0 an isometry or simply note that we have given to us:
Y:E--~MxH which isometrically maps each E~ onto a subspace of H: so we can take Ho = H. In this second way we can simply treat F(E) as a subspace of the space of maps of M into H.) Finally, to show that all such metric connections arise this way, let {(,)~ : x E M} be a smooth metric on E, with a metric connection ~7. By Theorem 1.1.2 there is a Euclidean space ll~"~ , 0 and an X : M • II~m --+ E whose L-W connection is ~. Let "Ym be the standard Gaussian measure of ~ m (,), and let "y be the image measure on F(E) o f ? under the map l~TM --+ F(E), e ~-~ X(.)e. We claim V ~ = ~7. Indeed if k is the reproducing kernel for "y then if u E E~
k(x,y)u
f
= /
JR
=
(X(x)e,u)~X(y)(e)d~m(e) m
x(y)Y(x)
for Y(x) = X(x)*. Thus by (1.1.12) the definitions of the vector fields Z ~ defined via X and via ~f agree and so V ~ = ~ by their defining property. | R e m a r k 1C. The proof above shows the essential equivalence between the "metric" and "Gaussian" forms. It also shows that the connection depends only on the law, % of the process (or equivalently on the subspace H of F(E) together with its inner product) not on the process itself. The case of H a Hilbert space of sections is more intrinsic than that of a mapping of a Hilbert space into the space of sections, and often the Gaussian formulation is simpler to use, especially when H is infinite dimensional. However it is often the "metric" form which arises in practice, for example in the gradient systems of example lB. E x a m p l e 1C: Gaussian vector fields on I~'~ are said to be isotropic if they are invariant in law under Euclidean transformations. The covariance of an isotropic Gaussian vector field on l~n is determined by two spectral measures FL and FN on ~+. It is given by the formula
E (Wi(x)WJ(y)) = ciJ(x - y)
(1.1.15)
with
C ij(z) = f JR
f ~- g s n - 1
e ip
Construction of connections
13
a being the uniform distribution on S ~-1. The vector field W(x,w) has an almost sure C ~ version when the measures FL and FN have moments of all orders. From formula (10) in [ELJL97] or remarks in w B below (or rather from its generalization to the Gaussian field case) we see t h a t =
o.
The connection so constructed is therefore trivial. The isotropic stochastic flows, associated with W was studied in [LJ85] and [BH86]. A special case had first been introduced in [Har81], using an approximation by discrete vortices. However, even in that isotropic case, nontrivial connections arise if you consider the motion of several points: In general, the Gaussian field W can be extended into a Gaussian field W (d) on M (d) = { ( X l , X 2 , . . . ,Xd) E Mdlxi • xj, for i # j} as follows. There is a d canonical isomorphism between T(xl,X2,...za)M (d) and the direct sum Oc,=IT,, M. With this identification, we can set W (d) (xl, x 2 , . . . , Xd) = @d=lW(X~). For the isotropic fields on ~n,
W(~)(xl,x2,... ,xd) = e~=ld ~--~ W~(x~) Ox~'O i=1
It is always a nondegenerate Gaussian vector except in dimension one, when the spectral measure is atomic. See e.g. Darling [Dar92]. We can define an associated metric on (II~")(d)
(x, ..... xd) ---- ,
where (, } denotes the Euclidean metric and K -1 is the inverse of the matrix of covariances:
K(~,i),(Z,y) = c i J ( x ~ - x z ) ,
l ~ _ a , ~ _ n , l ~ i , j ~d.
The L-W connection ~(d) is a metric connection on (lt~n)(d) for the metric constructed above. The computations below show it is not the Levi-Civita connection. From the result of w we can see it is related to a solution to a filtering problem: Given the paths of d points, the restriction to the path of one of the points of the d a m p e d adjoint parallel transport along the path of the d-point motion is the restriction of the derivative flow of d-point motion conditioned on the d-point motion. The connection can be computed explicitly. For example when d = 1, by (1.1.4), the Christoffel symbols of ~7(d) a r e ~,~ =
]t--o E{W(x.y)IW(xa + tsar) = 5 ~ for all ,~}.
-y For a r 7, F~,~ clearly vanishes. Moreover 0 r~,~ = ~-~lt=o ([K(t,c~)]-l)~,~ C(x~ - x~ + t ~ )
Construction of connections
14
with ([K(t, a)])~ A, = C (xx - x~, + tS~ - tS~,). Hence, setting K ( 0 ) ~ , = C(x~ - x~, ) 0 and K ' ( 0 , c~) = - ~ K ( t , c~)lt=o F~,Z = ( K ( 0 ) - I K ' ( 0 , a))~,
.
For d = 2 one checks that in particular,
- C ' ( x l - x2) r~2 = 1 - C2(xl - x2) and F~ 1 :
-C(xl
- x2).C'(Xl
- x2)
1 - C2(zl - x2) 1.2
Basic
Classes
of Examples
E x a m p l e A: S u m s o f s q u a r e s o f v e c t o r fields: o p e r a t o r s in H S r m a n d e r f o r m . For a particular manifestation of our basic class of examples consider a second order differential semi-elliptic operator .4 on M given in the HSrmander form 1 m `4= -~ E L x J L x J + LA. (1.2.1) 1
where X 1 , . . . , X m , A are smooth vector fields on M, with L v denoting Lie differentiation in the direction of a vector field V. We obtain X :~_~m - ~ T M
by
X(x)e= J for e l , . . . , e r a the standard orthonormal base for ~m. If we assume Ex := X(x)[ll~m] has constant rank we are in the metric form situation, obtaining a connection V on the resulting subbundle E -- Image X of T M . The geometry of such operators have been examined from a different viewpoint (see e.g. [Str86]), usually with a hypoellipticity assumption. T h e y are usually considered as operators on functions. However the presentation of .4 in the form (1.2.1) shows how to extend it to more general tensors, in particular to differential forms on M. We show below in section 2.4 the relevance of V to the analysis of these operators.
Basic Classes of Examples
15
E x a m p l e B: C o n n e c t i o n s arising in the theory of stochastic flows. Let 79 be the space of C a diffeomorphisms of M with C a topology making it a Polish topological space, (see [Bax84]). Following Baxendale [Bax84] consider a Brownian motion, {~t : t > 0}, on 7), i.e. a stochastic process on D satisfying 1. almost sure continuity in t, 2. independent increments on the left, i.e. ~ t ~ -1 and r162 1 are independent
ifO <_s < t ~_u < v, 3. time homogeneity; 4. ~o=identity.
For each x 9 M there is the process {r : t > 0} on M which was shown to be a diffusion with generator .4, say. Similarly on M • M there is a process {(~t(x), ~t(Y)) : t >_ 0} for each (x, y) 9 M x M giving a diffusion with generator A 2 say. It turns out that for f, g : M ~ l~ both C a with compact support then on f | g : M x M -+ ll~, defined by (x, y) ~-~ f ( x ) g ( y ) ,
,42 ( f | g) (x, y) = `4(f)(x)g(y) + f ( x ) A ( g ) ( y ) + 2 Fr ((df)~, (dg)y) where F r : T * M x T * M ~ I~ is the symmetric bilinear m a p given by
F ~ ((df)~, (dg)y)
limE(f(~h(X)) - f ( x ) ) (g(r h~O h
- g(Y))
(1.2.2)
If we assume that .4 is elliptic, its symbol, which is quadratic and given by F~ ((dr)x, (dr)x), will be non-degenerate and so determines a metric on T M . Raising and lowering indices of F~using this metric gives a section k of the bundle L ( T M ; T M ) over M • M,
(k(x, y)vl, y2)y = F~ ((Vl, -)x, (v2, -)y),
Vl C TxM, v2 9 TyM.
(1.2.3)
Following LeJan and W a t a n a b e [LJW84] we can define a connection ~7~ on
T M by V ~ Z = -~tk(a(t),xo)Z(a(t)) [t=o
(1.2.4)
just as in equation (1.1.10). Indeed it was shown in [BaxS1] (without assuming ellipticity) that Fr is the covariance for a Gaussian measure 7 on F ( T M ) , mean ~, and t h a t there is a correspondence between Brownian motions on 7) and such Gaussian measures, at least for M compact, see also [Kun90]. Given sufficient regularity the correspondence is obtained via the stochastic differential equation
dxt = Px, o dWt + ~(xt)dt
Construction of connections
16
where {Wt : t >_ 0} is the Wiener process on F ( T M ) associated to 7, and p~ the evaluation m a p at x E M, or equivalently for the stochastic differential equation on 7:) for the flow {~t : t _> 0}
d~t = ( T h e , ) o dWt + TTtr (~/)dt where 7r refers to right translation (i.e. composition) by the diffeomorphism h. It was shown in [LJW84] that the generator A is given by 1
A ( f ) ( x ) = ~tr~7~ (df)(x) + df(;/(x)). Z
In particular ~ is determined by F r and ~. Given our non-degeneracy assumptions we see V r = V 7, (of course ellipticity can be replaced by constancy of the rank of the symbol). Remark: Recall that ~ vanishes if and only if ~t is equal in law to ~-1 for any fixed t.
E x a m p l e C: S t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s ( s . d . e . ) . The form in which we will most frequently be using the theory will come from stochastic differential equations
dxt = X ( x t ) o dBt + A(xt)dt
(1.2.5)
where {Bt} is a Rm-valued Brownian motion, A is a smooth vector field and o indicates the integral involved being Stratonovich. For xo in M let {~(x0) : 0 ~ t < p(xo) be a maximal solution from xo, so that p(xo) E (0,~c] is the explosion time from xo. There is the associated semigroup Pt : t > 0: for bounded measurable functions P t f ( x ) = Ef(~t (x))xt
X:M• given by =
d (expte-x)It=o
Basic Classes of Examples
17
so t h a t e ~-~ X e := X(.)e is a Lie algebra h o m o m o r p h i s m . For g E K let Lg denote left multiplication by g acting on K or on M and let R 0 be right multiplication by g acting on K . There is the adjoint action
ad - adK : K --+ GL(~_) given by
ad(g)e = (TRg) -1 TLg(e),
eE~,gEK.
Note t h a t if x E M , g E K and e E ~ then d
X(gx)e-
d
dtexpte.gxit=o=TLg-~g
-1
(expte)g.xit=o
(1.2.6)
=
d TLg-~ e x p t a d (g-1)(e). xlt=o
(1.2.7)
=
TLgX(x)(ad(g-1)e).
(1.2.8)
Suppose further t h a t the action is transitive, and also fixing x0 E M the mapping K ~ M , g ~ g . x 0 identifies M with K / H for H -- {g E K : g - x0 = xo} m a k i n g M a homogeneous space which is also reductive i.e. there is a linear splitting e = 0_ + m, (1.2.9) with 0_n m = 0, where 0_ is the Lie algebra of H and m some linear subspace of which is adH invariant: ad(g) [m] = m,
all g E H.
Note t h a t O_= ker X ( x 0 ) , so by (1.2.8), for g E K , ker X(gxo) = ad (g)[0__].
(1.2.10)
T h e reductive p r o p e r t y allows us to define _mx : = ad (g) [m] C _~for x = gxo since if x = ghxo, so h E H , we have
ad(gh) [m] = ad (g)ad (h)[m] = ad (g)[m]. T h u s we have the splitting
~_= K e r X ( x ) + m__x for each x E M and we can define Y ( x ) : T~M --~ ~_ to be the inverse of the restriction of X ( x ) to _m~. 1.2.1 The connection x7 induced on T M by X , Y is K invariant. If ~_ admits an adK-invariant inner product (,) for which m is orthogonal to [} then x7 is the L- W connection for X, (,). Proposition
Construction of connections
18
This applies to spheres and Grassmanian manifolds in which cases the connections are the Levi-Civita connections. See Theorem 1.4.8 below. Remark:
P r o o f . First observe from (1.2.8) that ad (g)Y(x)TLg-1 : TgxM --+ ~ has image in rag, and is a right inverse to X(gx). Thus
Y(gx) = ad (g)Y(x)TLg-,.
(1.2.11)
From this and (1.2.8), for g 9 K , Z a vector field on M, and v 9 T , M we have
=
X(gx)d ( Y ( . ) T L g Z
(g--1.))
TLg(v)
=
T L g X ( x ) a d (g-1)d ( Y ( . ) T L g Z (g-1.))TLg(v)
=
T L g X ( x ) d ( Y ( g - 1 . ) Z (g-1.)) TLg(v)
=
TLgX(x)d(Y(.)Z(.))(v)
:
Thus V is K-invariant. When (,) is adK-invariant, if m_ is orthogonal to [~ then _mx .1_ ker X ( x ) for each x and we see Y(x) = X(x)*, from which it follows t h a t ~7 is the L-W connection for X, (). | These connections on homogeneous spaces are discussed further in w
1.3
below.
Adjoint connections, torsion skew s y m m e try, basic formulae
A . From now on in this section we shall assume that E is a subbundle of the tangent bundle T M of M. Then for any linear connection V on E there is an operation which gives a differentiation of arbitrary smooth vector fields V in E-directions: define
fT"V = fTV(zo)U + [U, V](x0),
u 9 Ez 0
(1.3.1)
where U is any section of E with U(xo) = u. In terms of the Lie derivative L v , mappings sections U of E into vector fields,
fT'V = fYy - L v ,
(1.3.2)
c.f. the tensor Av defined on p.235 of [KN69a]. It is easy to see that V ' V 9 FHom(E, T M ) , or see [KN69a], p.235, and also that V~ is a derivation on
A d j o i n t connections, torsion s k e w s y m m e t r y , basic formulae
19
F ( M ) over C a ( M ) for each U E F(E). We shall call such an operation a semi-connection on E and call ~7~ the adjoint (semi)connection to V. Adjoint connections were introduced in the case E = T M by Driver in [Dri92]. When E = T M it is a genuine connection: indeed, by definition, in this case for vector fields U, V we have
[U, V]
= V~'u V
- VvU
(1.3.3)
whence
~7~V = ~7uV - T(u, V(xo)).
(1.3.4)
Here 2 : T M x T M ~ T M is the torsion tensor of ~7, defined by - T ( u , v ) = IV, Y](x0) - VuY + ~ U ,
(1.3.5)
where v = V(x0), c.f. Proposition 2.3 of [KN69a]. When E is a genuine subbundle there is still a skew symmetric 2:E|
defined by (1.3.5) with U, V now sections of E, (and so, also vector fields on M). For our semi-connection V~ we can again define 2 ~ : E | E ~ T M by equation (1.3.5). It is immediate from (1.3.3) and (1.3.5) that T' = - 2 . Comin~ back to the specific case of a connection ~7 as in w1.1 note for any v, u E E , V v Z ~ = (7~Z ~ - O. So
2(u, v) = - [ z u, zv].
(1.3.6)
The canonical example of adjoint connections are given on Lie groups G: the adjoint of the fiat right invariant connection on T G is the fiat left invariant connection (and conversely). More examples are given in w below. The name 'semi-connection' is justified by: P r o p o s i t i o n 1.3.1 Let E • be a complementary subbundle to E in T M , so TM = E @ E • Let V • be any linear connection on E • and let ~1 be the direct s u m connection induced on T M : V 1 = ~7 @ V • . Let ~71' be the adjoint connection to V 1. Then r
1 , = (V)uV,
u e E,0, V E F(TM).
P r o o f . This follows from equation (1.3.1) since if U E F(E) and V E F ( T M ) then ~YvU = V ~ U . |
Construction of connections
20
From this proposition we see immediately that it is possible to define operab' tions -~ on smooth vector fields {vt : t E [0, T] } along pieeewise C 1 curves a in M with h(t) E E~(t) for each t ("horizontar curves) such that: if V E F ( T M ) and vt = V ( a ( t ) ) then
blvt Ot - V~(t)V" For example simply take V • as in the proposition to obtain (V1) ' with corresponding -0T DI' and then restrict to horizontal curves observing the result is independent of the choice of E • and V • Alternatively there are partially defined "Christoffel symbols", see Remarks in the next subsection. Similarly there are parallel translations
"//~ : T~(o)M --~ Tz(t)M, which are linear isomorphisms, but only defined along horizontal curves and indeed, as usual,
b'v, =/-/, d -,-, Ot
- ~ / / ~ v~.
A vector field {vt : t E [0, T]} along such a curve is parallel if and only if b'w ot = 0. Proposition 1.3.1 will enable us to be confident in applying the usual rules of connections to V' in consequence. The following extension of (1.3.3) will be of basic importance.
L e m m a 1.3.2 For S > O, T > O, let a : [0, S] x [0, T] --~ M be C 1 with a ( s , . ) horizontal for each s E [0, S]. Then [9' Oa [9 Oa Ot Os - Os Ot
P r o o f . By Proposition 1.3.1, we can assume E = T M . Then
ba~
bao- ~.a,:, ao-)
at as - as at +
(-o-['as "
But, from (1.3.4),
b'a~
bao- ~.ao- ao-)
at as - at as
-
( a t ' as "
!
In g e n e r a l / / t does not m a p fibres of E to fibres of E:
(1.3.7)
Adjoint connections, torsion skew symmetry, basic formulae
21
P r o p o s i t i o n 1.3.3 For the adjoint semi-connection ~Y' the following are equivalent: 1. V~ 1u V E E whenever V E F(E) and u E E. ~
!
2. / / t maps Ec,(o) -'~ Ea(t) whenever a is a horizontal path. 3. T maps E @ E t o E . 4. E is an integrable foliation and (7 restricts to a connection on each of its leaves. I f so V' restricts to a connection on the leaves of E.
P r o o f . Certainly (4) implies (1). Also (1) implies (3) by (1.3.4) and if (3) holds then [U, Z] E F(E) when E, Z E F(E) by (1.3.5) and (1) holds by (1.3.4), so (3) implies (4). To take in (2) observe that (2) implies (3) using (1.3.4) and the I formula ~a t = / / t ' ddt/ I-/ /t ' - i vt, while (4) immediately implies (2). -
R e m a r k : If P : T M --~ T M / E bundle then by (1.3.5)
is the projection of T M onto the quotient
P ( T ( u l , u 2 ) ) = - P ([U1,U2](x))
(1.3.8)
for U 1 , U 2 E F(E) with U l ( x ) = u 1, U2(x) = u 2. Thus P o T ~ i s i n d e p e n d e n t of the connection on E. It is a well known invariant of the subbundle, e.g. see Strichartz [Str86]. Y B. Suppose now we have H_ ~ E ---+ 0 as in w but still with E a subbundle of T M . The adjoint V' of the associated connection ~7 will be denoted by ~7, with parallel t r a n s l a t i o n / / e t c . It takes on a particularly simple form using the sections {Z ~ : u E E} of E defined in (1.1.1): L e m m a 1.3.4
1. For any vector field V and u E Ezo we have ~7uV = L z ~ V ,
2. Let a : [0, T] --~ M be a horizontal curve with a(O) = xo. Then /~/tvo = TxoS~(vo) all vo E Txo M , where S [ : M -+ M,O < t < T is the flow of the time dependent vector field Z a(t) = X (.)Y (a(t) )&(t).
P r o o f . Part (1) is immediate from (1.3.1) or (1.3.2), the defining property of V, (that V Z ~ vanishes at x0), and the skew symmetry of Lie differentiation of vector fields L z , V = [Z ", V] = - L v Z u.
22
Construction o f connections
For part (2) set xt = S [ ( x o ) and vt = TxoS~(vo). Since ddts t~ (Xo)
=
X(S[(Xo))Y(a(t))8(t)
s~(~o)
=
xo
we see xt = or(t). Also by Lemma 1.3.2 b Vt "~ Vtlt=O
=- (Yv, X ( V ( a ( t ) ) i r ( t ) ) = 0 = Vo
by the defining property of V. Thus {vt : 0 < t < T} is parallel for V.
9
Remarks: (1). Since Lie differentiation with respect to a fixed vector field obeys the usual derivation rules a corollary of Lemma 1.3.4 (1) is that, when E = T M , for any smooth tensor field A on M
(1.3.9)
VuA = Lz.A.
For the general case, for u 9 E we could define V~A, for example, by using Proposition 1.3.1 and then (1.3.9) will still hold; or more directly we could use (1.3.9) as the definition. (2). By expanding U over the basis, Lemma 1.3.4 (1) reads m
~TuY = [Zu, V] = y~[X~, V](X~,U),
U 9 F(E), V 9
F(TM);
(1.3.10)
1
or in the Gaussian form,
~ v v = N w , v ] ( w , u).
(1.3.11)
By (1.3.3),
~:vU =
[v, u] + Nw, v](w, g).
(1.3.12)
(3). For X : ~m __+ E C T M as described let ~7 be the Levi-Civita connection for some Riemannian metric on M (or indeed any torsion free connection on T M ) . Then for U E r(E), V 9 F(TM), V(xo) = v, U(xo) = u,
X(xo)d (Y(.)U(.))(v) CvU + X(x0)%Y (g(x0)) %U + ~(v, u),
~vU
say. Also by definition, (1.3.1),
CuV = = =
% u + [u, V](xo) r162162 r + ~(v,u).
A djoint connections, torsion skew symmetry, basic formulae
Working in a chart (U, r
of M with X r the local representation of X
X r = Tr o X : r
x ll~m ~-~ Tr
C r
x I~l~
etc. and taking V to be the usual differentiation in ~ "Christoffel symbols" given by Fr
, fi) = X r 1 6 2
where now 9 E ll~TM and fi E Txor
s
23
this shows that V has
)
(1.3.13)
E U. Moreover
= D V r (r
fi + Fr o (Tzor
Txor
(1.3.14)
Equivalently in the nondegenerate case
OxJ
X(x~
(1.3.15)
r-~l /=1
where {X(z)r'i}, {1 < i < n}, {1 _< r < m} is the matrix representing X ( x ) : IEm --+ ~, i.e. X ( x ) r,i = (X(e~), fi) for {ei} and {f/} orthonormal bases for ~m 9and T~M respectively, and {gke} the metric tensor. This shows that V is the L-W connection defined in [LJW84]. C. It will be important to know when V is adapted to some Riemannian metric (,)~ on T M , (see also w in the sense that '
for u E Ez and all ZI, Z2 E F ( T M ) , or equivalently that parallel translation/~/t along any smooth horizontal curve preserves (,)'. In the case that E = T M and (,)' = (,), when this holds ~7 is said to be torsion skew symmetric . See w Let V ~ be the Levi-Civita connection for (,)1. P r o p o s i t i o n 1.3.5 The adjoint (7 is adapted to (,)~ if and only i f V ~ Z ~ is skew
symmetric on T~M, (,)Ix for all u E E~,x E M. P r o o f . By Lemma 1.3.4 (2), V is adapted to (,)' if and only if d
~
1
a
2
t
-~ (TS~ (Vo),TS t (Vo)),~(t) t=o = 0 for all horizontal curves a and v~, v 2 in T~(0)M. But this is precisely the condition
'
+ (v~, V~'V 2 /~J ( o ) \ ' o
/o o>
=
O,
D'q.q,a[ 1~ since ~---~'t ~Voj : V'Z~(~ if D' is differentiation with respect to ~7'.
9
24
Construction of connections
C o r o l l a r y 1.3.6 Suppose X ( x ) is injective for each x E M . Then f7 is adapted to some Riemannian metric (,)t on T M if and only if X e - X(.)(e) is an infinitesimal isometry for each e E H. P r o o f . Injectivity implies that X r = Z ~ for u = X(xo)e, any x0 E M. But skew-symmetry of 27'X ~ is equivalent to X ~ being an infinitesimal isometry (e.g. see [KN69a], p.237). | R e m a r k 2C. Note that in the Gaussian form, w the injectivity hypothesis becomes the assumption that the vector fields in the reproducing kernel Hilbert space H~ of 3' never vanish. In this case V is the trivial connection determined by the trivialization Y of E and so the curvature/~ of ~7 vanishes (alternatively ~7W vanishes and /~ is seen to vanish by the expression given in Appendix I of [ELJL97], see also Proposition C.4 in Appendix C). In the stochastic flow picture, w it implies that for any x0 E M, and T > 0 the infinite dimensional process {~t : 0 < t _< T} can be expressed in terms of {~t(x0) : 0 < t < T}. The standard probabilistic approach to second order elliptic operators on ~ is to use such X taken to be the positive square root of the symbol of the operator considered as a map of ~n into the positive definite symmetric matrices. C o r o l l a r y 1.3.7 Suppose X is injective but the Lie algebra generated by {Xe^e E H } has dimension greater than l n ( n + 1) when n = d i m M < ~ . Then V is not adapted to any Riemannian metric on M . P r o o f . The Lie algebra of infinitesimal isometries of a connected Riemannian manifold has dimension at most 89 + 1) ([KN69a],Theorem 3.3 on p238). | E x a m p l e 2C. For M = ~2 define vector fields X a, X 2 by X l ( x , y ) = o-~, X 2 ( x, Y) = x 3 ~ + o-~" This gives an injective X as in w and the Lie algebra generated by X 1, X 2 is easily seen to be infinite dimensional. Thus the induced adjoint connection ~7 is not adapted to any metric. This example could easily be modified outside of a compact set to make it periodic, and so project to a compact surface. D. Consider the stochastic differential equation dxt = X ( x t ) o dBt + A(xt)dt,
(1.3.17)
where {Bt} is a Rm-valued Brownian motion, A is a smooth vector field and o indicates the integral involved being Stratonovich. Let {~t (x0)} be the solution with initial value x0 and ~7 the connection constructed from X. Then roughly speaking the parallel translation along the paths of {~t} is bounded if and only if (7 is adapted to some Riemannian metric. (This statement is true for smooth paths as easily seen from the proof of the next theorem). See also Theorem 5.0.7 for the corresponding result on the derivative flow.
Adjoint connections, torsion skew symmetry, basic formulae
25
1 . 3 . 8 Assume that X is nondegenerate and the stochastic di~erential equation (1.3.17) does not explode. For xo,Yo E M and T > O, let {~t 'U~ : 0 < t < T } be the process conditioned to be Yo at time T of {~t(Xo)} and let
Theorem
l i t =: /~/t T'z~176 be parallel translation along {~tT'Y~ If ~7 is adapted to some Riemannian metric then l I T is a bounded L(TxoM, TuoM ) valued random variable for such Xo, Yo and T > O. Conversely if just for one set of Xo, Yo and 7" > 0 the parallel translation process l I T along the path of {~T'Y~ is bounded then ~7 is adapted to some Riemannian metric on M. P r o o f . The 'if' part is clear since ~IT would be an isometry. Suppose that l I T is bounded. Let u0 be a frame at x0 and P(uo) the holonomy bundle through u0, i.e.
P(uo) = {u E G L ( M ) I there exists a horizontal curve from u0 to u.}, with structure group
'~(uo) = {g E GL(n) I uo "g C P(u0)}. We can reduce ~7, now a genuine connection, to a connection on P(uo). Let (at) b e the solution to
dut = X (ut) o dBt + .4(at)tit with initial value u0. Here )( and A are the horizontal lifts in P(uo) of X and A. Then {at : 0 < t < T} is the horizontal lift of {~t(x0) : 0 < t < T}. The support of the law #t(P) of UT is all of P(uo) by the Stroock-Varadhan support theorem and the definition of P(uo). Consequently by Carverhill [Car88] the support of UT when {~T(X0)} is conditioned to have ~T(X0) = Y0 is 7to 1 (Y0), where 7~0 : P(uo) --+ M is the projection. Thus parallel translations from Txo M to TyoM along the paths of the conditioned process {~~ : 0 < t < T} are dense in the space of parallel translations along smooth paths. So the latter is a bounded set and ~(u0) is bounded in GL(n). As a consequence there is an inner product on ~n, (,), say, invariant under O(Uo). The required metric at a point z of M is then (vl,v2)z = ( u - l v l , u - l v 2 ) ! for any u E 7rol(Z). 9 E. Proposition 1.3.3 was concerned with the case when E is integrable. At the other extreme is the situation where the vector fields X 1 , . . . , X m together with their iterated brackets span T z M for each x in M, giving hypoellipticity of the operator A by Hhrmander's theorem. Bismut showed how this hypoellipticity was reflected in the behaviour of the derivative of the associated stochastic flow so that L e m m a 1.3.4 (2) makes it not surprising that it is also reflected in the behaviour of parallel t r a n s l a t i o n / / s along the paths of the associated diffusion, as we see next. As before we consider the stochastic differential equation (1.3.17) and assume it to be regular with A(x) E E~ for each x in M. Our discussion is an adaptation of t h a t in [Bel87] which was in turn based on [Dis81]. ^
--1
Let Rt(w) = s p a n { / / s X ( x s ) e : e C ll~m,o < s <_ t A T } C TzoM w h e r e T is a fixed, positive predictable stopping time less than the explosion time for
Construction of connections
26
our stochastic differential equation. By Proposition 1.3.3 in the integrable case Rt(w) = E~ o for each t _> 0. In general let Exo C T~oM be the linear span of X 1( x 0 ) , . . . , Xm(xo) together with all the brackets and iterated brackets of the vector fields X t , . . . , X m evaluated at x0 (this depends only on E ~ T M , not on V or a choice of X 1 , . . . , X m determining E): Theorem
1.3.9 For each t > O,
Exo C Rt(w)
almost all w in ~.
P r o o f . Set R(w) = N t > o R t ( w ) . By the Blumenthal 0-1 law there exists a nonr a n d o m Ro C Txo M with Ro = R(w) almost surely. Moreover there therefore exists a predictable stopping time T1 with 0 < T1 < r such t h a t Rt (w) = Ro for 0 < t < T1 almost surely. Suppose e 6 T~oM annihilates Ro. Then for e E I~m, with X e := X(.)e, ^
--1
x
e
= 0,
O<s<~.
^-1
Now if Z is a vector field with g(//s Z(xs)) = 0, 0 < s < 71, taking the local martingale part of its canonical decomposition we have ^
--1^
vx(
.)sz) = 0
all f E ll~m, 0 < s < T1. By L e m m a 1.3.4 (1) this gives ^-1
(IL
[z, x S ] ( x . ) ) = 0,
0 < s <
(1.3.18)
since [Z, X I] = [Z, Z x'] + ~TzXf a n d / / 8 V z X f 6 Ro because ~TzXf 6 Ex,. Taking Z = X e show that [Xe,Xl](Xo) E Ro each e , f 6 R m. Repeating the argument, using (1.3.18) with Z = [X e, X I] g i v e s [ [ X e , X S ] , X g ](Xo) E Ro for all e, f, g in ]~m, and the full result follows by induction. 9
1.4
Example: Homogeneous spaces continued
A. For an important class of examples we will go back to situation of a reductive homogeneous space M = K / H described in w using the notation there. In Proposition 1.2.1 we saw from our construction that the action of K determines a K-invariant connection on M and that given an adK-invariant inner product on the L-W connection is K-invariant: in particular the metric < ) x (i.e. induced by X ( x ) : ~_--+ T M ) is K-invariant, as is seen from (1.2.8). In general given an inner product ( ) on _ethe metric ( )x induced on T M will not be K-invariant. 1.4.1 A K-invariant connection ~7 on M is uniquely determined by the mapping m -+ L(T~oM; T~oM )
Proposition
Example: Homogeneous spaces continued
27
given by e ~ fT'X ~
T~oM
where X7' is the adjoint connection. P r o o f . Observe t h a t for any vector field Z on M
f t z : T M -+ T M defined by ftz(v) := - ( 7 v Z - T ( Z , v ) as defined in [[SN69a], p255, [SN69b], p188] is given by ftz(v) = V~Z. -' The result is then a reformulation of Corollary 2.2 of [[Ki69b], p191]. 9 B. T h e K-invariant connection on M corresponding to the identically zero map: _m-+ L(TxoM; T~oM ) is the canonical connection 1.4.2 Let (7 be the connection on M determined as in w by its reductive homogeneous space structure and let f7 be its adjoint. Then (7 is the canonical connection. In particular (i). R ( u , v ) w = - [[Z~, Z v] - z[Z~,Z~],ZW] , u , v , w 9 T~oM, (ii). r = o, (iii). (TR = O.
Theorem
P r o o f . Take V = V in Proposition 1.4.1 and use the defining property of V to see ~ is the canonical connection. For (i), (ii) and (iii) see Theorem 2.6 p193 [KN69b] and use the fact that e ~-~ Xe(.) is a homomorphism of Lie algebras. 9 C o r o l l a r y 1.4.3 Any K-invariant tensvr on M is XT-parallel. P r o o f . See Proposition 2.7 p192 of [KN(i9b]. From this we immediately have C o r o l l a r y 1.4.4 The connection X7 is metric for any K-invariant metric on M. Taking V = V we can obtain in this way a class of metric connections whose adjoints are not metric for any metric on M. C. Suppose next that as well as being reductive there is an adH-invariant inner product B on m. We then obtain a K-invariant Riemannian metric on M which agrees with B under the isomorphism _m ~ T~oM. The space, together with B, is called naturally reductive if it has a decomposition as before with also (1.4.1) where the subscript m refers to the projection in _~onto m.
Construction of connections
28
Proposition 1.4.5 The decomposition ~ = m + ~ together with 13 is naturally reductive if and only if ~Y is torsion skew symmetric for the induced K-invariant metric on M .
Proof.
g
~
( l ,vl, Lo : -IE ~
Ixol, l o
by (1.3.6). Set a = Y(xo)u, fl = Y(Xo)V, "y = Y(xo)w. Then
[zu, zv](xo)
=
[xo,x
=
X(xo)
](xo)
and the result follows from (1.4.1).
C o r o l l a r y 1.4.6 In the naturally reductive case ~7 is a torsion skew symmetric connection. Note however that the metric involved may not be induced, via X, by an inner product on ~. However from [KN69b] p203 Theorem 3.5 we see that if has an adK-invariant inner product ( ) , we can let m = ker X ( x o ) • = O_• to have K / H naturally reductive with B(a, fl) = (c~, fl), cr, fl E m. Thus T h e o r e m 1.4.7 Let ~_ have an adg-invariant inner product ( ) . Then the LW connection on M determined by X , 0 is torsion skew symmetric and K invariant. Its adjoint connection is the canonical connection of the corresponding reductive homogeneous space structure.
D. Specializing further suppose that we have a symmetric space (K, H, a): so K, H are as before with a reductive decomposition [ = m + O such t h a t m and are, respectively, the - 1 and +1 eigenspaces for the involution a of ~. T h e o r e m 1.4.8 For a symmetric space ~7 and ~7 are torsion free. In particular they are the Levi-Civita connection of any K-invariant Riemannian metric on M. P r o o f . T h a t T = _~b = 0 follows from (1.3.6) and the fact that [__m,m] CO, see Proposition 2.1 p226 [KN69b], or from the corresponding fact for the canonical connection [KN69b] p231 Theorem 3.2. The result follows from Corollary 1.4.4. 9 C o r o l l a r y 1.4.9 Let the symmetric space (K, H, a) be such that ~_has an adginvariant inner product ( ) invariant under a. Then the L- W connection for X , 0 is the Levi-Civita connection for the induced metric (which is K-invariant).
Example: Homogeneous spaces continued
29
P r o o f . In this situation m 2 It, see [[KN69b], p233], and so m_= 2 K e r X ( x ) for each x E M : thus Y = X* and ~7 is tlhe L - W connection and hence metric. Since it is torsion free it is Levi-Civita. 9
Example
1.4.1 Lie groups as symmetric spaces
Recall the s t a n d a r d s y m m e t r i c space structure for a Lie group G. AG={(g,g) EGxG:gEG}. Let G x G a c t o n G b y
Let
(g, h) 9x = gxh -1. T h e stabilizer of e is A G and G has the homogeneous space s t r u c t u r e G = G x G / A G , and s y m m e t r i c space structure with involution induced from a : G x G -+ G x G given by a(g, h) = (h, g). For example the s y m m e t r y se : G -+ G is just x ~-~ x -1, see [KN69b] (p228). T h e relevant decomposition of _~= 9 @ 9 is
where A9 = {(a, a) : a E g}, m = { ( a , - r : a E g}, see [KN69b] (p198). Now suppose 9 and hence g x 9 has an a d a - i n v a r i a n t inner p r o d u c t ( ) . T h e stochastic differential equation on G is
dxt = TL=, o dB, - TR=, o dB~ for (B.), (B!) independent BM(g). T h e flow is given by ~t(x) = gtxg~ -1 for
dgt = T Rg, o dBt
dg'~ = T Rg, o dB; with go = e = g~. T h e flow consists of isometries so t h a t the m o m e n t exponents are zero. However for G with bi-invariant metric (as we are considering), if g has trivial centre then Ric > 0. T h u s we obtain a class of stochastic differential equations such that
(i) ni~ > 0; (ii) all m o m e n t exponents vanish; (iii) ~. consists of isometries; (iv) V is the Levi-Civita connection. In particular (i) and (ii) can be contrasted with results which says t h a t negative c u r v a t u r e implies first m o m e n t exponent positive. T h e y also give an e x a m p l e where the hypotheses of Corollary 6.4.7 irL [Li92] hold. R e m a r k : A m o n g other homogeneous spaces with a R i e m a n n i a n space structure are the spheres S n = S O ( n + 1)~SO(n), oriented G r a s s m a n n i a n manifolds S O ( p + q)/SO(p) x SO(q) and hyperbolic spaces O(1, n ) / S O ( n ) .
Chapter 2
The infinitesimal generators and associated operators For any second order elliptic operator s with smooth coefficients and s --- 0 on a manifold M of dimension greater than 1, we construct X : ~ m __+ T M , 1 m for some m, such that s = ~-'~1 L x J L x J , the differential generator for the solutions of the stochastic differential equation without drift dxt = X ( x t ) o dBt. This result is, in fact, proved for a class of semi-elliptic operators. In section 2.4 the HSlmander form operator ..4q = ~1 ~ j Lx~ L x J + LA on differential q-forms is analysed. Section w discusses a special class of connections on T M and also the natural generalization (~ of 5 to our situation. We show in w that .Aq = _ 1(d5 + 5d) + LA so that when the L-W connection associated to X is the Levi-Civita connection and A = 0 then .,4q is the De Rham-Hodge Laplacian . In the regular case there is the Weitzenbock formula : Aqr
=
~trEV.(~7r
+ LAr
Rqr
for /~q a zero order term, the 'Weitzenbock curvature' related to V. In the last section we give conditions for the leading order terms of .4 q to be symmetrizable as an operator on a suitable L 2 space. In particular we show that this holds if .4 0 is symmetrizable and V is metric for some metric on T M .
2.1
T h e irrelevance of drift in d i m e n s i o n g r e a t e r than 1
A. The first application of the construction in section w is that any elliptic differential operator can be considered as an infinitesimal generator to some stochastic differential equation with zero drift: dxt = X ( x t ) o dBt. Let E be a subbundle of T M with fibre dimension p. Let Z be a E-valued vector field, and
T h e irrelevance o f drift in dimension greater than 1
31
V : ~m __.} E C T M a C a surjection. Consider the operator s on M: 1
m
= ~-~Lv,
(2.1.1)
Lv, +Lz,
1
where V i = V ( e i) for an orthonormal base of ~m. T h e o r e m 2.1.1 A s s u m e p > 1. For s as given above, there is a map X : M x I~m --} T M linear in the second variable such that the solution to dxt = X ( x t ) o dBt has s as infinitesimal generator, i.e. s = ~1 ~-~j L x ~ L x J . Given E the Riemannian metric induced by V. Recall that a diffusion {~t (x) : 0 < t < p(x)} generator s is a ~7 martingale if f ( ~ t ( x ) ) - t r a c e E V d f ( ~ t ( x ) ) is a local martingale for any smooth f : M -+ R. (e.g. [Eme89]). An immediate corollary of the Theorem is that a diffusion {~t(x) : 0 <_ t < p ( x ) , x e M } , whose generator satisfies the conditions of Theorem 2.1.1 is a V-martingale for some metric connection V on E. In fact (as will be seen explicitly in w ~t(x0) will be the stochastic development of a Brownian motion on Ex0 using V. The existence of a connection on T M for which this is true for a nondegenerate diffusion was shown by Ikeda-Watanabe [IW89] and a modification of their construction for the case E = T M is one of the main ingredients of our proof. The other key ingredient is Narasimhan & R a m a n a n ' s theorem on universal connections used via Theorem 1.1.2. The proof is given in w167below. To a d a p t Ikeda ~z W a t a n a b e ' s construction we need the modification of the classical result on connections given in Proposition 2.1.2 below. Its proof taken up w167 Some of the notation in w167will be used later. B . Let E be a subbundle of T M with a given metric. Two connections V a and ~Tb o n E are associated with a bilinear m a p D ab : T M • E -+ E such t h a t V~u=vbU+Dab(v,u),
VeTM,
UeE.
Let T a and T b be respectively the torsions, defined by (1.3.5), for the two connections, then Ta(u,v)) = Tb(u,v) + Dab(u,v)-
Dab(v,u),
u, v e E.
(2.1.2)
In this section we use uppercase letters for vector fields and lowercase letters for tangent vectors. It will be convenient to have a class of connections on E with which to relate metric connections. In the non-degenerate case the obvious base connection is the Levi-Civita connection V for the given metric. In the degenerate, regular, case let (,)o be an extension to T M of the metric (,) on E and let PE : T M -+ E
32
T h e infinitesimal generators and associated operators
be the corresponding orthogonal projection. Let ~7~ be the connection on E which is the push forward by PE of the Levi-Civita connection V of (,): ~~
: PEVvU,
U 9 F(E), V 9 F ( T M ) .
(2.1.3)
In fact V ~ is a metric connection. Note that if T O : T x E --+ T M is the torsion for ~70, defined by (1.3.5) or rather its modification for connections on the subbundle E, T~
V) = P E ( V u V
- V v U ) - [V, V] = - ( I - PE)[U, V].
(2.1.4)
Let T be the torsion for a connection ~7 on E and take (I - PE) of (1.3.5) to see: (I -- P E ) T ( U , V) = - ( I - PE)[U, V] = T~ V), (2.1.5) and so T(U, Y ) = P E T ( U , V) - (I - PE)[U, V].
(2.1.6)
The converse also holds: P r o p o s i t i o n 2.1.2 Let T : E x E ~ T M be a skew s y m m e t r i c map satisfying (2.1.5). There is a metric connection ~7 on E with T as its torsion. The proof of Proposition 2.1.2 will be given after Lemma 2.1.3. First we introduce the tensors b and S. Define b : T M x E --+ E by (TvU = V~
(2.1.7)
+ D(V, U).
For U, V 9 FE, w r i t e / ) as the sum of its symmetric part S and antisymmetric part A: [9(u,v) = A ( u , v ) + S ( u , v ) . By (2.1.2), T ( u , v ) = T ~
+ 2 A ( u , v ) . So A(u, v) = ~ P E T ( u , v),
so that fYvU = VvU~ + [ P E T ( u , v )
+ S(u,v),
u,v e rE.
(2.1.8)
Let Cyl denote cyclic sum. L e m m a 2.1.3 A connection f7 on E is metric if and only if the map D ( v , .) : E ~ E is skew s y m m e t r i c for each v E T M ,
For V E F E , (2.1.9) is equivalent to
The irrelevance of drift in dimension greater than 1
(S(Ul,U2) ,v) ~---~I
(pE~['I(V,IL1),U2)--[-I (PET(v, u2),IL1)
33
(2.1.10)
and (2.1.10) implies Cyl(S(., .), .) = 0.
(2.1.11)
Consequently for U1,U2, and V in rE,
( D(V, U1), U2) (2.1.12) P r o o f . First take V E r(TM) and Ui E r(E). The equivalence of ~7 being metric and the skew symmetricity o f / ) ( - , - ) follows from differentiating (U1, U2}:
dv (U1, U2)
= (V~/UI,U2)-I.-(UI,V~ So V is metric if and only if
(L)(V, U1),U2) -}.-(UI,~)(V, U2)) -..~0. Next suppose V and Ui E FE, writing/9 = A + S to get:
(A(V, U1), U2) + (A(V, U2), U1) = - (s(v,
u,),
u2> -
(s(v, u2), u,>
(2.1.13)
Suppose (2:1.13) holds. Observe for an alternating bilinear map L : E x E --+ E: Cyl [(L(v, Ul), u2} -[- (L(v, u2), ul)] = 0.
(2.1.14)
Take the cyclic sums of equation (2.1.13) and apply (2.1.14) to A to obtain Cyl (S(V, U1), U~) = 0. Substitute the above back to (2.1.13) to see:
(A(V, U1), U2) + (A(V, Uz), g~) = (S(U~, Uz), V). So (2.1.13), and therefore (2.1.9), implies (2.1.10). On the other hand (2.1.11) clearly follows from (2.1.10) and the two give (2.1.9). 9 P r o o f of P r o p o s i t i o n 2.1.2. Define S : E x E --+ T M by (2.1.10). Set b(v,u) =- 0 for v e ( r E ) • and D(v,u) = 1PET(V,U ) + S(v,u) for v E r E . Define V by
CvU = v ~
+ D(v, u).
34
The infinitesimal generators and associated operators
Then the equality (2.1.9) holds for v E FE and extends to F T M since by the construction/9(v, u) = D(PEv, u). The connection V is the required connection.
C. P r o o f of T h e o r e m 2.1.1. Let V be the Levi-Civita connection for a metric extending the metric induced on E by the map V and V ~ its projection to E as before. Set 2 = 89~ V ~ i + Z and choose T such that traceE(PET(u,--),--} = -2 (Z(x),u},
e E~.
(2.1.15)
One choice of such T is given by 2 PET(v,u) = ~_l(UAv)Z,(x),
u , v e Ex.
Recall ( u A v ) Z ( x ) = ( Z,(x), u} v - ( Z(x), v I u. Let ~ be the metric connection on E with torsion T : E x E --~ T M as constructed in Proposition 2.1.2. We show the associated stochastic differential equation has s as infinitesimal generator. Let X : __~m _.4. E C T M be a bundle map, as in Theorem 1.1.2, which gives rise to the metric connection ~. The solution to the following stochastic differential equation dxt = X ( x t ) o dBt (2.1.16) has generator 1
"4o = traceE[V~
m
+ ~E
V~
i:l
On the other hand from (2.1.1) m
/: = 1 traceE[V~
- ' V ~ v, Vi + Z) = ~ t r a c e E [ V ~ + ( 5l ~A.., 1
The required result then follows after we show m
v~
b ( x * , x ~) = t r ~ b ( - , - )
i) = -
i=1
1
equals Z. For this note that for all v E E,
= - E
xi} i
Consequently
: -2
Torsion Skew Symmetry
35
1
traceE[V~
m
+ 2 Z
V~
= traceE[V~
+ 2
i=l
and the X so constructed is the required map.
2.2
Torsion Skew S y m m e t r y
A metric connection ~7 on the tangent bundle T M can be expanded in terms of another connection V and its defining map X or W as introduced in Theorem 1.1.2. More precisely by (C.2) in the appendix,
% u = CvU + Z r
(u, XJ)
J However this process is not reversible: we do not seem to be able to write the second term of the right hand side in terms of V and X. For example if ~7 is taken to be the Levi-Civita connection, the special case when
ZVvX (U,X )xo=-ST(v,.), 1 v
J turns out to be particularly interesting. This is just the torsion skew symmetric case. See Proposition 2.2.2 below. In this section we shall explore this situation. A metric connection ~ on T M is torsion skew symmetric if ~ is adapted to the same metric. Here is a corollary of Lemma 2.1.3. C o r o l l a r y 2.2.1 Suppose ~7 is a metric connection on T M . The following are equivalent: (a). ~7 is torsion skew symmetric; (b). the symmetric part of D(., .)
vanishes; (c). (\ T ( u/ , . ) , . ) is skew symmetric for each u E T M . In the following we put together the equivalent conditions for a connection to be torsion skew symmetric, in terms of X. P r o p o s i t i o n 2.2.2 Let X be a defining map for a metric connection ~7 on T M as in Theorem 1.1.2. Then V is torsion skew symmetric if and only if
i=1
i=1
,i=1
i=l
36
The infinitesimal generators and associated operators rn
equivalently 2F(v, u) = 2 E
X~(x) ( V v X i ' u}. In this case,
i=l
lv V v U = (TvU - -~T(v, u).
(2.2.1)
In particular, V x ~ X i = 0 for each i.
P r o o f . The first identity comes from (C.4) with V replaced by V: m
~(v,u) = Z x'(x) (~,vvx~> - ~x~(x) (v, VuX~). i=1
(2.2.2)
i=1
The second comes from the first and (C.3). The third is a consequence of (2.2.2) and the second. Finally (2.2.1) follows from (C.2) and the third identity. 9 Recall the definition of ZU: ZU(x) = X ( x ) Y ( T r ( u ) ) u as in w P r o p o s i t i o n 2.2.3 Let (7 be a metric connection on T M with defining map X . In terms of the adjoint Y of X ,
T(Vl,V2) = X ( x o ) d Y ( v l , v 2 ) ,
(2.2.3)
vi E T~oM.
Furthermore the connection (7 is * the Levi-Civita connection if and only if X ( x ) d Y ( u , v ) = 0 for all u , v E T z M , all x E M , or V Z v vanishes at xo for all v C TxoM. 9 torsion skew symmetric if VZWI%o M : Txo M ~ T x o M is skew symmetric for all w E T M , or V v Z u + V u Z v = 0 for any u, v E T M , equivalently ~7uV + ~TvU = V u V + V v U for all vector fields U and V.
P r o o f . Recall Z ~' = X ( x ) Y ( x o ) v ~ as in (1.1.1). By Proposition 1.1.1, the definition of V,
~(vl,v~)
= %,z~-%:z"-[Z~l,Z 31- V Z V 2 ( V l )
~]
=
X(xo)VvlY(V2)
=
X(zo)V~aY(v2) - X(xo)V,~Y(Vl) = X(xo)dY(vl,v2).
-
X(xo)Vv2V(Vl)
-Jff V Z vl
(v2)
-
[ Z vl , Z TM]
That VZ ~ vanishes for all u iff V is the Levi-Civita connection follows from (1.1.2), the defining property of the connection. For the equivalent conditions of torsion skew symmetricity: the first is exactly Proposition 1.3.5: V is torsion skew symmetric if and only if (v, V u Z w) + (u, VvZ w) = 0
u,v E TM.
From the equivalence of the first two identities in Proposition 2.2.2:
(w, v u z ~) + (w, v ~ z ~) = 0.
The 'divergence operator' 5
37
Adding VvU with V u V and use (C.2) to obtain the last equivalence.
9
Finally if V is a metric connection on T M we can define a differential 3-form D # by D#(-,-,
-):=
Alt((D(-,-),-))=-
Cyl((D(-,-),-))K~Cyl(T(-,-),-),
where Alt is the alternating mapping. In the torsion skew symmetric case, there is a differential 3-form T # from the torsion tensor:
T # ( u , v , w ) = (T(u,v),w).
(2.2.4)
Indeed there is a bijection between torsion skew symmetric connection for a given metric and 3-forms given by T ~ T # using L e m m a 2.1.3. In Appendix C we see t h a t dT # and 5T # appear in curvature identities.
2.3
The 'divergence operator'
A. Let X : ~m _+ T M be a smooth bundle m a p (not necessarily of constant rank). Let A*T*M be the space of differential forms on M. Define 5 : AqT*M --+ Aq-IT*M by
5r = - ~
txJLxJr
(2.3.1)
j=l
Here rye is the interior product of r by Y: r y e ( - ) = r On smooth functions ty = 0. In the case t h a t X comes from an isometric immersion of M to ll~m , ~ is the usual divergence operator 5. Let A be a smooth vector field on M. Consider our operator ,4 in H5rmander form 1
.A = ~ ~
Lx~ Lx~ + LA.
(2.3.2)
Since Lie differentiations also act on forms we can also extend .A to operators on forms and will use .,4q when we want to emphasize that we consider it acting on q-forms. One of the observations which demonstrates the role of ~ is the following proposition: Proposition
2.3.1 1
m
Ar = ~ Z j=l
LxJ Lx~ r + LAr = --
1
(Sd + dS) r + LAr
(2.3.3)
38
The infinitesimal generators and associated operators
P r o o f . Just observe that the Lie differentiation LxJ is given by LxJ r -- ~x~ d~ + d (~xJ dp) and d commutes with the differentiation.
9
B. Assume that X has constant rank. The covariant derivative V.r of a q-form is a linear map r = V . r : E --+ A q T * M over M, i.e. a section of L(E; AqT*M). It is not obvious in the degenerate case how to apply V to it again: we would want
%
(r
= Vwr (V(x)) + ~p ( V w U )
9 AaT*M
(2.3.4)
for U 9 F(E) and w 9 Ex, but in general V ~ U will not lie in Ex. However we can use (2.3.4) to define " t r E V . r by m
trEV.r
:= E
~x~ (~(XJ))
(2.3.5)
j=l m
since ~'~-j=l V x j X j = O. Since this agrees with (2.3.4) it will coincide with the result obtained by taking any extension r : T M -+ A q T * M and using (2.3.4) as the definition of V r or extending V as in Proposition 1.3.1 to some V 1 on T M and using (V1) ' in the usual way. P r o p o s i t i o n 2.3.2 Let V be a metric connection on a subbundle E of T M and X its defining map as in Theorem 1.1.2. Then ~, defined by (2.3.1), does not depend on the choice of X . In fact,
= -tr
r r
.)
P r o o f . Let E • be a complementary bundle to E so that T M = E @ E • Let V • be a connection on E • and set V 1 -= ~7@ V • Observe that for a connection on T M with adjoint connection V', q
...,
..
-' Y
,vq) (2.3.6)
j=l
for v = ( v l , . . . ,vq) E A q T M . Take V' to be V 1. By Proposition 1.3.1, q
Lxpr162162
l ~_p~_m, j=l
(2.3.7)
The 'divergence operator' ~
39
and V• is actually not involved. By the defining property of the L-W connection m
~r
= - Z ~x,
Cx, r
= --trEe_C(-, .).
(2.3.8)
1
In the Gaussian field formulation of w
we have (2.3.9)
,4 = ~ E L w L w + LA
with ~ defined accordingly. The extension of (2.3.3) to this case holds in the same way as does Proposition 2.3.2. An important consequence which follows using Theorem 2.1.1 is: C o r o l l a r y 2.3.3 The operator A on forms defined by (2.3.2) or (2.3.9) depend on X or the field W only through the associated connection ~7 and the induced metric on E. In particular an operator Jl given by (2.3.9) can always be written in the form (2.3.2) using a finite set of vector fields X 1 , . . . , X m.
C. The rest of this section will be on the comparison of 5 and the usual divergence 5. R e m a r k 2.3.1 Let V be a metric connection on T M with defining map X . 1. Assume Z
V x J X j = O. Then 5 = 5, the usual divergence, on differential
1-forms. (E.g. this holds in the gradient Brownian system case, or if V is torsion skew symmetric ). 2. Assume y~j V x J X j = ~ h for some smooth h : M -~ IR. Then ~ i Lx~Lx~ = A + Lvh on differential 1-forms. Here A is the Laplace-Beltrami operator. 3. ~ ' ~ X i A V X i = 0 if and only if V = V. In particular for Z-forms, = ~ - ~ Vx~X~ if and only if ~ is the Levi-Civita connection. P r o o f . Remarks 1 a n d 2 are readily seen by ~r
= -
tx, V x J r - r 1
VxJXJ)"
(2.3.10)
1
(i.e. (2.3.6) using V). Similarly on differential q (> 1) forms, 3r
= - E ~ ~x, V x , r
m
--~j=l
[ ~-~ ~k=l
- r
Vx, X~,v~,...,~_l)
~)(XJ,vl,'",Vv~ x j , ' ' ' , v q - 1 )
]
,
(2.3.11)
The infinitesimal generators and associated operators
40
for any V = (Vl,... For q = 2,
,Vq-1) ~-.Aq-ITM.
~)(V) : L tXJ V X'~)(Vl) - E ~)(xi' VvaXi) - E ~)(VxiXi' vl)" 1 This leads to the second statement of Remark 3 (assuming the first). Now we show the first. Firstly ~7 = V implies the vanishing of ~ X ' A V X i by the characterization of V. Now assume ~ X i A V X i = 0, i.e. for any vectors wl, w2,
Since E j V~ X j <w,, XJ> = - ~_,j
E <Wl'XJ> <w2, VvXJ> = 0 J and therefore ( w 2 , V , x J ) = 0 if X j r O. This is exactly the characteristic property of the connection ~7 associated to X. 9 D. Let V be a metric connection on E. Using the Levi-Civita connection V ~ for some metric on T M extending the metric of E as in section 1.3, set 60
= - ~ ~x~V% j=l
acting on q-forms, 1 < q < n, and annihilating smooth functions. So 6~ = 6 when E = T M . Let Kg : AqT*M to Aq-IT*M be defined by K ~ r _= 0 and for q>l, Koqr = (~r 6~162 + LET v o x j ( r (2.3.12) -
Then f o r / 9 defined by/9(V, U) = V v U - V ~
= -+
x')(x,^-))
Set AO =
1 m
j-----1 It follows from (2.3.6) and (2.3.3), for q > 1,
m
ELxiLx~r j----1
= -l(d6~ +6~162162
1
q
~d(K~r
1 Kq+l
- ~
o
t d ~j. ~
(2.3.13)
The 'divergence operator' 5
41
P r o p o s i t i o n 2.3.4 For nondegenerate X , A q = 1 A q + LAo if and only if dKq + Kq+i d = O. Also r 1 = 1n -~ LAO only when ~7 = ~7. P r o o f . The first statement of the theorem is clear. The second statement m Kg(dr = - ~2j=1 de(X J, VvX~). 9
follows from part 3 of Remark 2.3.1 and
Note also J 1
m
= ~zx~r
v~) + LAor
~) + + ~ r
~,R(~,, v~)X ~)
1 m
m
+2 ~ Vx,r
X~), v~) + 2 ~ V~, r
1
D(v~, X~)) + 2r
X ~, V~X~).
1
Finally as an example we calculate the divergence and .4 q for the connection used in the proof of Theorem 2.1.1: E x a m p l e 2.3.5. Consider the connection constructed in the proof of Theorem 2.1.1 of w Set W = 2. Let ~7 be the associated connection. Recall
b(v, u) - p -2 1 [<w, u> v - w]. m
Then - E
m
V~
= E b(X~' Xi) = - 2 W and for each j,
i:1
1
m
E r
V l ' " " D(vj, x i ) , . . . , Vq_1)
i--1
2
m
,vq-l)
p-1 Er 1 2 P- lr
vl,...,vj,...,Vq-l)
2 P- lr
4 P-- lr
Vl,...,vj,...,Vq_l).
Consequently by (2.3.12), for a q-form r 5r176162
xv~ ' rx,
=
4(q--1)~wCp-1 50r
_
r4(q-_i) +2]~wr
L p_l
and in the nondegenerate case m
E L x J L x j = i2 A + L[4(22~)+21W 1
n -4 l~Wd"
The in~nitesimal generators and associated operators
42
Thus if W is a gradient, W = Vh say, in the non-degenerate case .Aq restricted to closed forms is the Bismut-Witten Laplacian corresponding to the measure exp([
2.4
S(q- 1) n-
~ +4]h(x)) dx
H5rmander
generators
on
differential
rms We will treat in detail the case of finite dimensional noise (or equivalently a finite sum of squares of vector fields); the infinite dimensional situation can be reduced to this by Corollary 2.3.3. Let {~t(x)} be the solution flow to the s.d.e. (1.2.5) and Pc the induced semigroup on measurable forms defined by
= when the expectation exists. Its differential generator A is given by (2.3.2) and hence (2.3.3), as seen by It6's formula. See e.g. [Elw92].
Proposition 2.4.1 Let M be a Riemannian manifold. Assume non-explosion and Esups
Ptr = r +
Ps ACds.
However by Theorem 2.3.1
PsACds
as required.
=
~0t Ps(dSr
=
d
9
R e m a r k : The condition dPs = Psd holds by differentiation under the expectation sign if M is compact and somewhat more generally see [Li92], [Li94b], and [EL94]. The fact that Ptr is cohomologous to r for closed q-forms on compact M was noted in [Elw92] and on non-compact M under the related hypothesis that the flow is strongly q-complete in [Li94b]Theorem 2.4, using the fact that fo ~ r = f(~). ~ r for any q-simplex a.
43
HSrmander form generators on differential forms
Now assume X has constant rank. Then .A~ is given by A~
- L A f = tr~TgradEf = tr~Tdf = tr(Tdf ,
but not V g r a d E f in general. Here g r a d e r is the gradient of f with respect to the metric on E induced by X: (gradEr , u) = dr(u), Set
(2.4.1)
any u E E.
m
AX
1 Evx~x
j + A.
1
In the nondegenerate case .40 = -~A 1 q- LA x and for gradient Brownian systems Aq = 89 + LA, as can be seen in section 2.3, see also [Elw92], [KusS8]. In fact r = 89 + LA, each q, for gradient Brownian systems. The main theorem of this section is the following Weitzenbock formula (we use the notation in Appendix B for the linear operators dA and 52A): Theorem
2.4.2 ( W e i t z e n b o c k f o r m u l a ) Suppose X has constant rank. Let ~7 be the associated connection on its image bundle E with adjoint ~7, R its curvature tensor and /~qr = - r
((m (dA) q
E
ff~(XP,-)(XP)
\p=l
52A(V.XP)(-)
-I-
)
(2.4.2)
p=l
the so called Weitzenbock term. Then for q > 1,
Aqr
=
-~trEV.(V.fb) + LAdp 1
^
(2.4.3)
Rqr
^
(2.4.4)
=
and
l < i < k < n , l <j
(2.4.5) Here Tt is the curvature tensor for V as below, with
/~ikj, = (/~(ei, e k ) e , , e j ) ,
1 < i , k < n, 1 < j , e < d i m ( E )
Also I~ic# : T M --+ E is defined by m
# (v) =
it(v, x ' (x) )xJ j=l
v E T~M,
44
Tile infinitesimal generators and associated operators
and a i, (ai) *, 1 ~_ i <_ n are the annihilation and creation operators corresponding to some base e l , . . . , e n of T~M which extends an orthonormal base el,...,edim(E) of Ex. In the nondegenerate case <1~ic# (vl), v2} is the Ricci curvature t~ic(vl, v2)
4~7. A. L e t / ~ : T M
x T M -~ L(E; E) be the curvature tensor for V on E given
by
CT~u,vlW
R(U,
for all vector fields U, V and E-valued vector fields W. Then we have an expression for/~ in terms of X: (2.4.6)
1 See Proposition C.4. L e m m a 2.4.3 Suppose X has constant rank. For a differential q-form r
m
Z Lx~Lxpr p=l m p----1
p=l
with the convention r mal generator is given by
JJ*q~
vanishes for q = 1 and so the infinitesi-
: ltrE~7.(~7.(/)) "Jr-LA~)'-~ 1Eprn 1 (~ ((di)q
(
(~(Xp,_)(Xp)))
(2.4.7)
)
+7 Ep=l r ~2A(VXP(-))
P r o o f . Let r be a q-form, and v a q-vector. By (2.3.7),
Lxp(Lxp
)(v)
=
+
+r
(VX p) o dA* (VX p) (v)).
Summing up from 1 to m, the second, and the fourth term disappear by the defining property. In Appendix B setting A ( - ) = V _ X p so that A2(-) = V~7_xpX p to obtain,
H6rmander form generators on differential forms
45
m
p=l
p=l m
m
p:l
p=l
m
Observe ~p=l ~xp(~.XP)("v)
m
:
Zp:i
~xp(~.XP)(v),
again by the defining
property of the connection V. m
Let U E F(E). Then since Z V u ( V x ,
XP) = 0 ,
p--=l m
m
~[~Tv~x.XP + Cx,(~uxp)] = ~ [ V v ~ x , x ~ + Vx~(Vux')] p=l
p=l
p--=l m
p=l
p----1
m
= Z ~(x,, u)x~. p=l
The stated result now follows. P r o o f of T h e o r e m 2.4.2. Let a i and (ai) * be respectively the annihilation and creation operators as given in Appendix B. By Corollary B.3 in the Appendix,
r (52A(gT.XP(-)) = -2
Z
(~Te'XPA~Te~XP'ejAet)(ai)*(ak)*aJatr
l
and by (2.4.6), 7~
p=l
giving 7~
E,
p=l
The theorem follows.
E
l <_i
9
46
The infinitesimal generators and associated operators
R e m a r k : The last term agrees with the term R(4) in [CFKS87] (on page 260) if ~7 is the Levi-Civita connection, after applying Bianchi's identity. Indeed: in the Levi-Civita case: --
ktji (a ) (a k)* a j a l i,j,k,l=l n -- E lr~i j k l (~a i \)* '~a k ' *) a j a I i,j,k,l=l
=
-
E
[Rijkt -- nkjil(ai)*(ak)*aJa z
l(_i
-- E E [Rijkl -- R k j i l -- R i l k j -t- R k l i j ] ( a i ) * ( a k ) * aJa I. i
However by Bianchi's identity: R i j k l - R k j i l ~-- - - R k i j l = R i k j l , - - R i l k j -1- R k l i j ~- - - R i k l j = R i k j l
and thus R(4) = - 2 E
E Rikjl(ai)*(ak)*aJal"
i
N o t e 3B In terms of the Gaussian field in the non-degenerate case the second term in/~q can be written 2E (a W('))*(aW(2))* aW(3)a W(4) where W ( Z ) , . . . , a W(4) are independent copies of W and (aW(1)) * etc. the corresponding creation and annihilation operators, e.g. (aWO))*r r
are =
W(I) A V).
B. Finally we give a formula for the adjoint of/~q in terms of the vector fields X 1 , . . . , X m, extending those given in [Elw92]. This result is to be used in section 5 for analyzing the dynamical properties of the stochastic flows. Let (1;~q)* : AqTxM -4 AqTzM be given by
L e m m a 2.4.4 Let S[ : M -+ M be the solution flow of the vector fields X i, i = 0 , 1 , . . . , m . Here X ~ is taken to be A. Ifvo E AqTxoM, then .
1
:
(vo)
r:O
-
* (vo).
(2.4.9)
47
On the infinitesimal generator
P r o o f . In fact this follows from It6's formula for the flow of dxt = X ( x t ) o dBt + A(xt)dt,
e.f. [Elw88], if r is a q-form, then
r 1
1 9s T
d2
t=0
From here we conclude the infinitesimal generator on q forms in terms of V is given by:
Ar
= ~ r
~
) +r
Aq TSVvo) r:0
/ lus r t
r=O/
order terms and second order terms, since
dt
Compare this with (2.3.4) to obtain the required (2.4.9).
2.5
[]
On the infinitesimal generator
It is well known that the Laplacian is a symmetric operator. We consider the question of the self-adjointness of the operators A q, coming from a general s.d.e. rather than a gradient Brownian system. We conclude, in the case of a gradient drift, A q minus a zero order term is symmetric provided the associated L-W connection is torsion skew symmetric (Corollary 2.5.6). 2.5.1
Example
There is an important class of examples for which ,4 q is self adjoint although not the de Rham-Hodge (or Witten) Laplacian and here we do not even need to assume regularity of our stochastic differential equation. Let F0 refer to C a sections with compact support. E x a m p l e 2.5.2A. Suppose the X e are Killing fields for a complete Riemannian metric (,)~ on M . Assume A = O. Let # be the volume element for (,)~. Then
The infinitesimal generators and associated operators
48
.Aq with domain Fo( AqT*M) using the measure # and inner product induced by (,)~ is symmetric. P r o o f . Let r ~ 6 F0/k q T*M. Then
M(Ptr162 d# = EL
= /M ( E ( ; r 1 6 2 T x ( t - ) , ~b~(-,..., - ) ) ~ u(dx).
(r162
Let f l , . . . , fn be measurable vector fields forming an orthonormal base for
TzM at each x of M. Our hypothesis implies that ]i(x) : : T(/-~(x)(~ (fi ((/-l(x))),
i = 1 to n,
also gives an orthonormal basis. (It also implies that we have a flow (. of diffeomorphisms either from the fact that we can lift our equation to the, finite dimensional, isometry group of M, (,)~, see [gunS0], or by [Li94b] since T(t consists of isometries.)
fM (Ptr r = E fM ~
dp (Tx(t(ft, (x)),... ,T~(t(feq(x))).
r
~'z (fh (x),..., ftq (x)) #(dx),
r
summed over all gl < g2 < ... < gq,
(fh(~,--1 (Y)),. '',ftq((t -1 (Y))) .(dy)
r
(T~t- 1 -Ie,(Y), . . . , T~t- t - ftq ( y ) ) # ( d y ) ,
since(t preserves#. Now, since A - 0, the law of {~/-1 : 0 < t < oc} on the group of diffeomorphisms of M is the same as that of {~t : 0 < t < oc}, [LJW84], [Nun80], [CE83]. Thus we have
M
= JM
d~
d#
as before. Thus Pt is symmetric on F0(AqTM), consequently so is its generator A q, [RS80].
49
On the infinitesimal generator
2.5.2
Symmetricity
of the generator
fl, q
A. Consider the regular case X : ~ m __+ E C T M for E a subbundle with A E F(E). Recall the definition of trE~.~b(.) in w C. Set
where/~q is as given in (2.4.5). In the following which wilt allow us to deduce symmetricity of A q - g2q from that of A ~ part IV is essentially a finite dimensional specialization of results by Bogachev &Roeckner [BR95] and by H. Long
[Lon]. T h e o r e m 2.5.1 Let q~(x) E (AqT*M; AqT*M) be the zero order term in (2.4.4) and # be a Borel measure on M . Assume (i) the s.d.e. (1.2.5) is regular with A(x) E Ex for each x, (ii) the adjoint semi-connection (7 for (1.2.5) is adapted to a Riemannian metric (,) on T M . Then the following are equivalent: L The generator A ~ with domain restricted to C ~ ( M ; R) is symmetric on L 2 ( T M ) . II. For a given q E {1,2,... ,n - 1}, A q - q~ with domain F0 (AqT*M) is symmetric on the space of L 2_ q-forms using the measure # and the inner product induced on A q T * M by (,). P r o p o s i t i o n 2.5.2 Under the conditions of Theorem 2.5.1, statements I and II are equivalent to each of the following: IlL The adjoint of gradE with domain C ~ ( M ; R) as an operator from L2(M, #; R) to L 2 sections of E using the metric product ( , ) x is given by (gradE)*(U) = --trE~Y.U -- 2 (A, U) x , # almost all x E M , for U E Fo(E). III'. For f E C ~ ( M ; R ) let d E f be the restriction of df to E. Then as an operator from L 2 ( M , # ; R) to L 2 section of E* using ( , ) x the adjoint of dE is given by
d~r = 8r - 2r IV. There exists a 'logarithmic derivative' for # : R m --+ L~
R),
a linear map of R m into the space of measurable functions on M such that for each e E R TM, fMfLx~g
d# = - f M ( L X ~ f +(~(e)f) g d # m
all f , g smooth with compact support for fg, and A = Z j=l
~
(2.5.1)
50
The infinitesimal generators and associated operators
R e m a r k s : (i). If u = e 2hit is an equivalent measure to a Borel measure it, then the adjoint of V with respect to v is given by: V r + 2r ) where is the adjoint using it. (ii). Example 2.3.5 gives an example when A q is symmetric when restricted to closed q-forms, using the given metric on T M and the measure exp ( [ ~
+ 4]h(x)) dx.
In many geometrical situations it is the behaviour on closed forms which is important. B. Here we give the proof of Theorem 2.5.1 and Proposition 2.5.2. We first give three lemmas. L e m m a 2.5.3 Let A1 be a section of E and it be a Borel measure on M . The equation fM (trEV.r
+ 2r
(2.5.2)
holds for all r E Fo(E*) if and only if for all compactly supported q-forms 0 and r in FoL(E, AqT*M).
In particular if the equation (2.5.3) holds for some q E { 1 , . . . , n - 1} it holds for all such q. P r o o f . Clearly (2.5.3) implies (2.5.2). Now assume that (2.5.3) holds for some q. Let r E F0(E*). Take X 1 , . . . , X q to obtain the elements of FoL(E; AqT*M) r
= r
Set 8 ( - ) = (X 1 A . . . A X q , - } ~ . implies (2.5.3) for all q.
(X 1 A . . . A X q , - ) ~ . Then (2.5.3) reduces to (2.5.2) which in turn 9
L e m m a 2.5.4 Let q belong to {1,... , n - 1 } and A1 be a section o r E . Assume ~7 is adapted to a Riemannian metric (,)~ on T M . Let # be a Borel measure on M. Then the adjoint V* : FoL(E; AqT*M) --+ FoAqT*M of ~J, in the sense of the L 2 spaces using it, the inner product (,)~ on T M , (,) on E and corresponding inner product on AqT*M and L(E; AqT*M), is given by ~7"~(-) = - trE~7.r if and only if for all r E
-- 2r
(2.5.4)
to(E*)
/M (trEe r
+ 2 (A1)) ,(dx) = 0.
(2.5.5)
On the infinitesimal generator
51
P r o o f . Let r : T M --~ AqT*M restrict to r on E. consider the 1-form r ~ defined by
For 0 E F/~q T ' M ,
vET~M. Let S j denote the (possibly partial) flow of XJ, for some basis e l , . . . , e m of ~m which we shall choose to be adapted at a point Xo of M in the sense t h a t { e l , . . . , ep} span KerX(xo) • Recall from L e m m a 1.3.4 that Txo Sj is the parallel t r a n s l a t i o n / / , along S~(xo). Therefore for v C TxoM,
Thus by (2.3.1), the definition of (~ :
_~O(xo): Z
~(~o)~
(~(.)(-),0(-))
~
J
d
*
t=o
J
Ixo
j
J
Then for any vector field A1,
fM{_~r - (trE~7.r
2r -- 2r
(x))(-),O) x #(dx).
Thus ~7"r = --trE~7.r
-- 2r
(.))
The infinitesimal generators and associated operators
52
if and only if for all 8,
/M ((~DO"{-2~/)~
(2.5.6)
# ( d x ) = O.
This is (2.5.3). But, by Lemma 2.5.3, (2.5.3) is equivalent to (2.5.5).
9
R e m a r k : In Lemma 2.5.4 we used two metrics and we now show this is essential unless we have torsion skew symmetricity: Let ~ be the adjoint of a metric connection ~. Let 0 be a one form, then
r
= Cvo + o(~(v,-)),
(2.5.7)
and trace~!0(-) = trace~.0(.). So condition (2.5.5) does not change if we use the adjoint ~ replacing ~. Consequently if both connections are metric connections (for the same metric) and the adjoint of one of the covariant derivatives is given in the form of (2.5.4) then so is the other. The converse is also true as seen in the following lemma: L e m m a 2.5.5 Let ~7 be a metric connection with ~7" given by
~7"r = -traee~7.r
- 2r
for 1-forms. Then the adjoint connection has (~7')* given by (~')*r = -trace~7~.r
- 2r
if and only if ~7 is torsion skew symmetric. P r o o f . We only need to prove that if both ~7. and (V')* are given in the prescribed form, then ~ ' must be a metric connection. Without loss of generality assume A = 0. Take ~ C F L ( T M , T ' M ) . Then tr~'r = Z
i
Vx'r
r
+ Z
i
-))"
Thus for 0 a 1-form,
/ =
,, I
On the other hand
=
- (~, t r V . r
+ (8(T(.,-)), r
On the infinitesimM generator
53
giving that for all such 0 and r
Take 0 = (U,-} and r Then
= (V, .} (W, -} for vector fields U, V and W.
(o(~(.,-)), r ?,,J
and
= /M E o(xJ)r
<0, ~ / ~ ) ( x i ) ( T ( X i , - ) ) >
i,j
z,J
So
and V is torsion skew symmetric.
|
P r o o f o f T h e o r e m 2.5.1 and P r o p o s i t i o n 2.5.2.
(1). Assume I. Let f E C~(M; R). Then taking a suitable sequence {g~} in [0, 1]) converging to 1 we have
C~(M;
MA(f)d# = O. i.e.
/M (-~trE~.(gradEf) + (A, gradEf}X) d# = 0. For any A e
C~(M;R)
we see
]M { ltrE~V.(AgradEf) + (A, AgradEf)X d#} 1 (gradEA , gradEf} x d# = /MAA~
(2.5.9)
The infinitesimal generators and associated operators
54
which is symmetric in A and f . Since fM A~
dlz = O. this gives
1
/M (-~trEV.(A gradEf) + (A, A gradEflX) d# =O. Now any U ~ F0(E) has the form N
U(x) = y~ A~(x) gradEfi(x) i~-I
where the Ai and the f i are in C~(M; R); this is true when E = TM using local coordinates and a partition of unity and follows for general E by applying any projection PE : TM --~ E. Thus (2.5.9) holds with g r a d E f replaced by an arbitrary element of F0(E). Replacing it by AU where A E C~(M) and U C F0(E) yields III. Thus I implies III and similarly, using A ~ = - 89 + LA, we see I gives III'. (2). Next assume III and apply it to u(x) = f(x)Xe(x)
for e E R m and f E C~(M;R) to obtain, for any g E C~~
- / M g(z) (df(Xe(x)) + 2 (Y(x)A(x), e)R~ f(x)) d#(z) giving (2.5.1) with
a(e)x = 2 (Y(x)A(x), e)Rm and hence
1
A(x) = ~ Z a(ej)XJ(x). J (3). Thus III gives rise to IV. Similarly III' gives IV by applying III' to the section of E* given by
v
(xe(x), v):
or observe that since (~ = -tr(7. on 1-forms III and III' are essentially the same. (4). Next assume IV. Let r E Fo(E*). Then for each j, if r is a 1-form extending r (taking g - 1)
0 = / ~ {LxJ (r
+
o~(cj)~9(xJ)}
d#
On the infinitesimal generator
55
giving i2.5.5) with m
A = AI= E c~(ej)XJ j:l
By Lemma 2.5.4 and the Weitzenbock formula i2.4.4) we see IV implies II. (4). To show II implies I, take f,g E C~(M;R) and r E F0(AqT*M). Using
A = ~1 ~ Lx~ Lx~ + LA we have (,4(fr
gr
=
M,4(f) g (1r 2 d# + / M fg (,4r162 + 2 / M E g nxJ f (nxJ r r J
d#.
Let
qt(x) E L (AqT*M, AqT*M) be the zero order term of ,4q. Since (ko(fO),gr symmetric in f, g statement I I implies that this is also symmetric in f , g. If (1r 2 = 1 for x E Supp g then
g E LxJ f (LxJ r r J vanishes identically using i2.3.6), the adaptness of ~ to (,)" and the defining property of V. In this case we therefore do have
(`4f, g)L~ = (f,,4g)L~ o0 of M with a partition of unity However we can always find an open cover {U i}i=o { )~i)i=o subordinate to an open { i)i=o and with a q-form r in Fo(AqT*M) ~ U ~ with Ir = 1 for x E Ui. Then for any g E C~(M; 1~),
(,4f,g)L2 = ~ iAf, )~ig)L2 = ~ if,`4i)~ig))L2= i
proving I, and completing the proof.
if,`4g)L 2
i
[]
C. As an application we return to our stochastic differential equation assuming nondegeneracy. Let i, ) be the induced metric. Denote by dx the Riemannian volume measure and V the Levi-Civita connection. Recall that the infinitesimal generator ,40 for the s.d.e, on functions is symmetric if and only if the drift term A X = ~1 ~ T x J X J + A is gradient, e.g. see [IW89]. Thus in the torsion skew symmetric case, since then A X = A, we only need to consider A of the form Vh.
56
The infinitesimal generators and associated operators
C o r o l l a r y 2.5.6 Let h be a smooth ]unction. Consider a nondegenerate s.d.e.: g~, = x ( ~ , ) o d B , +
Vh(x,)gt.
Suppose the associated connection (7 is torsion skew symmetric. Then A q is symmetric with respect to the measure e 2hdx for dx the volume element of the Riemannian metric induced on T M . E x a m p l e 2.5.2C. Let M = G be a Lie group and (1.2.5) a left invariant s.d.e, as in w with #R right invariant Haar measure. Then the hypotheses of the theorem are satisfied if A - 0 with ( )" any right invariant metric which agrees with that given on Eid by X ( i d ) . (This is a special case of Example 2.5.2A.) If left invariant Haar measure #L is used then we must take A to be the
dpL
logarithmic E-derivative of the Radon-Nikodym derivative d#R Finally we observe that if A = gradh.
Then (dA)qVA is symmetric for
all q if and only if ( T ( u , v ) , ~Yh} = O, i.e. V h is ortho.qonal to the image of T. However the /~q term is in general not symmetric. symmetricity o f / ~ 1 the Ricci curvature, is discussed.
In Appendix w the
Chapter 3
Decomposition of noise and filtering T h r o u g h o u t this chapter we will assume that we are in the non-singular situation with a subbundle E of T M and surjective vector bundle homomorphism X : ~m _+ E over M or a mean zero Gaussian field W of sections of E with the non-degeneracy assumption as in w (which in particular allows infinite dimensional noise). In the first case, given a smooth vector field A on M there is the S.D.E. dxt = X ( x t ) o dBt + A ( x t ) d t (3.0.1) as Example C in w We will be interested here in the case where X ( x ) is not injective for each x, in which case some of the noise given by { B t : t >_ 0} is 'redundant' from the point of view of a solution xt = ~t(x0); t >_ 0: we will make this notion precise, giving a decomposition of { B t : t >_ 0} into redundant and relevant noise (see Theorem 3.1.2). This will enable redundant noise to be filtered out in a variety of situations, see w below. We follow very closely the method used in [EY93] for gradient systems. We also describe the corresponding decomposition of { W t : t > 0} in the Gaussian field picture. As an application we show t h a t the adjoint connection is metric with respect to some Riemannian metric for which R i c # is bounded if and only if the derivative process and its inverse are uniformly bounded in t when conditioned on the end point.
3.1
A canonical d e c o m p o s i t i o n of the noise driving a s t o c h a s t i c differential e q u a t i o n
The trivial bundle __Rm is decomposed into the sum of the subbundles k e r X and k e r X • with total spaces ] ] { k e r X ( x ) : x E M } and U { ( k e _ r X ( x ) ) • : x E M } where [_] refers to disjoint unions. Define the connection V on _Rm to be the direct sum connection of the push forward connections induced on k e r X and on ( k e r X ) -L by the orthogonal projections K : _Rm --+ k e r X and K • : _Rm
Decomposition of noise and filtering
58
(kerX) • respectively as in w Given any suitable process or path a in M the corresponding parallel translation ~It will preserve the decomposition of RTM and if e E [ K e r X (a(0))] • then
(e).
= f/,X
(3.1.1)
for the LkW connection ~ on E. Fix x0 in M, and define the following processes:
/~t
f f f
:=
tAp
~ --1
//,
K(x,)dB~
J0
/~t
:=
tAp ~ - 1
/ / s K i ( x ' ) dB~
J0 tAp
~ --1
,/0
where the parallel translations are along the solution {x~ := ~(Xo) : 0 _< s < p} of (3.0.1) defined up to explosion time p := p(xo). These processes are Brown• motions on kerX(xo), (kerX(xo)) • and Exo, respectively, stopped at p(x0), by Levy's characterization. Note that
[~t = X (xo)[~t.
(3.1.2)
We first show that on [0, p) it is the martingale part of the stochastic antidevelopment of {xt : 0 _< t < p} using ~7, (or, in case A does not have image in E, to be more precise using any connection V 1 on T M extending V as in Proposition 2A): L e m m a 3.1.1 Take a Riemannian metric on T M which extends the given one on E, giving the decomposition T M = E e E L. Take any metric connection V • for this metric, on E L and let V 1 be the direct sum connection ~7 + V • on T M , with parallel translations denoted b y / / 1 . The corresponding stochastic anti- development Zt
(//1)--1 o dxs,
:=
0 < t <~ p
(3.1.3)
of x. := ~.(x0) is then given by zt
:=/)t +
/o
0 <_ t < p.
P r o o f . Since the stochastic differentials along (xs : 0 < s < p) satisfy
od ( ( I / D - 1 Z ( x , ) )
=
(//1D-1 o m l X ( x , )
=
(lllJ-lV1X(odx~)(-)
(3.1.4)
A canonical decomposition of the noise driving a stochastic differential equation59
by definition of the covariant differential o D 1 X ( x s ) ( - ) , s e e [Elw88], we see that for0
=
/o
/o
~(//~)-~X(xs)dB~
=
+ =
(//Is)-1 o dxs =
/o
+ -trace
2
Jo
/o
(//I)-lA(xs)ds
(//1.)-IV1X (X(x~)-)(-)as
(//~)-~A(xs)ds
I ~(//~1
= ~t +
(//ls)-lX(xs) o dBs+
+
/oo(//i~)-IA(~)ds
/o (/fs)-~A(x~)&,
since V t X ( X ~ x ) - ) = V X ( X ( x ) - , definition of Bt.
and by the defining properties of V and the 9
Recall that for any stopping time ~- the a-algebra j r _ is defined to be that generated by 9v0 together with the sets {t < 7 } A A for A E ~-t. See [RY91] (4.18) (p.44). For any process {ys : 0 _< s < p} let 9vty be the a-algebra generated by {t < p} together with the elements of the a-algebra on {t < p} generated by y~, s _< t, augmented as usual. Note that then F ~(tAp)_ = 9vtY_. Furthermore if {Tn}nCr is any increasing sequence of ~-Y -stopping times converging to p, then ([RY91], Ex 4.18) ~'Yp- = V 5v~: " (3.1.5) n
When y~ = ~s (xo), we set 9vtu = .T~~, etc. In the situations we will be considering we will have left continuity with ~YT - = ~'~. However we will not need to use this result and will keep to the more intuitively correct notation with the minus sign. The following gives direct analogues of the corresponding decomposition in [EY93]. B. T h e o r e m 3.1.2 (1). 99v~Ap(xo)_ = 5vt~p(xo)_,
0 < t < oo
and Jzfixo) -
xo = Y:~(~o)-" (2). I] l~t = fit + [~t then {[~t : t >_ 0} is a Brownian motion on I~TM stopped at p(xo) with
BtA,(,o) =
f
tAp(.'~O)
//sdBs.
(3.1.6)
JO
In particular {fit : t > 0} and {[lt : t > 0} are orthogonal martingales (and independent Brownian motions when there is no explosion).
60
Decomposition o f noise and filtering
P r o o f . For part (1), it is clear from (3.1.3) and (3.1.4) that ~-B C 5t'~~ The opposite inclusion comes from the fact that the stochastic anti-development is the inverse of the stochastic development and so is essentially known. In detail let O M denote the orthonormal frame bundle for M, with Uo E O M a frame at x0 adapted to the splitting T z o M = E~ o + E x0 • with u0 restricted to ~P • {0} C ~'~ identified with the restriction of X ( x o ) to ( k e r X ( x o ) ) t . For u a frame at x let H~, : T ~ M ~ T u O M be the horizontal lift (using ~71). Let (&s : 0 < s < p} be the horizontal lift of x. starting from uo, so (3.1.7)
d~:~ = H~ X ( x ~ ) o dB~ + H~ A(x~)ds.
Using (3.1.3) and (3.1.4) and parallel translation along {xs : 0 < s < p}:
5CsUo1 o dBs
=
//1s o df3s = / / ~ o dzs - A ( x s ) d s
(3.1.8)
=
odx~ - A ( x ~ ) d s = Z ( x ~ ) o dB~.
(3.1.9)
Thus, if 7r : O M ~ M denotes the projection, 2. satisfies the SDE on M dSc~ = H~ 5:~u~ t o df~s + H~ A (Tr(~)) ds
(3.1.10)
which is driven by {/3s : 0 _< s < co}. Since the explosion time for ~. is p(x0), e.g. see [Elw82] we have that p(xo) is an ~ # -stopping time. To complete the proof of (1) it is enough to observe that, for any s.d.e, with smooth coefficients driven by a continuous martingale M., if {Yt : 0 < t < 7(yo)} is a solution with explosion time T(yo) then 9c ~AT(yo) y C .~-M. tAr(y0)" (This is easily seen by choosing ~-n to be the first exit time ofy. from a ball of radius n about Y0; then r ~ / z 7"(y0) M. and 9rtU,(rn C 5riAT~.) P a r t (2) is immediate.
3.2
C a n o n i c a l d e c o m p o s i t i o n of the G a u s s i a n field
Wt For the corresponding results in the Gaussian form we use the notation of w w We have a splitting H = ker p | (ker p)• of the trivial H-bundle of M with k e r p = U {ker px : x E M} for pz the evaluation m a p at x. The projections K(.) and K • of H onto these subbundles determine connections on the subbundles as in w and hence a direct sum connection V on H . The restriction of V to (kerp) • and the connection V on E are intertwined by p, as therefore are the corresponding parallel translations. There are two main complications when H is infinite dimensional. The first is that the driving process {Wt : t :> 0} is a process on F (E) not on H , whereas
Canonical decomposition of the Gaussian field Wt
61
our connections is on H. To construct the analogues of the processes ft., /3. to decompose W. we will therefore either have to extend the parallel translation in some sense over some space on which W, lies (with no obvious choice for non-compact M) or consider the cylindrical Wiener process {W~ : t > 0} on H , corresponding to W. in the sense that if i : H -+ F(E) is the inclusion, then (as cylindrical processes) W = i W c, t >_ O. The theory of these generalized processes and stochastic integrals with respect to them is now standard, e.g. see [095] and [DPZ92] and we will use this approach. The second point is that in general solutions of linear stochastic differential equations in infinite dimensions, such as the equation for parallel transport given by a connection are not known to have versions which are almost surely linear in their initial conditions. This is more an aesthetic than a serious obstacle. However both this and the potential problems arising from the use of cylindrical processes disappear because of the finite codimensionality of ker p. Indeed if u0 9 ker p(x0) and a : [0, co) -+ M is a smooth curve with a(0) = x0 let ut = /-/tuo for /~/t := ~-It(a) parallel translation along a(t) : t >_ O. By definition of the connection d _ bus Ot - K (a(t)) ~ u t and so, since us = K (a(t)) ut, --=
dt
K(o(t
us=-
K l(~(t
us.
(3.2.1)
The corresponding equation for u0~ 9 (kerp) • is
du~ = ( d K • (a(t)))
(3.2.2)
Now d K • (a(t)) has finite rank for each t and hence is Hilbert-Schmidt, as an operator on H. From this we see t h a t / - / t considered hs a map from H to H will lie in the group 02 (H) which is the intersection of the orthogonal group of H with the 'Fredholm group' GL2(H):
GL2(H) = {T 9 G L ( H ) : T v = v + av where a 9 L ( H ; H ) is Hilbert Schmidt}. Our diffusion on M is given by
dxt = p~ o dWt + ~/(xt)dt as in w The parallel translation (uo,u +) E k e r p ( x 0 ) + (kerp(x0)) • = H along {xt : 0 < t < p} is obtained by solving the analogues of (3.2.1), (3.2.2) in their Stratonovich form. However the e v o l u t i o n / / t can be obtained now as the solution of an equation on the separable Hilbert Lie group O2(H), (or in the standard way by taking the horizontal lift of {xt : 0 <_ t < p} to the principal
Decomposition of noise and filtering
62
bundle of H , taken to be the trivial 02(H) bundle). Thus /~/t(x.(w)) E 02(H) and parallel translation on H , and hence on the subbundle, is almost surely linear. We can follow w above to define
•f
:
J0 t
:
Bt
:
K(zo)
f
_ -i
tAp
c
//~ dW~
dO
0tap/~/;1K• ~-
c
fotA/?;lp.dW,.
Here /~.~ is a cylindrical Wiener process on kerpx0 with ~. = iOc a genuine Wiener process on F ( E ) since
f tap /-/-~1 d W c = W[^p + f t A p )%dW: dO
ao
where A. is a continuous adapted Hilbert-Schmidt operator valued process, so that the stochastic integral gives a genuine continuous process in H. Note t h a t we could also write
~f =
f
tAp
/~/-~IK(xs)dW:
JO
to agree with w above. Since /). and /). take values in (kerPx0) • and Ex o respectively they are finite dimensional. As before we can, and, will write
dW[ =
12,(dZf + dBt)
(3.2.3)
observing t h a t / ~ + jot is a cylindrical Wiener process on H and ft. and JO. are orthogonal martingales. Thus the analogue of Theorem 3.1.2 holds true in this case, and (3.2.3) is our canonical decomposition. R e m a r k : The real point behind this discussion is t h a t since ker p has finite codimension in H it has an induced Fredholm structure: a reduction of its structure group to GL2(H ~176 where H = H ce-p @ H p is a fixed orthogonal splitting of H with d i m H p = p = dimE, [ET70]; and the connection on k e r p is compatible with this structure. To see this, first use the fact that ker p is a subbundle of H to take an open covering {uj : j E T}, say, of M with subbundle trivializations
8j : uj ~ GL(H) such t h a t 0j(x)[ker px] -- H ~176 and Oj(x)[(ker px) • -= H p. A simple argument using the fact that G L ( H ~176 is open in L(H~ H~ allows us to modify the 0j so as to choose them so that Oj (x) differs from the identity m a p of H by
Filtering out redundant noise
63
a finite rank operator; see lET70]. In particular Oj(x) E GL2(H). The restrictions of the 0j(x) to kerp give the Fredholm structure. Because of the special form of these trivializations the fact that the connection is compatible with this Fredholm structure is immediate from (3.2.1) which, as we have observed shows that parallel translates of u0 E ker Pxo differ from u0 by the action of HilbertSchmidt operators. The frame bundle of kerp can be taken to be an 02(H ~176 bundle: in particular a separable Hilbert manifold, so that the horizontal lift of {xt : 0 <_ t < p} to it can be defined as usual, to give a direct version of parallel translation in kerp (which will of course agree with that given by (3.2.1)). From the point of view of the connection the structure group of H is most naturally taken to be 02(H ~-p) • O(H p) C 02(H), but it will not be trivial as such a bundle in general.
3.3
Filtering
out redundant
noise
The following is an abstraction of the method used in [EY93] to take conditional expectations of solutions to certain linear stochastic differential equations with random coefficients. We first prove a general result: Lemma 3.3.1 on filtering of linear stochastic differential equations. It will be used in the proof of the main theorem. By 'filtration' we shall mean a filtration satisfying the usual conditions. Let (~, 5r , (~'t)t>0, P ) be a filtered probability space, and {N~, 0 _< t < cxD} an R pvalued continuous local iT.-martingale. Denote by ff.N the filtration generated by (Nt,t < cx~) and let r be an ~-.N -stopping time. Let M. be an RP-valued continuous local ~.-martingale which can be decomposed as M, = 2~/t + N ~ (3.3.1) where 1~/. is an ~-N-local martingale and N.• is orthogonal to N. (i.e. the quadratic variation dN. | tiN. -L = 0). Assume (vs, 0 < s < ~) is a solution, starting from vo E An , of the equation
dvs = Ps(vs)dM8 + Qs(v~)ds + RsdN~ + rsds,
(3.3.2)
where P: [0,~)• Q : [0, r x fl
R: [0, 0 x and
r : [0,0 •
--+ L(R n, RP; R~), -+ L(RP; R'~), -+ L(RP; R '~) -~ L(RP; R n)
are respectively .P.N, 9v,N , ~'., and iT. progressively measurable processes. It turns out that the equation which governs the conditional expectation of v. is of the same type as (3.3.2):
Decomposition of noise and filtering
64
L e m m a 3.3.1 Assume { Nt } has the predictable representation property. Let T be a j:.g stopping time with 0 < T < ~ such that the stopped processes P~, Q~, R ~, r ~, and v ~ satisfy
1. p r and Q~ are bounded, (so E fo IRsrl2d < N , N > s < oc.
2. R r belongs to s
3. r ~ belongs to s ([0, T) x f~), ~. v ~ belongs
to
s
N/:2(NT) C)s ([0, T) X f~).
Set ~ . , s = E{v~^~ 17 N }.
Then {0~As} is jz.g adapted and satisfies the equation up to time ~dos Oo
= Ps(O~)d~/I~ + Qs(o~)ds + [~sdN~ + ~ d s ,
(3.3.3)
: Vo~
where /~rAs = E{RrAs[gvN}, ~r^s = E{rrAs[J-~ }, and 2Yl is defined in (3.3.1). P r o o f . Take r E L ~ ( f t , ~ - g , P ) . The representation property gives an ~Npredictable 9 : [0, oc) x ~ -+ L(RV; R) with = Er +
/0=
+~(aXs).
Set r = E O + f t Os(dNs). Let ~r^s = E {v~A~12-~'}- Note that since 0. is both an ~'.-martingale and an ~.N martingale ~-)VrAt = E ~ r A t V T A t = ~ ) r A t V t A r = ~ f ) r A t
so that ~At = 0~At. Next by It6's formula, using the orthogonality of N • to N
~)tA~-VtAT
(EC)vo + vo(r +
+ (EC)(v~A~ - Vo)
~s(dNs)vs + JO
ftAr +
- Er
JO 1
+~
Cs(Ps(vs)dM~ + R~dNs) JO
i/t^~ r (Qs(vs) + rs) ds + ~ +o
ftAr
P~(v~)(dlfIs)es(dN~)
Rs(dNs)Os(dN~).
JO
Then ~tAr
VtAr
= - ( I ~ ) v o + voI~t^~ + (1~)Ev~Ar
+ E f ~ "~ 0~ (Q~("~) + ~ ) d~ + l E f t A r Ps(vs)(dMs)~s(dNs) + 8 9 t ^ r R s ( d N s ) ~ s ( d N s ) (3.3.4)
Filtering out redundant noise
65
using (1), (2), (3), (4). Using (1), (2), (3), (4) again, the boundedness of r and the K u n i t a - W a t a n a b e inequality we obtain from (3.3.4) t h a t
]T.,Ct^~-'~t^~- -- - ( E r 1~
+ voECt^r + (EC)Evt^r + E f o ,,tAr
+~l~Jo = ECt^,
1
tAT
tAr
Ps(~8)(di~/ls)~8(dNs) + ~E fo
Vo + fo ^" Ps(O8)d~/Is + f~^" Qs(s
+ fo
~
r (Qs(vs) +
-
r,)
ds Rs(dNs)~s(dNs),
RsdN8 + Jo
rsas, (3.3.5)
whence
vt/~. = vo +
P~(f;~)dlfl~ + JO
Q~(f;s)ds + JO
f4dN~ + JO
~ds. JO
|
To apply the previous results we will first make a general definition. Note first t h a t if p : K -+ M is a vector bundle over M with possibly infinite dimensional, but separable, Banach spaces as fibres, then there are measurable trivializations 8 : K --+ M • E where E is a linear space, with 8x = 81p-l(x ) a continuous linear isomorphism from p - l ( x ) to E. Any two such, 81,82 say, will have 81 o 82 : M x E -+ M x E measurable. Let v : fl --+ K be }'-measurable. Set pov = y : ~ --+ M and let G be some a sub-algebra of ~- containing t h a t generated by y. We can define the conditional expectation of v given ~ by = o
whenever there exists a measurable trivialization 8 and a F-measurable a : l~ --+ (0, co) such t h a t w ~-+ a(~)t?~(~)(v(w)) : 12 --+ E is integrable. From the 6measurability of y. this definition is independent of the choice of suitable trivialization; from standard results it does not depend on the choice of a: see also the proof of the next lemma. The introduction of a is helpful particularly because the trivialization does not necessarily have any relationship to any norm on E (and in practice we will want to use ones which do not). When we have a continuous process {Vt : 0 < t < T} in K over Yt := pVt in M we can consider ourselves to have a random variable with values in the total space of the vector bundle
C([O,T];K) po C([O,T];M) of continuous paths with values in K over these with values in M. When p is smooth and y. is a semi-martingale (starting from a fixed Y0 E M for simplicity) any connection on p gives a parallel t r a n s l a t i o n / / t along the paths of y. and hence a measurable trivialization, almost surely defined for the law of y. :
C([O,T];K) po C([O,T];M) x C ([O,T];p-l(yo))
Decomposition of noise a n d filtering
66
//, v, W h e n discussing conditional expectations for such processes as v. it will be particularly useful to use such a trivialization, especially since in this context we will often want to take a predictable projection for some filtration {Gt : 0 < t < cx~}. In this case we can also use localization in time to further extend the flexibility of this procedure. Moreover we also want to include processes defined only up to some stopping time: Let {yt : 0 < t < r} be a continuous process starting from a point Y0. Let p : K --+ M be a s m o o t h vector bundle over M , possibly infinite dimensional, but separable. Let {Vt : 0 _< t < T} be a process in K over y , i.e. pVt = Yt. Assume T is an St'.y - p r e d i c t a b l e stopping time. Let G be a a - s u b a l g e b r a of 5r containing ~ uT - - "
Definition 3 . 3 . 2 We say that V. has a local conditional expectation with respect to G, denoted by V., a process in K along {y.}, if there exist 1. an affine connection on K (or semi-connection) with parallel translation
l i t : TuoM --+ Tu, M,O <- t < 7" defined along {Yt : 0 < t < 7}, 2. a C-measurable process c~. : [0, T) • f~ ~ R ( > 0), 3. ~.~r_-stopping times {rn : n >_ O} increasing to v such that ~--1
//tA~ atAr. Vt^~. E L 1 (f~,Y:,P;p-l(yo)),
(3.3.6)
and =
(3.3.7)
for all t > O,n = 1,2,.... We will see the local conditional expectation is well defined in the following lemma, in the sense t h a t if there are a n o t h e r set of parallel translations / / , function a ' and stopping times T" increasing to ~- such t h a t the relevant r a n d o m variables are integrable, then the corresponding conditional expectations m u s t be the same:
Proposition 3 . 3 . 3 If (3.3.6) holds there is (up to modification) a unique V. over y. satisfying (3.3.7). Moreover ~ is independent of the choice of the ~, the 7n, and the connection satisfying (3.3.6). P r o o f . For fixed n set
{__1
}
t>0.
For m < n we see ~ ttTrn n ~ Y t m" T h e n Vt is well defined up to equivalence and is similarly seen to be independent of the choice of stopping times {Vn}n~__l satisfying (3.3.6). Suppose now we
67
Filtering out redundant noise ~
!
have another set up 1/c, a!t, Tn, ! n = 1, 2 , . . . as in the definition and defining 12.!
by the analogue of (a.3.7). Set ~a' = rn ^ r
Since
//,^,.Z
//~^,.,: is ~'~_
measurable, we see, from above,
,
_,
_,
t^,.. ,, v,^,.., = I h^~., E
,^,..,
ash,-:, Vt^,-., Ig
}
I-/:^,.-, ~.{ (Ih^~-,) I1,^,.,,//,^,..,~,^.:v,^,.., ~^,.t,
)
//~^,.'.'E{//~^,.*'~^,.zv~^rt'IG}
!
a tAr,~,, Vt^r-,. from above and (3.3.7). Thus 1~' = 1~. as required.
I
C o r o l l a r y 3.3.4 Suppose v = oo. If for some R i e m a n n i a n or Finsler metric on K , IIVtllyt E L 1 each t, then the local conditional expectation ezists and is j u s t the conditional expectation in the sense of (3.3. 7).
C o r o l l a r y 3.3.5 With the notation above, suppose that K , as above, has a local conditional expectation, V. over y.. Let Ct : 0 < t < 7- be a G-measurable process over {y.} in the dual bundle K*. If Ct(Vt)Xt
E {r
16} =
=
r
Thus Esgn(r (~))r
= E let (?t)lXt<,.~
Hence by the monotone convergence theorem Ct(~)Xt<,. is integrable. Since
for each n the result follows.
[]
Remarks: (i). The independence from the choice of connections was originally pointed out by Emery.
Decomposition of noise and filtering
68
(ii). We are not assuming any Riemannian structure, and the connections are not necessarily metric for any metric. C. The following generalizes the main result of [EY93]: we consider first a s.d.e. (3.0.1) with solution (partial) flow {~t(x0) : 0 < t < p(xo),Xo E M}, explosion time p : M • f~ ~ (0, c~], and derivative (partial) flow T~o~t : Tzo M --+ T M , xo E M. with the same explosion time. Assume the non-singularity condition that we have the subbundle E of T M . Recall the definition (2.4.2), (2.4.5) of the Weitzenbock curvature for the L-W connection, /~q : AqT, M --~ AqTxM from w First we have a crucial lemma; for the case A is not in F(E) see (3.3.14) below. L e m m a 3 . 3 . 6 Let 1 <_ q < n. Assume A(x) E E~ for each x E M. Let Vo E AqTzoM and Vt = AqT~t(Vo). Then Vt is a solution of the following equation (in It5 form):
J~Vt = dn q ( V X (-)dJ~t) (Vt)-~(Rq,)* (Vt )dt +dAqVA(Vt )dt,
o < t < p(zo).
(3.3.8) P r o o f . The process V: satisfies the equation
9v~ =
dA q /~ V X ( - ) o d B t ) " (Vt) + dA qfTA(Vt )dt,
0 <_t < p(xo),
(3.3.9)
which in It6 form (using V) is
D V t = dh q ( ~ X ( - ) d Y t )
v
(Yt)- -~(Rqt)* (Vt)dt+dAq~A(Vt)dt,
o < t < p(zo),
(3.3.10) since the It6 correction term for dA q ( ~ T X ( - ) o d B t ) ( V t ) i s :
2
2 =
i=l
1
- ~ (/~~ (Vt)dt,
Here we used the notation from Appendix B, (2.4.7) in w before (2.4.7).
and the calculations 9
Filtering out redundant noise
69
3.3.7 Let 1 < q <_ n. Assume A(x) E Ez for each x E M . Then ~o the local conditional expectation of {Vt : 0 < t < p(x0)} with respect to .7:p(xo)_ exists and is equal to the solution {Vt : 0 < t < p(xo)} to
Theorem
^
_
DVt
Ot -
1 2 ( / ~ ' ) * ( ~ ) +dAqfYA(Vt)'
0 < t < p(xo)
(3.3.11)
where xt = ~t (Xo). P r o o f . Using any complete metric on M let vn be the first exit time of x. from the ball radius n a b o u t Xo, n = 1, 2 .... Since these balls are compact (3.3.6) holds with at - 1 for a n y / / . . Now, with the notation of w 7X(.)dBsAp(
)
=
=
+ d[7,)
=
by the defining property of V, since f i s t s is orthogonal to K e r X ( x s ) for each ^
s. U s e / / t
--1
to pull (3.3.8) back to AqTyo M. We then have an equation of the
form (3.3.2) for vs = /-/[lvs, with R = 0 and Ms = ~s. Taking N = /7. and 7 = Tn in L e m m a 3.3.1, since a stopped Brown• motion has the predictable representation property and ~'p(~o)- = "T~(~0)- by Theorem 3.1.2, we see
Pt
o)--
for ~ given by (3.3.11), as required.
3.3.1
II
W h e n A d o e s n o t belong to the image of X
In the next result we give a version of (3.3.9) without the assumption t h a t A(x) E E~ and also translate (3.3.9) into an equation using a Levi-Civita connection. In it we keep the non-singularity condition, and the notation, of w above. We take any Riemannian metric on M and let V be its Levi-Civita connection, with corresponding covariant differentiation D along xt = ~t(Xo), 0 < t < ~(Xo). We also let E • be the orthogonal bundle to E in T M with V• a metric connection on E • and as in w set V 1 = V + V • to get a connection on T M with adjoint V 1' say. 3.3.8 For lit = AqT~t(Vo) as in Theorem 3.3.7, the local conditional •o expectation Vt, 0 < t < p(xo), with respect to Jr~(~o), is adapted to 5~.z~ and satisfies
Theorem
Ot
-
1 2 (/~q')*(~) + d h q ( V ' l A ) ( ~ )
(3.3.12)
Decomposition of noise and filtering
70
or equivalently
(3.3.13)
for l i t and Bt as in w P r o o f . We have DI'Vt
=
dA q (W}X o dBt) (Vt) + dA q (V}A) Vtdt
o dB )
dA q
+
dAq(V} A ) Vtdt
by the calculations in the proof of Lemma 3.3.6, where the It6 equation is with respect to ~71. As in the proof of Theorem 3.3.7 this gives
from which the local conditional expectation {Vt satisfies (3.3.12). To obtain this result in terms of X • : R t -+ E • which induces V • and set X 1 = so that X 1 induces V 1. By (C.8) in Proposition T M --+ R_t @ R_TM is the adjoint of X 1 we have
D " ~ = D ~ - d a q (VZ ~
: 0 < t < p(xo)} exists and V it is convenient to take an X ~ + X : R e q) R TM -+ T M , C.3, in Appendix C, if y1 .
(~)
with Z ~ = Xl(.)Yl(xt)(odxt). By (3.1.4)
zodxt
~. xl(.)Yl(:gt) (//~ odUt-- ~-A(xt)d~) =
Thus WZ ~
X ( ' ) Y ( x t ) / / t o df~t + zA(~')dt
= V , X ( Y ( x t ) / / t o d B t ) + vzA(~*)dt. Since, by w
1 = V v A - ~v ZA(y), V~,A
if v E TuM, y E M
equation (3.3.13) follows. We will let
q,A
W~,xo : AqTzoM ~ AqT~M,
0 < t < p(xo)
(3.3.15)
Filtering out redundant noise
71
be the solution map determined by (3.3.11); and that given by (3.3.12) will be 1 q,A
written Wt,~o. We will also let W ~ 0 denote the local conditional expectation of AqTxo~t with respect to ~-~o p(xo)-" For our regular s.d.e, define ]D
Ot -
~:)A,q
Ot
to be the operator on q-vector fields Vt : 0 _< t < r ~(x0)} defined 15y ID Ot V t -
along {(t(x0) : 0 _< t <
DI'Vt 1 (/~q ~*lp cgt + 2 ' ~'~ v t - d A q ( V ! A ) ( V t )
(3.3.16)
for V 1 as above. P r o p o s i t i o n 3.3.9 The operator ~ and its related modified parallel transport 1
operator W ~ 2 are independent of the choice of E • and V z used to define V 1. 1
P r o o f . Clearly W ~ 2 depends only on our s.d.e, since it is just Wtq~o. However
ohVt=w:2
[w:,r -iv,
In general b-7 ~ will depend on V • if considered as an operator on arbitrary smooth paths in M. For example if a(t) satisfies ~(t) = X ( a ( t ) ) a ( t ) for some continuous paths a and if V is a vector field on M, setting V(t) = V(a(t)) D 11
Ot V(t) _ VI/(t)A = V~(t)V _ Vv(t)A, 1 1 (Since R~c depends only on by Proposition 1.3.1, which will depend on Vv(t)A. ~7 it cannot help to make ~7 ~) intrinsic.) However if h(t) = X ( a ( t ) ) a ( t ) + A(a(t)) and V(t) = V(a(t)) then D1 ~ Ot V(t) - V I v ( t ) A
^
11
~x(a(t))~(t)V -t- "~ A(a(t))V -- V v ( t ) A
=
Vx((,(t))(~(t)V + [A,V](a(t))
^
which is intrinsic.
1
=
72
Decomposition of noise and filtering
3.3.2
The
inverse
derivative
flow
In the regular case the local conditional expectation E { iq(Txo~t) -1 [.~-'xo } is just (ulq,~ Indeed using the notation of Theorem 3.3.8 to allow A(x) f[Ex. ,r t,xo ~-1 ] .
D 1' Aq(T~t) -1 = - A q ( T ( t ) -1 o D 1' (Aq(T~t)) Aq(T~t) -1
whence, by (3.3.14) D it Aq(Txo~t) -1
=
-Aq(Txo~t)-ldA q (~7.X//dBt) + 2Aq(Txo~t) -1 (Rq )*dt - A q (T(t)
dAq(V! A)dt
from which, as before, the local conditional expectation Zt : 0 <_ t < p(xo) is defined and satisfies 1 D 1' Zt "Or -- ~Z -R (tq*)t2
(3.3.17)
- Zt d A q ( V I A )
1 q,A which is just the equation for ( W t )-1.
3.3.3
Integrability
of certain
C r norms
for compact
M
For further reference we quote the following result of Kifer with a corollary obtained as an application of the filtering result given above. Kifer's proof was an elegant and quick application of Baxendale's integrability theorem [Bax84]. The results could also be obtained by the direct inductive proof in [Nor86] given for certain non-compact situations. For r = 0, 1 , . . . , and a Riemannian metric on M define
IT~tlc.
= sup I(VrT~)I~, xcM
where V is a connection on T M and T~t is treated as a section of the bundle L ( T M ; ~ ( T M ) ) over M given its induced connection. By compactness different metrics and connections will give equivalent norms and equivalent norms would also be obtained using local coordinates systems as in [Kif88].
Proposition 3.3.10 [Kif88] For compact M and r = O, 1 , . . . , both sup IT~tlc ~ 0
and sup I T ~ l l c . lie in f~P for 1 < p < 00, all T > O. O
P r o p o s i t i o n 3.3.11 For a regular stochastic differential equation on a compact 1 q,A Riemannian manifold M let Wt,xo be given by (3.3.12), xo E M . Then both 1 q'A I sup sup Wt,xol
0<s
x0
and
f 1 q,A~-I sup sup 0 < s < T xo ~Wt'x~
73
Filtering out redundant noise
lie in s for 1 < p < 0% all T > O. If f7 is adapted to some metric on M they both lie in s ]or each T > O.
P r o o f . The case of 1 < p < c~ follows from the previous proposition and w167 3.3.2 above together with Jensen's inequality for conditional expectations. When V is adapted to a Riemannian metric (,)' choose such a metric in Theorem 3.3.8. Then by Proposition 1.3.5 the Levi-Civita covariant derivative VX(e) satisfies (V~,X(e),v)' = -(V,X(e),u)'~,
any u,v E T ~ M , x E M , e E K e r •
2 If we apply It6's formula to [ t[x, in (3.3.13) we see that the It6 stochastic 1 q,A
integral vanishes and a n / 2 ~ bound for W follows by compactness of M. The same argument shows that there exists c E 1~ such that for all v0 E AqT, oM , all Xo E M and 0 < t < T
1 q,A Wt,zo
(Vo) ,2x
>
e
cTxT
2
v o xo
(, From this we obtain a uniform bound on
a.s..
q,A -I
Wt,~oJ ]
,O
xEM.
A more precise result, obtained by the argument above, gives the key estimate for semigroup domination: P r o p o s i t i o n 3.3.12 For general M with a regular s.d.e, suppose ~Y is adapted to a metric (,)'. Set
<
pq(x) = inf{ /~q(V) - 2 d A q ( V A ) ( V ) , V
>'
: Y ~ AqrxM,
Ivl' ___ 1}
x
where V is the Levi-Civita connection for (,)'. Then 1 q,A
I w~,,o (Vo)l:, < e
-89
oq(~)as ~T ,
I~Ot,o
all Vo E AqTzo M .
3.3.4
The semigroup theorems
on
forms:
Bochner
type
vanishing
For any stochastic flow the 'semi-group' on q-forms P t r E ~ (r162 was considered in w From Corollary 3.3,5 and Theorem 3.3.7 we know that /~ 1 q,A (Vo)) ~(~
(3.3.18)
74
Decomposition of noise and filtering
whenever r is a q-form for which E ~ (r exists, Vo E T~oM. In case ~7 is adapted to some metric and pq : M --+ R, defined in Proposition 3.3.12, is bounded below (i.e. R q - l d A q ( V A ) is bounded below) this shows that Ptr has an 'extension' to a semigroup on bounded measurable forms defined by the right hand side of (3.3.18): which we will also denote by Pt. For future reference note 1 q,A
It6's formula for 0 ( ~ ) when ~ =Wt,=o (V0): = r
+ f to ~7r (3.3.19)
+ J3 Aq(r for X7 a Levi-Civita connection. P r o p o s i t i o n 3.3.13 Assume nonexplosion. Suppose X7 is adapted to some metric (,)t and the corresponding pq : M --~ ]~ is bounded below and satisfies lim sup sup Ee - ~ fg' Pq(=~)ds Xt
(3.3.20)
x~U
for all compact sets U and K of M . Then IPtr --+ 0 as t --~ oc for all ] - ['bounded q-forms r uniformly on compact subsets of M . In particular, if also there is no explosion, there are no non-zero bounded C 2 forms r with Aqr = O. If further pq-1 is bounded below and dPtO = PtdO for all compactly supported ( q - 1)-forms then every closed compactly supported q-form vanishes in De Rham cohomology. P r o o f . The convergence to 0 is immediate from (3.3.18) and Proposition 3.3.12 which gives the semigroup domination
EIr
IPtr _<
sup 1r
Xt
(3.3.21)
xEM
The vanishing of Aq-harmonic bounded forms then follows from the fact (which comes from It6's formula (3.3.19) and (3.3.21)) that Aqr = 0 for r bounded implies Ptr = r Just as in the proof of Proposition 2.4.1 the commutativity of d and Pt implies that Ptr is cohomologous to r when de = 0 using (3.3.19). However if a is a closed q-simplex and r a closed compactly supported q-form the decay of Ptr implies the decay of f~ Ptr Thus f~ r = 0 and so by De Rham's theorem [r = 0 in Hq(M; ~). 9 C o r o l l a r y 3.3.14 Suppose M is compact and X7 is adapted to a metric for which satisfies (3.3.20). Then Hq(M; ~) = O. In particular H q ( M ; ~) = 0 if there is a subbundle E of T M and a metric connection ~7 on E with ~7 adapted to some metric for which Rq is positive.
"]~q
Filtering out redundant noise
75
P r o o f . For M compact Ptr E ( ; r for bounded r and dPt = Ptd follows by differentiating under the expectation using Theorem 3.3.8. The first part then follows from the proposition and the second also using Theorem 1.1.2 to know that ~7 is the L-W connection for some s.d.e.. 9 For various versions of such results and their consequences when V is the Levi-Civita connection on T M see [ER88] and [ER91]. For relationships between Ric and R~c see Corollary C.7 of the Appendix and Remark (ii) following it. 3.3.5
Bismut
formulae
The simplest application of the filtering procedure is to obtain Bismut formulae (in fact [EY93] originated from considering (3.3.23) below). We briefly describe a simple case, assuming that M is compact and that we have a regular s.d.e. (1.2.5) such that E is integrable and A E F(E). Then E is preserved by T~t. If f : M --+ ~ is C 2 and T > 0, ItS's formula for PT-tf(~t(xo)) gives /.
T
f(XT) = PTf(Xo) + ]0 d(PT-sf)Z(xs)dBs.
(3.3.22)
Multiply both sides by f : (Y(xs)T~8(vo), X(xs)dBs)~o where v0 E E~ o and 0 _< c~ < ~ < T, and taking expectations giving
Ef(xT) =
/:
(Y(xs)T~s(vo), X(xs)dBs)xo = E
d(PT-sf)(T~s(Vo))ds
d(P~(Pr-sI))(vo)ds
by differentiating under the expectation sign. Thus if vo E E~ o
d(Prf)(vo) - ~ - aEf(XT)
(T(s(Vo),X(xs)dBs)
(3.3.23)
OL
and so by Theorem 3.3.7,
since X (xs)dBs =/~/sdb,. Formulae (3.3.24) now extends by continuity to continuous f : M -~ II~ and exhibits the smoothing properties of PT along the leaves of our foliation . From it come formulae for the logarithmic gradient of the heat kernel, proved for Brownian motions and the Levi-Civita connection by Bismut [Bis84]. For this, variations, and non-compact cases, see [EL94], [TW98] and [SZ96]. It is a primitive form of integration by parts formula like (4.1.2) below and can be proved from it (and implies it in the integrable case, as in [EL96]). Similarly (3.3.22) is an explicit form of the Clark-Ocone formula (4.1.3) below.
Chapter 5
Stability of stochastic dynamical systems A. Consider SDE (3.0.1). Let {~t} be the solution flow and Txo~t : TxoM T x o M the derivative flow for ~t(Xo). For Vo 6 T~oM, the almost sure limit l i m t ~ log IT~t(vo)l, called the sample Lyapunov exponent, describes the rates of convergence or divergence of solutions initiated from nearby points. We are also interested in the moment stability determined by the moment exponents:
PK(P)
=
lim sup I log sup Elrx~t[ p t-~oo
~
(5.0.1)
xE K
for a subset K of M. The system (3.0.1) is strongly pth-moment stable if #K (P) < (x~ for all compact sets K . It is pth moment stable if #x(P) - #{x}(P) < 0 for all x, pth moment unstable if #x (p) _> 0 for every x. Under suitable hypoelliptic conditions for compact manifolds, tL~(p) is independent of x [BS88]. See e.g.
[ElwSS]. There are generalizations of the moment exponents to q-vectors: # q ( p ) = lim sup 1 log sup E I Aq T~tl p t--+oo
t
(5.0.2)
xEK
with the related concept of (q,p)-moment stability. We shall apply the technique of filtering to obtain estimates on the moment exponents, extending that in Li [Li94a] for gradient systems, (with corresponding homotopy vanishing result extending Elworthy-Rosenberg [ER96]). We also use the L-W connection to give a neat form to a Carverhill's version of Khasminskii's formula, and show t h a t in certain situations an L ~ condition on the derivative flow implies t h a t V is metric form some metric on M. B. Assume t h a t X has constant rank and image E C T M with V the associated L-W connection for X. Write E = Ira(X) and define H i : AqE --+ AqE by
Stability of stochastic dynamical systems
88
Hq(V, V) =
Ei~=, F~ldAq(VXi)(V)t '2 t2
(5.0.3) '
((dAqVA)(V),V)
'2
Let Pq be the set of primitive vectors in AqEx and set
hap(x) = inf{pHq(V, V): V 6 7~q, IV]' = 1} and
hap(x) = sup{pHq(V,V): V 6 T~q, IV I' = 1}. Then, T h e o r e m 5.0.5 Assume the stochastic differential equation does not explode, A(x) 6 E~ for each x, and that ~7 is metric with respect to a metric (,)' on T M . Then for p 6 R, 17o 6 "Pqo and Vt = AqT~t(Vo), ]Vo]'P E exp ( ~ ~othq(x,)ds)<E]Vt]'P<]Vo]'PE exp ( 1 fothap(xs)ds). (5.0.4)
Here the norm ]-]' corresponds to the metric (,)'. P r o o f . First note that by (3.3.9) DVt 6 AqE. Also AqT~t maps primitive vectors to primitive vectors. By It6's formula and Lemma 3.3.6, =
m
( ~)}
r
(5.0.5)
+~ fo~ lVsl":'H~(V,,, V~)ds, where /?q(v, v ) =
Ei~=l N~]dAq(~YXi)(V)[ ''2
+ ( p - ) E,_
"
v, aa~(~Xi)(v)
(5.o.6)
Then
HI(V, V) =/}pq (V, V) + 2 Let W, = T ~
fo ((AqVAl(vl,v)'ds. IVI,2
(5.0.7)
be the process projected on the unit sphere S(AqTM) and
"~ fot<W,,(dAq(gxi)(Ws)}'dB~. Then set Mt = p ~i=l
89
1 (M} t -t- pro IVtl 'p ---IVol 'p exp ( Mt - -~ ~ t Hq(Ws, Ws)ds ) .
(5.0.8)
Let (xt, ~ ) be the solution and the derivative process of the stochastic differential equation
d~t = X(~t) o d[~t + A(2t)dt where
[~ = B~ - p
Ws, (dAq)VXi(-))(Ws)
/' ds.
Then x. and ~. have the same distribution from the defining property of ~7. By the Girsanov-Cameron-Martin theorem, if 11d8 = IV,l' '
and so it follows that
E exp (~ fo
_<E[~]_<E'Vt"P e x p ( ~ f 0 t h q ( x s ) d s )
"
Remarks : (i). The quantity V X ~ which appears here and below is a generalization of the shape operator of a submanifold of RTM. Indeed in the gradient system case
where a is the second fundamental form of the embedding determining X. See [Li94a], [ER96] for geometric implications of Corollary 5.0.6 in the gradient case. (ii). Fix x E M and let (,) be an inner product on TzM extending that given by our s.d.e, on Ez. Let { e l , . . . ,en} be an orthonormal base for T=oM, (, }. As observed for the Levi-Civita connection in [ER96], see also [Ros97], the Weitzenbock curvatures acting on primitive vectors V = el A . . . A eq, say, satisfy q
j=l
)~
I>_q-F1
w h e r e / ( is the sectional curvature defined by
(TY-~(vAw),vAw}
= (= 0
K(v,w) unless
v, w E E= )
Stability of stochastic dynamical systems
90
for 7~ : AqTM --+ AqEx the curvature operator and {e~,..., e~} together with Ex 7) { e l , . . . , eq} gives an orthonormal base for Ex. P r o o f . First by Corollaries B.2 and C.5 of the Appendix,
r=l
:-2~
E r=l
= 2
(V~J X~ A V ~ X ~ ' e i A e t }
(ajakV, ataiV)
i
l<_j
where ~ E refers to summation only over these j, k with ej, ek in Ex. On the other hand
((dAq)R(X r, -)XrV, V) = r=l
m
q
r=l
j=l
yr(_~ldim(E) The result follows by choose It.~ ~x)tT= 1 to be the given orthonormal base for Ex. 9
The following corollary is immediate using [ER96] for the last part. C o r o l l a r y 5.0.6 Assume the stochastic differential equation has no explosion
and Suppose that ~7 is metric with respect to a metric (,)' on T M . Then the stochastic dynamical system is strongly (q,p)-moment stable, with respect to the metric (,)', if limsup 1 sup logE exp t--+c~
-t x E K
(tfo
h~(~8(x))ds
)
<: 0,
all K compact.
It is not strongly (q,p)-moment stable if h_q > O. In particular if limsup sup t1 log Eexp ( ~ f 0 t h~(~8(x))ds t--*~ xeM
)
<0
and M is compact then the homology group Hq(M, Z) vanishes.
R e m a r k s : (i). In fact we have a more general formula than (5.0.5): Let be a semi-connection, metric with respect to (,)~. Write X ~ =- A for simplicity
91
and denote by Si(t,x) the flow of the vector fields X i, and TSi(t,x)(v) derivative flow. Then
IVrl
=
IVol'" + :g,
for lV,
Pro
IV, I ''-2
the
v,,,
V,)d,, (5.0.9)
where
[iq(v, v) = E,~=I F ~ ] D AqTS~(V)I~=ol '2 .
12
1 + ( P - 2 ) E l rn lV~ Iv, ~D A q TSt(i V )lt=o / -2
-
(5.0.10)
t
This is deduced by an It6's formula in Elworthy ~Elw88], see also ElworthyRosenberg [ER96] for gradient systems. Now take ~' = V and observe that
b -~TSi(t,x)(v)
= VXi(Si(t,x)) (TSi(t,x)(v)) - ~ (Xi(Si(t,x)), TSi(t,x)(v)) = V X i (Si(t,x)) (TS'(t,x)(v)),
and similarly ^
~--~(AqTS~(t, x)(Y)) = (dAq~TZ i) (AaTSi(t, x)(Y)) .
(5.0.11)
Formula (5.0.5) now follows from Lemma 2.4.4, and (5.0.11). (ii). If A r E, let V 1 be any extension of ~ as in w and V 1' be its adjoint. Suppose ~711 is metric with respect to a metric (,) 'on TM. Then the above result holds with Hpq replaced by
gq(v, V) = [-I~(V, V) + 2 fo' (((dAq)VIA) iVi, 2 (V), V ) ' ds.
C. From Theorem 3.3.8 and (3.3.17) we see that in the nonsingular case with M compact if ~7 is adapted to some Pdemannian metric on M then the conditional expectations E{ Tx0 ~ I ~ o ) and E{ Tzo ~t-11 ~-~o) are bounded processes in (t,~o) E [0, T] • ~, any T > 0. This boundedness can be with respect to any Riemannian metric on M, by compactness. In particular in the nondegenerate case these two processes conditioned on ~T(xo) ----Yo, will also be almost surely bounded on [0, T] x ~t, any Y0 E M. (As described in the proof of theorem below we can make sense of this for all, not just almost all Y0.) The following partial converse to this shows that the existence or not of a metric to which V is adapted is reflected rather drastically in the behaviour of the flow.
92
Stability of stochastic dynamical systems
T h e o r e m 5.0.7 Suppose our s.d.e (3.0.1) is nondegenerate and has no explosion. Fix xo C M and T > O. Let {~t(x0) : 0 < t < oc} be the solution from xo, with {Txo~t : t > 0} the derivative process at xo. Suppose from Yo E M the local conditional expectations E{Txo~tl~ x~
and E { T x o ~ t l ] ~ z ~
0
when conditioned on ~T(xo) = Yo are in L ~ uniformly in t E [0, T], for some Riemannian metric on M for which 1~c # - V A is bounded. Then #7 is a metric connection for some metric on M .
N o t e : The conditioned process are the same in law as the conditional expectations, given the bridge process, of the processes T~o~t and T xo~t-1 , respectively, conditioned on ~T(xo) = Yo. (Their expectations at time T are E{ Tx o ~t I ~T(X0) = x0} and E{ T~o ~ t l l ~T(Xo) = X0}. ) P r o o f . By Theorem 3.3.7 and (3.3.17) the local conditional expectations are given by Wt and Wt-1 respectively. From [Car88], for example, the corresponding conditioned processes {Wuo,t : 0 < t < T}, {W~olt : 0 < t < T} are given by the equations (3.3.11) and (3.3.17), for q = 0, but with xt = ~t(x0) replaced by the bridge, ~yo,t(xo), i.e. ~.(x0) conditioned to be at Y0 at time T. From these equations they are seen to be mutually inverse. Let { / / t : 0 < t < T} be parallel translation along the bridge using XT. We have
d ( W y--1 o j J /^t )
= 1
1 I~C#//t
-
WyooltCA//t
-1 ^ is in LOO uniformly in t E [0, T] for the given metric. From Whence Wyo,t//t this our assumption on Wt implies that { / / t : 0 < t < T} is in L ~ . The result follows from Theorem 1.3.8. 9 D. Finally we give the following Khasminski formula; c.f. [Car85], [Elw88], P r o p o s i t i o n 5.0.8 Suppose M is compact and our s.d.e, is nondegenerate and X7 is metric with respect to a metric (,)'. Let S q ( T M ) be the unit sphere subbundle of A q T M , v an ergodic invariant measure for the process induced on S q ( T M ) , and Xn < ... < )~1 the corresponding Lyapunov exponents of the solution flow. Then for some choice of v,
dv
+.=_ L(TM) It
+-~
~
1
I ( d A ) a ( V X i ) ( V ) l '2dr
(TM) i=l ,
2
(TM)
93
P r o o f . From [Car85], there exists an ergodic v such t h a t )~1 + . . . + )~q = limt-~o~ ~ log IVt(w)l for v * P almost all (Vo,w) in Sq(TM) x 12. For such V0 we apply It6's formula to log IVt I' and use (5.0.3)
log IV~l' = log IVol' + = log IVol +
f
t (]Vs[') -1
E1
l ~ot d'V~"dlVs"
2
]Vsl'2
'dBi lYsl' ds
Iv l t
+ 2 ~o ~ ~ i=l 2
[v~l'
]o
2 Jo 1
~ot dlV,' '
i=1 ~
+7l for ~
'dAq(~yXi)(Vs)''2ds
1
ds
1 'ds
i=-i ~ - ~
=
Ws,(dAqVXi)Ws dBis
log IVol + 1
-
W~,(dAq~Txi)w~
ds + -~
1
+2
Z ]dAq(~YXi)(Ws)]'2ds i=1
Ws,-(nq)*(W,)
>,ds.
Taking the limit and by the standard ergodic theorem we have the result.
9
5.0.9 Suppose X is injective, and (7 is adapted to some metric on M and A(x) EEx for each x with (TA - 0 (or more generally A is a Killing field for that metric). Then all the Lyapunov exponents vanish. Proposition
P r o o f . In the non-degenerate case with VA - 0 this is immediate from Proposition 5.0.8 since all terms in the integrand there vanish. The general case comes from Corollary 1.3.6 which implies that the flows ~t will consist of isometries. Note t h a t V A = 0 implies that A(x) = Z?=I ~ some a j E 1~ and hence that A is a Killing field if each X j is. 9 R e m a r k s (i). Under certain hypoellipticity condition on the stochastic differential equation and its derivative flow Baxendale showed in [Bax86] t h a t vanishing of all the Lyapunov exponents implies t h a t the flow consists of isometries for some Riemannian metric on M. He also showed that equality of the exponents holds if and only if there is a metric on M for which the flow consists of conformal transformations. See also [BS88].
94
Stability of stochastic dynamical systems
(ii). We do not have examples satisfying the hypothesis of the proposition other than left or right invariant systems on Lie groups.
Chapter 6
Appendices A
Universal Connections
as L - W c o n n e c t i o n
A . Let ~r : E -4 M be a vector bundle over a manifold M with a surjective bundle m a p X from the trivial bundle 7to : M x ]1~m -4 M, to 71". T h e adjoint of the bundle m a p is denoted by Y : E* -4 M x I~m. We give E the induced metric so X Y : E* -4 E is an isometry, and use the metric to identify E* with E , so 7r can be considered as a subbundle of 7to via Y. T h e fibre of E above x will be written Ex, and similarly for other bundles. Let O(~r) be the bundle of o r t h o n o r m a l frames of zr with structure g r o u p
O(p) for p = d i m Ex, O(Tro) the principal bundle O(~r0) : M x O(m) -4 M with s t r u c t u r e g r o u p O(m). T h e a d a p t e d frame bundle O(Tr0,Tr) : O(M; M ) -4 M is the bundle of a d a p t e d frames in O(z~0), i.e. O(M; M ) x consists of frames (x, u), u E O(m) such t h a t u (R p • O) --~ Y ( E ~ ) .
M xO(m)
O(M,M)
,
h
, O(E)
O(p)
O(p) x O ( m - p)
M
,
id
,
M
,
id
.M
B . Denote by Wo the trivial connection form on O(Tr0), i.e. wo(w,v) = TL-~lv = a - i v for each (w,v) E T(x,a)(M x O(m)). T h e induced connection Woo on O(zr, 7r0) is the restriction of wo to O(Tr, Tro) followed by the projection o n t o the Lie algebra o(p) x o(m - p ) . By Proposition 1.2 of volume II a n d P r o p o sition 6.4 of volume I of K o b a y a s h i and N o m i z u ([KN69a],[KN69b]) we see t h a t
Appendices
96
woo is a connection because ad(O(p) • O ( m - p ) sends the complement o(p, m - p ) of o(p) + o(m - p) to itself. Define
h: O(~'o,7c) u
~
O(~r),
by
~
X(x) oulR~.
Here we are using u for both the frame and its principal part. Let w be the unique connection on O(Ir) satisfying (c.f.p.79 of Kobayashi-Nomizu [KN69a]): 1. It is the only connection such that the horizontal subspaces of Woo are m a p p e d into the horizontal subspaces of w by h.
2. h*w = h. woo, where h . Woo is the o(p) component of woo. This unique connection w on O(Tr) (related to X ) is the universal connection in the sense of Narisimhan and R a m a n a n when M is the Grassmann manifold and 7r the universal bundle, as is discussed later. C. With no loss of generality we assume Y ( E x ) = ll~p • 0. Let a be a curve with a(0) = x and &(0) = v, with horizontal lifts in O(~ro), O(Tro,~r) and O(Tr) respectively ~o(t) = (a(t), 5~(t)), ~oo(t) = (a(t), ~oio(t)), and ~(t) = ( a ( t ) , ~ l ( t ) ) , for ~ 0 ( 0 ) = ~1(0) = Id, and 5i (0) = h (~oo(0)). The covariant differentiation coming from w will be denoted by V. We shall show this is in fact the L-W connection corresponding to X. Recall that if ~ is a section of E and v E T~M, the L-W connection determined by X is given by
(Try(x) = X ( x ) d [Y(-)~(')] (v)
(A.1)
and is the unique metric connection such that if e E [KerX(x)] l ,
then (VXe)~ = 0.
Here Xe(.) = X(.)(e). P r o p o s i t i o n A.1 The L - W connection V induced by X is the unique connec-
tion w on 0 ( ~ ) related to X as in w P r o o f . Fix x E M. Take e EIm(Y(x)). We only need to show that (VXe)~ = 0. Here X ( x ) e = X(x)(e). By definition,
(vx~
(v) = ~(0) d (~(t))* x e (o(t)),~=o.
Universal Connections as L - W connection
97
But by the definition of the induced connection w, 5(t) = h (5oo(t)) = X (a(t))(5oo(t)lRp ) . So
(~(t))* = PR, (~oo(t))* Y (o(t)), where PRp is the projection from R m to Rp. Consequently
( v x e ) ~ (v)
=
~ ( 0 ) d p ~ , (~oo(t))* Y (o(t)) X e (~(t))l~=o
=
5(0)Prop [(5o0(0))*] d y ( a ( t ) ) X
(a(t)) (e) t=0
The last step comes from the fact that Y ( x ) X ( x ) is the projection P to Y(E~) and P(O)/5(O)P(O) = O. The required conclusion will now follow from the following lemma: Lemma
[d
A.2
]
~(0)PR, ~ (~oo(t))* I~=oY(~) = 0. P r o o f . By skew adjointness, d d--t [500 (t)]* ]t=o = -&~o(O)" Since 5oo(t) is horizontal, 0
woo (~oIo(t))
~1
=
o ( p ) • o ( m - p ) - component of w0 (al0o(t))
--
o(p) x o(m - p) - component of (Coo(t))
-1
-1:1
Coo(t).
--1:1
So (Coo(t)) aoo(t ) belongs to o ( p , m - p). In particular so does aloo(0). The result follows. 9 D. Let M = G(p, q) be the Grassmann manifold of p-planes in ]RTM for m = p+q, and ~r : B --+ M the vector bundle over M where the fibre BL at L E G(n,p) is the set of points in L. It is a Riemannian bundle. Let y V be the vector bundle map from 7r to the trivial bundle _RTM defined by y U ( L ) ( x ) = i L ( x ) for each x E BL. Here iL is the inclusion of L onto ]Rm. Its adjoint map X v is given by X U ( L ) -- PL, the orthogonal projection. Thus B is identified with a subbundle of _.Rm.
Appendices
98
Let V(p, q) be the set of p-frames in ~m ,i.e.
V(p,q) = { ( e l , . . . , e p ) l e i e ~m, (ei, ej) = 6i,j }. This is the Stiefel manifold of p-frames in II~TM . W i t h structure group O(p) it is the Stiefel bundle over G(p, q). It is in fact the associated principal bundle of B. Consider the principal bundle Ir : O(m) --4 G(p,q) with structure g r o u p O(p) • O(q), where ~r(T) - T [~P • O] C IRTM for each T E O(m). It turns out to be the bundle of a d a p t e d frames of 7r : V(p, q) --4 G(p, q) in 7r0 : G(p, q) x O(m) ~ G(p, q). This is because T G O(m) is a d a p t e d if and only if T [~P • 0] = L = BL. We are now in the picture of earlier discussions.
M•
o(m)
,
O(m)
M
,
h
,
O(B)
O(p)
O(p) x O(q)
id
,
M
,
id
,M
Now h : O(m) --4 V(p, q) is given by h(T) = T ( ~ n ) . As before there is the connection w on O 0 r ). Now V(p, q) --~ G(p, q) has a canonical connection wu as described by N a r a s i m h a n and R a m a n a n [NR61] as "the universal connection" via the connection form STdS, where S maps V(p, q) to m rows and p columns matrices in the following m way: if a E V(p, q) with ai = ~ j = l vi,jej, where (ej) is an o r t h o n o r m a l basis of Rm, S(a) = (vj,i). Denote by S(a) T its transpose. Set wU = S T d S .
T h e n wu is a o(p)-valued 1-form (since S ( o l ) T S ( o l )
-- Idp•
A . 3 The L-W connection w induced by X U : ~__m_+ B is given by the universal connection STdS on the Stiefel bundle.
Lemma
P r o o f . We only need to show t h a t the connection w is given by STdS, i.e. t h a t h*(STdS) is the o(p)-component of Woo. This is clear since for A E o(m),
h*(ST dS)(A) = (So)T d(S o h)(A) = PRpAiRp.
9
Given our surjection X : ~_m __+ E with adjoint Y so t h a t Y(x) : E~ ---+~m, there is a bundle h o m o m o r p h i s m (I) from O(E) to the Stiefel bundle,
Universal Connections as L - W connection
9:
O(E)
~o
M
99
,
V(p, q)
.
G(p, q)
defined by: (I)o(X)
=
Image Y ( x ) ,
(~(u)
=
(Y(x)u(el) .... ,Y(x)u(%)),
where (el) is an orthonormal basis for [r Conversely any such bundle homomorphism ~ : O ( E ) -~ V(p, q) over ~o : M -4 G(p, q) comes from a surjective map: indeed for x E M, take a frame u E O ( E ) at x, and set
Y(x) = r
o u -1 : T~:M -~ II~TM .
This is independent of the choice of u. Here we have used r transformation: ~P --+ I~m. Set X ( x ) = Y ( x ) * .
for the induced
On the vector bundle level let E 1 -~ M1 and E 2 -~ M2 be two vector bundles over manifolds and let f be a vector bundle map which is a 1-1 map on fibres over f0 : M1 -+ M2. Given a vector bundle map X2 : _~m _~ E 2 there is an induced map X1 : __~m__~ E 1 given by --1
= (f)io(*)
E1
f
9
E2
M1
fo
,
M2
,
X2
/-
~
This gives E 1 an induced metric and makes f an isometry on fibres. Let F x2 be the L-W connection on O ( E 2) from X2 and F xl be the L-W connection on O ( E 1) from X1. Let f* (F x2) be the pull back of F x2.
1O0
Lemma
Appendices
A.4
s* (r x~) = (rx'). P r o o f . Let Y1 and Y2 be respectively the adjoint of X1 and X2. Take e E I m a g e ( Y l (x) ), say e = Yl(v~). We need only to show
(vxf) x
=
o.
Here V is the covariant differentiation corresponding to the pulled back connection. Take a curve al E M1 with al (0) = x. Let ~1 be the horizontal lift of al (-) in the frame bundle of E 1, and ~2 the horizontal lift of f0 (ax(')). Then
S (~(t)) = ~ ( t ) , By definition,
(vx~)x
=
5.1(o) d (5..(t)) -1 x ~ (~(t))I,=o
=
d f - 1 (5"2(0)) ~ (5"2(t)) -1 X~ (fo(a(t)))I~=o
The right hand side is zero from the characterization of L-W connections and the observation that e is in the image of Y2 since e = Yl(x)(ve) = Y2(fo(X)) (f(ve)).
T h e o r e m A.5 Every metric connection on E is a L - W connection for some X : ~ m __~ E , some finite m.
P r o o f . By Narasimhan and Ramanan, any metric connection form wg is given by aJg = O*(wv) from some bundle homomorphism 9 : O(E) --~ V(p, q). Define X, Y from 9 as before.
E
M
f
,
,
B
9
Xu
~_m
G, p,q)
Set f = ( x U y ) . Then ,I~ is induced by f and so, by lemma A.3, ~*(wu) is the connection form for f ' ( F X V ) . By Lemma A.4 f * (F Xv) = F X, So wg is the connection form for the L-W connection F X.
9
Creation and Annihilation operators (notation for section 2.4)
B
101
Creation and Annihilation operators (notation for section 2.4)
Let A : V --+ V be a linear map on an inner product space V and ( e l , . . . , e ~ ) an orthonormal base for V. There are the operators (dA)* (A) on the space of tensor products of A'V, which restricts to APV to give (dA)P(A): P
ul A . . . A Uj--1 /~ Auj A uj+l A . . . A up,
(dA)P(A) (ul A . . . A up) = ~ 1
and (52A) * defined by: (52A)*A = (dA)*A o (dA)*A - (dA)*A 2.
If r is a linear map on APV, we define Ar transformations we defined on APV give:
= r
and so the last linear
P
(d-A)P(A)(r
u,) = Z
Auj,..., u,),
r
j~-I
and for p > 1 (52A~"~)A(r
up) -- E
r
Aui,..., Auj,..., up).
And so (52AI-'~)A(r = (dA ~'~) o (dA~';)A(r - (~'~)A2(r Let a~ be the "creation operator'on A'V, a~v = ej A v if ( e l , . . . , e n ) is an orthonormal basis for A'V, and aj its adjoint, the "annihilation operator". For linear forms we have the corresponding operators: (aJ)*r = r and (aJr = r In particular aJr = r A V) and (aJ)*r -- e; A r Then dA*(A) = Z
(Aej, ei) a;aj.
i,j
See [CFKS87]. Lemma
B.1
(52A)'A = - ~ i,j,k,!
(Aej, ei) (Ae~, el) a~a~ajak.
(B.1)
Appendices
102
P r o o f . By (B.1), (dA)* o (dA)*(A)(-) = Z
(Aej, ei) a~aj ((dA)*(A)(-))
i,j
= Z
(Aej,ei)(Aek,el) a*aja~aa(--)
i,j,k,l = _ }--~
(Aej, ei) (Aek, el) a* a~ajak (-)
i,j,k,l
+Z
(Aej, ei) (Aek, ej) a*ak(--)
i,j,k
The last step comes from the identity aja~ = -a~aj + 5jl, as used in [CFKS87]. However
(Aej, ei) (Aek, ej) a* ak(--) = Z i,j,k
( A2ek' ei) a;ak(--) = (dA*)(A2).
i,j,k
Consequently, (dA)* o (dA)* (A)(-)
=- Z
(Aej, ei) (Aek, el) a~a~ajak (--) + (dA)* (AS)(-),
i,j,k,l
C o r o l l a r y B.2 The map ((f2A)*A : V --~ V can be written (5~A)*A = - 2
y~
as:
(Aej A Aek,ei A el) aial aj k.
i
P r o o f . Note a~a~ = O. We split the sum in (52A)*A into two parts: i < l and i > l. After rearrangements, we have:
(52A)*(A) =-
~
[(Aej, ei) (Aek, el) - (Aej, el) (Aek, ei)] a~a~ajak
i
=-
* * (Aej AAek,ei Ael) aiala ja k
~_, i
= -2
Z
(Aej A Ae~,ei A el) a~a~ajak.
i
There is a similar expression for linear forms from (d'-A*A)(r = Z i,j
r
ej) a;aj) = Z (Aej, ei) (ai)*aJr : i,j
Basic formulae
103 A
*
C o r o l l a r y B . 3 The map ( 5 2 A ) A : A'V* ~ APV * is given by: (5~)'A(r
=
(Aei'ej)(Aek'e')(ai)'(ak)*aJalr
- E i,j,k,l
=
-2
E
(AeiAAek,ejAel)(ai)*(ak)*aJa t r
i(k,j.(l
C
Basic
formulae
In this section we give some basic formulae for V and ~b in terms of the defining m a p X or the Gaussian field W. Let W be a mean zero Gaussian field of sections F ( E ) of a vector bundle E, the associated L-W connection, and let (,)x be the metric on E induced from the Gaussian structure, as given in section 1.1C. Recall (1.1.5):
~u
= EW(x) ~d (U(a(t)),W(z(t))L(t)
t=o
v 9 T~M,U 9 rE,
(C.1)
where a is any C x smooth curve with ~r(O) = v, and also recall the expansion
v = EW (u, W) for V 9 r(E). Proposition
C.1
1. For any connection s on E, ~7~U = E~TvW (U, W)xo + ~TvU.
2. E V w W
(C.2)
= O.
3. A connection ~7 on E is adapted to (,)x if and only ]or any U E F ( E ) and tangent vector v,
ECvw (~, w) + ~ w (u, % w } = 0
(c3)
P r o o f . Take v E T, oM and let a be a C 1 smooth curve with #(0) = v. Expanding U (a(t)) in W we see:
Cvu = ~ w
(~(t))(u(~(t)),w(~(t))L(~) ~=0
= E%W(U,W)=o+EW(xo) d (U(a(t)),W(a(t)))t=o = ECvw(u, WLo+~vu. Thus (C.2). Take V = V in (C.2)) to see part (2) (or use the defining property of Proposition 1.1.3).
104
Appendices
Next suppose that V is adapted to the metric. Applying Vv in the second line of the earlier calculation, we have
r
: E % w + Ew (u, CvW} + r
and thus (C.3). On the other hand (C.3) implies
E
v } = o.
Consequently, d (U, U) (v)
i.e. V is metric. Corollary C.2 Let f7 be a connection on a subbundle E of T M. Denote by :F and T respectively the torsions for ~7 and ~7. Then for sections U, Z of E with U(xo) = u and Z(xo) = z, T(z,u)-T(z,u)
=
EW(xo)(U, fTzW}-
EW(xo)(z,~TuW}
=
- E (W(xo), u) C z W + E (W(xo), z} r
(C.4)
(c.5)
and
~(z,~) = [z,u](~0) +E[w,z](~0)(w,u>~o - E [ w , v ] ( ~ 0 ) ( w , z ) ~ o .
(c.6)
Proof. The first formula follows from (1.3.12) and the second from Proposition C.1 and (C.5). 9 Finally we write V in terms of the vector fields Z" as defined by (1.1.1): P r o p o s i t i o n C.3 Let ~7 be a connection on a subbundle E of T M containing E. Then for U E F(E) and V E F(TM) with U(xo) = u and V(xo) = v,
~7~u = C v U - r r
=
r
r
u, u.
(c.7) (c.8)
Proof. The first identity is just (1.1.3). The second follows from (1.3.1) and (C.7):
r
= ~u
+ [u, v](xo) = r
- % z ~ + [c, v](zo) = v~v-' - Cvz".
105
Basic formulae
Formulae related to curvatures First we give a formula for the curvature tensor/~ in terms of X or W. From this a series of identities and inequalities relating the quantities of V with those of are observed. In particular we give an interpretation of the form T # , derived from torsion T, being closed or co-closed. A . Let E be a subbundle of T M , R : T M • T M -~ L(E; E) the curvature tensor for a metric connection ~7, and W a Gaussian vector field which induces ~7 in the sense of section 1.1. Proposition
C..4 For u E E , o, and Vl,V2 ~. T, oM,
In the metric form,
i
i
P r o o f . First recall formula (1.3.12) from section 1.3: VvV = [V, U] + E[W, V] (W, U)
(C.11)
Let U be a horizontal vector field and V, Z vector fields,then
Ez,~.vl + EIw, zl (W,*vU)
~z[~vU]
[z, IV, u]] + E [Z, [W, V]] (W, U)
+E[w, v]~(w, u)(z(.)) + E[w,z] (w, ~vv}. Use Jacobi's identity twice to obtain
k(z,v)u
:= =
9 z g v U - 9 v g z U - 9[z,v]U E{ [W,V]d(W,U)(Z(.)) - [W, Z] d (W, U) (V(.)) }
Take U = Z ~ and V , Z with V(xo) : v, Z(xo) : z for z , v 6 T~oM, and u E E~ 0. Then
Appendices
106
using (1.3.1) and the independence of W(x) a n d ~.WlTxoM. Consequently for vi E TxM, ui E Ex,
'
/A2E~
C o r o l l a r y C . 5 Let ~ : A2TM -4 A2E be the curvature operator defined by ('~(V 1 A v2), Ul A u2) ) ---~ (/~(Vl,
v2)u2, ~tl).
Then In the metric form, m
= ~ ~7.x ~/x ~7.X i. i=l
Remark:
For A : T M @ K e r X -4 E the 'shape operator', defined by
A(u,e) = (TX(e)(u), the proposition gives
R(u,v)w = trace { A ( u , - ) ( A ( v , - ) , w ) - A ( v , - ) ( A ( u , - ) , w } } which reduces in the gradient case to Gauss's equation for the curvature of a sub-manifold in R TM ( e . g . p . 23 [KN69b]). B. Let V and V ~ be two connections on E with b : T M x E -4 E the bilinear m a p defined by :
fVvU = V ~
+ D(V, U).
(C.12)
Their curvature tensors are related by the following formula: for vl, v2 and u in
Ex, R(Vl,V2)U
: R~
u -~- ( ~ v0l D-) (v2,~t )
-
0 ~ (~v2D)_ (vl,u)
+ b ( v l , D(v2, u)) - D(v2, D ( v l , u)) + D (T~
(C.13)
v2), u ) ,
and the two covariant derivatives of b restricted to E • E are related by
In the nondegenerate case, we shall take V ~ --- V, the Levi-Civita connection for the induced metric. Recall the differential 3-form T # defined in section 2.2.
Basic formulae
107
P r o p o s i t i o n C.6 Let (7 be a torsion skew symmetric connection on T M . I] R, and [~ denote respectively the curvature tensors for V, (7 and (7 then:
R(Vl,V2)U+ v ~ ( ) v 89 V v l T
[r
+ 88
()
(vl'u)
(C.14)
(?(-,u), ~(-,v)},
(C.15)
( v 2 , u ) - 8 9 Vv2T
T(v2, u)) - 88
T(vl, u)).
In particular, 1.
mc(u, v) = nic(~, v) - 89
,) - ~tr
so I~ic(-, - ) is symmetric i] and only if 5T # = O. .
l~c(u, v) - t~ic(u, v) = - ~ r
v).
(C.16)
.
R(Vl,V2)U:/~(Ul,V2)U- ((7Vl~) (v2,u)"+"((Tv2T)(Vl,U).
(C.17)
If=~(Vl,V2)U,W) _ IR(U,W)Vl,V2) _~ ~dT 1 ~# (v~,v~, u, w)
(C.18)
.
and t~ic(u, v) = Ric(v, u),
(C.19)
R e m a r k : See also Lemma 3.5 in [Dri97] for symmetricity of/~ic in the torsion skew symmetric case. P r o o f . First note that (7 -- V + 89 in the torsion skew symmetric case. Equation (C.14) follows straightaway from (C.13). Let X : ]~m __+T M be a map which has (7 as its L-W connection. Recall l~c(u,v) = ~ ([~(X~,u)v,X' I so that
t~c(u, v) - Ric(u, v) =
~ 4
The first term of the right hand side is 1ST#. On the other hand the torsion skew symmetry gives
Appendices
108
and thus
It follows that the second term - 89 ((VuT)(Xi, Xi), v / vanishes. The last terms is now =
4
l ( T ( X i , v ) , T ( u , Xi) I 1
I
We have proved (C.15). Apply (C.15), and (C.14) respectively to both ~7 and to obtain (C.16) and (C.17). Equation (C.18) follows from
I(~uT)(W, Vl),V21 = I(~uT)(Vl,V2),wl, and (C.19) from (C.18). C o r o l l a r y C.7 If ~7 is a torsion skew symmetric connection on T M ,
1~ie(u,u)
(C.21)
Furthermore ilk, k, and k are respectively the scaler curvature of the connections (7, ~7 and V. Then 1 = ~ = k - ~IT(-,-)12.
Remarks:
(i) When M is a Lie group with ~7 the left invariant connection and (,) is bi-invariant then /~ = 0, VT _-- 0 and (C.14) reduces to the standard formula for the curvature 1 R(vi,v2)u = - ~ [[Z vl, ZV~], Z ~ ] by (1.3.6) and the Jacobi identity. In this case (C.18) shows that T# is a closed form, as is well known: for nonAbelian compact Lie groups it represents a non-trivial class in H3(G), [Car36], which is clear since (C.16) shows it is harmonic. Indeed we obtain the following: if M is compact with dimM > 3 and admits a torsion skew symmetric connection with nonzero torsion which is flat together with its adjoint connection then H3(M; R) ~ 0. This would be an extension of Cartan's result if the existence of such a connection were known on any manifolds other than Lie groups. (ii) The inequality (C.21) does not hold in general without the assumption of torsion skew symmetry, even when ~7 is adapted to some metric. A class of counter examples is provided by Lie groups with left invariant metrics having negative curvature, e.g. see [Mil76].
List of notation
D
109
List o f n o t a t i o n M H
(,I or
basic manifold, Hilbert space, = R '~ if finite dimensional bundle homomorphism between the trivial bundle M x H and E over M,
x(~) Y(~) N(~)
=
X(x)*:TxM-4H,
=
KerX(x) (often assumed of constant rank);
Z"
=
X(.)Y(xo)v,
(,)x
v e TxoM
metric induced on E from X,
(,)~
a Riemannian metric on T M = E G E • extending the metric induced on E by X, having E • orthogonal to E
{, )'
a Riemannian metric not necessarily coming from X; the connection associated to X , R, ftic, Flq its curvature, Ricci
~7,
curvature and Weitzenbock terms, and 2r its torsion tensor the adjoint semi-connection of ~7,/~, ftic, Ttq, its curvature, Ricci
~7
curvature, Weitzenbock terms and :T its torsion tensor, Levi-Civita connection, R, Ric, R q, its curvature, Ricci curvature,
V
and Weitzenbock terms V1
=
~7 G V •
direct sum connection of V on E and a connection
V • on E • T 1 its torsion any connection,/~ its curvature tensor,/~ie its Ricci curvature, /~q its Weitzenbock terms, /-/. the corresponding parallel translation, T its torsion tensor, /~Okl the associated curvatures,
R]c#(v)= ~ mc(v,x~(x))xi(x), 1
T#
the 3-form related to 7~,
W
Gaussian field of sections of E,
D
(2,0)-tensor, the difference between two linear connections, infinitesimal generator associated with a given sde and its
A, A q
restriction on q-forms,
Pt AX
~,(xo), p(zo)
semigroup associated with stochastic flows;
=
1~
V X j (X j ) + A, where A the drift coefficient of the s.d.e, involved, J 'divergence' operator associated to V, annihilation and creation operator, solution to sde with initial point x0 and life time p(xo), 2
110
Xt
=
~,(z0),
the derivative process associated to ~t,
T~t
often T~o~t(v0); for vo E T~oM,
Vt
W~,a
tile solution to covariant equation (3.3.11) involving/~q orthogonal decomposition of the Brownian motion Bt on •m using ~Tf / t l d B t = dflt + Bt, =
X(x0)/3t, the martingale part of the stochastic anti-development of
5(z0) dEf DiffM
the restriction of df to E C ~176 diffeomorphisms of M.
Chapter 4
Application: Analysis on spaces of paths A . F o r our m a n i f o l d M consider C~ o = Cxo ([0, T], M ) , t h e s p a c e of c o n t i n u o u s a : [0, T] --+ M w i t h a ( 0 ) = Xo, e q u i p p e d with t h e law # = #~o given b y o u r s t o c h a s t i c differential e q u a t i o n (3.0.1). Since C , o has a C ~ B a n a c h m a n i f o l d s t r u c t u r e we can c o n s i d e r C 1 f u n c t i o n s F : Cxo -+ ~. S m o o t h c y l i n d r i c a l functions are a subclass of these. T h e r e is t h e n t h e (Fr~chet) d e r i v a t i v e m a p d F : T C z o ~ 1~ with (dF)~ : T~Cxo ~ a b o u n d e d linear m a p for each a E Cxo. A n y R i e m a n n i a n m e t r i c on M , or F i n s l e r m e t r i c {[ [~ : x C M } , d e t e r m i n e s a F i n s l e r m e t r i c on C~ o w i t h n o r m on T~Cxo:
rIV ~F~ns=
sup IJV~ll~(~). 0
The norms on TzCxo which arise this way all determine the underlying Banachable structure of TaCzo and are all equivalent: though not uniformly so in a if M is not compact. We say F is BC I if both F and dF 6 L(TCxo;~) are continuous and bounded, using such a given Finsler norm. B. Consider first the regular case with A(x) 6 E~ for each x 6 M. Assume that there is no explosion. We shah define the 'tangent space' for #xo, relative to V, at a path a, to be the subspace Hz = / I ~ of TzCxo ([0, T]; M) defined for # almost all a by
Ha :=
(
V. E T~U~o
= WA
/o (wA) -1//shsds,
h 6 L~ '1
}
(E~o) (4.0.1)
77
where the translations W A, /~/8 are along a and (W A) is the solution map of (3.3.11). Give it the Hilbert space structure inherited from Lg '1 (E~0), so it has inner product
T d t \ o Vt' + zR]c# (Vt') -
=
(Vt'), ~ V t + ~Ric
(Vt') -
(Vt')
by (3.3.11). Note that if V E T~Czo then, almost surely, V. e H~ if and only if //t
Vt : 0 < t < T
is absolutely continuous and b 1 - # (Vt) - VA (Vt) E E~(t) ~-~Vt + ~Ric
(4.0.3)
for almost all t 9 [0, T] with I~l~ finite, for t I~ defined by (4.0.2). Since R]c# and VA both map T M to E, (4.0.3) can equivalently be expressed as
D
~-~Vt 9 E~(t),
0 < t < T,
(4.0.4)
which is a direct analogue for vector fields of the usual notion of 'horizontality' for paths. For a B C 1 function F : Cx0 --+ ll~ the gradient V F := ~ F := ~ H F is then defined as the measurable vector field, defined #-almost surely, by V F ( a ) 9 g~ ---/~ ( V F (a) , V)~, = d F ( V ) ,
V 9 g,.
C. For the regular case with A not necessarily a section of E we can define 1
H~ as in (4.0.1) and (4.0.2) with W A replacing WtA, using the notation of w The analogue of (4.0.3) will hold in the form ]D ~ V t --
1~_~ 1 ~# Vt + ~Ric (Vt) - V1A (Vt) E E~(t)
(4.0.5)
and ' Ot
I~(t)
But in general (4.0.4) will no longer be true. Thus (4.0.5) expresses the "horizontality' of these vector fields. Note that (4.0.5) is intrinsic by Proposition 3.3.9. The spaces H~ are determined by the measure #x0 (i.e. by the generator .4) and
78
Application: Analysis on spaces of paths
the choice of any metric connection on E with Riemannian metric induced by the principal symbol of ,4. Our basic assumptions in this section will be M is compact, possibly with boundary OM but if so A, X vanish on OM. The stochastic differential equation (1.2.5) is regular.
A(i) A(ii)
From Proposition 3.3.10 we have immediately
Under assumptions A(i) and A(ii) the gradient V F of a B C 1 function F : Cxo --4 ~ lies in s for 1 ~ p < oc, i.e. P r o p o s i t i o n 4.0.15
Iv
z0
IVFl~d~(a)< (X) .
If ~Y is adapted to some metric on M then [VF] is in s
4.1
Integration by parts and Clark-Ocone formulae
Clark-Ocone formulae and integration by parts formulae are closely connected e.g. see [095], [AM95] and [Hsu] and it will be efficient to prove them together. A vector field V on C , 0 with V(a) E ~I~ for almost all a will be said to be adapted if there is a version of { ~ V ( a ) s : 0 < s < T} adapted to {grt~~ : 0 < t <: T}. If so by -~V(a)8 we will always mean such a version. 4.1.1 ( I n t e g r a t i o n b y p a r t s ) Under assumptions A(i), A(ii) let F : Cxo -+ ~ be B C 1 and let V be a vector field on Cxo with V(a) E ~I~ almost surely which is adapted and has
Theorem
c
IY(a)t~+~ dPx~
(3O.
(4.1.1)
F(a) div, V(a) d#~o(a )
(4.1.2)
<
z0
for some c > O. Then /c
dF(V(a))d#~o(a)=-/c sO
~0
where div, V : Czo --+ ~ is given by divttV(o
)
=
-
W 2
~
(WA)-lY(6r)t
,
[/td[~t
where {/~t(a),0 < t < T , a e Cxo} is the martingale part of the stochastic antidevelopment of the canonical process given by # on Cxo, using ~Y. If ~Y is adapted to some Riemannian metric on M we can take e = 0 in (.~.1.1).
Integration by parts and C1ark-Ocone formulae
Theorem
79
4.1.2 ( C l a r k - O c o n e f o r m u l a for p o s s i b l y d e g e n e r a t e d i f f u s i o n s )
Let F : Cxo -+ I~ be B C 1. Under assumptions A(i), A(ii) for #xo almost all E Cxo
,4.:.,> for rD as in (3.3.16). P r o o f o f T h e o r e m 4.1.1 a n d 4.1.2. First we will prove Theorem 4.1.1 for a special class of V.. Let 0 ~ tj < tj+l < T and let c~j be bounded ~m-valued and 5r~~ Set kt = k j for
k~ = (t A tj+l - t A tj)oej to give a bounded, ~-~~ process, with paths in the Cameron-Martin space L~'l ([O, T]; I~m ). Let {~[ : 0 < t < T} be the solution to the stochastic flow of the perturbed stochastic differential equation
dyt = X(yt) o dBt + A(yt)dt + TX(yt)]gt dt
(4.1.4)
obtained by replacing B. by B. + 7k. in (1.2.5), 7 E ]~. Set x~ = ~[(Xo). By the Cameron-Martin theorem and Markov property of Brownian motion (or by the G i r s a n o v - M a r u y a m a theorem) the law of x.~ is equivalent to P~0 and
Differentiating for 7 at ~- = 0 under the expectation gives
EdF(v.) : EF(x.) /o T (k.s,dB,}
(4.1.5)
where vt = o-~x[]r=o, 0 < t < T, and so satisfies
Dl'vt = Vv, X o dBt + V 1 , A d t + X(xt)ktdt.
(4.1.6)
with v0 = 0. This goes back to Bismut's approach to Malliavin's calculus IBis81]; see also [Nor86] where the differentiation under the expectation is carefully justiffed in a more general case with M not compact. Let ~t = E{vt I.,w~~ }. Just as in the proof of Theorem 3.3.8 we see
1)1 ~vt 0t
-
1
~(Ricx,)(~t) + V~,A(xt) + X(xt)(kt)
(4.1.7)
with ~0 = 0, from which we see by 'variation of parameters' that
1 l)t ~-wA
~0t (wA)-lX(xs)]gs 1 ds.
(4.1.8)
Application: Analysis on spaces of paths
80
Set V(a)t = E{vt Ix. = a. } so V(a) is given by (4.1.8) with x. replaced by a. The left hand side of (4.1.5) therefore reduces to fC~o dF(V(a))d#~o(a ). For the right hand side, by Theorem 3.1.2,
(4.1.9) Thus (4.1.2) holds for V of the form
ljo'1
V(a)t =W A
(wA)-lX(x~)]~sds
with k = k j. By linearity it holds when k is any bounded elementary ~m-valued process, adapted to ~-.~0. Before completing the proof of Theorem 4.1.1 we will prove Theorem 4.1.2. We can assume
Iv
F(a)d#~(a)=O. 20
For this let g be a bounded elementary process with values in ~ m adapted to {Pt : t > 0}. For some c E ~m set
a = c+
(98, dBs)•m.
By the martingale representation theorem such G are dense in L 2 (f~, ~'T, P; ~m). As in (4.1.9)
~
i?~0 > : c + I0 ~ {~(x~>k~,
~/~~,}
where ]cs= E{gs ]9~ ~ } is again bounded and simple. It follows from our special case of Theorem 4.1.1 that T
EdF E
W.A
V~F, W.A
-1X(xs
>k8ds)
(wr X,
Integration by parts and Clark-Ocone formulae
E/0T/E{
81
x/x/k>x IXt
proving (4.1.3), and Theorem 4.1.2. To complete the proof of Theorem 4.1.1, simply multiply both sides of (4.1.3) by div~V and take expectations using Proposition 4.0.15 and (4.1.1). 9 Remarks:
For Brownian motion measures # these integration by parts results go back to Driver [Dri92] in the torsion skew symmetric case. As pointed out in [EL96] in the nondegenerate case our vector fields V are all "tangent processes" in the sense of Driver, for which integration by parts formulae are known see [Dri95], [CM96], [AM95], and [Aid97], [Dri99], and the monograph [Mal91] which gives further references. In the degenerate case a formula for a special class of hypoelliptic diffusions is given in [Lea]. It is shown in [EM97] that from Theorem 4.1.1 follows the closability of the form f , E(F,G) : = / _ a~
(VHF(a),VuG(a))~dp~o(Cr) ~0
with domain the B C 1 functions and the result that its closure, s G) say, is a quasi-regular local Dirichlet form on Cx0. In particular there is an associated sample continuous process on Czo: the generalized Ornstein-Uhlenbeck process determined by the #xo and the connection V on the Riemannian subbundle E, (,) of T~4 determined by #xo. The general results in [EM97] give an automatic extension of the integration by parts formula to a class of non-adapted vector fields with values in {Ho, a G Cxo} with an extended definition of divu. ~ . The Clark-Ocone formula extends to F in the domain D(E) of s and immediately gives "uniqueness of the ground states" for s C o r o l l a r y 4.1.3 If F E D(s and s F) = 0 then F is almost surely constant. In particular if F is B C 1 and V H F vanishes (or equivalently dF vanishes on H~ for almost all a) then F is almost surely constant. We have described a family of Hilbert spaces/t~ for each metric connection (and vector field A when A is not a section of E). The corollary shows that each family is sufficiently large to give at least the beginning of a Sobolev space theory. It would be interesting to know if each family is in any sense minimal with respect to the property that dF vanishes o n / t ~ for almost all a implies F almost surely constant. In [EM97] the set of all tangent processes is shown to be too big to give a gradient and hence a Dirichlet form theory in any obvious way.
Application: Analysis on spaces of paths
82
4.2
Logarithmic
Sobolev
Inequality
We can now follow the path mapped out by Capitaine-Hsu-Ledoux [CHL97] for the non-degenerate case to obtain the Logarithmic Sobolev inequality for our degenerate diffusions from the Clark-Ocone formula. We include the details, based on [CHL97], for completeness. T h e o r e m 4.2.1
equality
Underassumptions A(i) and A(ii), the logarithmic Sobolev in-
/C.o F2(a) log F2(a) d#(a)- fC~o F 2 ( a ) d # ( a ) l o g / C . o F2(a)dp(a) < 2 /;~o 'VHF'2ad#( holds for F E D(E). P r o o f . It is enough to prove it for a Ledoux [CHL97] set Ft
:=
BC 1 function F.
Following Capitaine-Hsu-
E{F(~.(Xo))[$: ~
Suppose first that F > e > 0. Then It6's formula applied to F l o g F gives:
1 s
E(F log F) - E F log EF = ~ E . u
dt
2 [E{~(VHF)t lgrt~~ }I~,(x0) Ft
Replace F by F 2 in the above and use the Cauchy-Schwartz inequality to estimate the right hand side: lD
e
IE{ ~ (VHF~)t 17: ~ } I~,(x0/
= 4 [ E{~---~(VHF)tF [9~t~~ } I~,(~o) < 4 E { F 2 [5rt~~ }E{
I~(VHF)tl 2 [9~ ~ }.
Consequently there is the logarithmic Sobolev inequality:
fC~o F2(a)log F 2 ( a ) d p ( a ) - fC.o F2(a)d#(a)log ac.of F2(a)dp(a) ___ 2E ~0 T =
2E
]D 2 [5-~o} dtE{ [-~(VHF)tl~(t) I
(VHF)t Io(t) 2 dt. = 2
IVHF[2 dp(a). ~0
For general F >_ 0 this holds by using (F + e) 2 instead of F 2 etc and taking the limit. 9 An immediate corollary of the Logarithmic Sobolev inequality is the spectral gap inequality (e.g. see [Bak97]).
Analysis on Cid(DiffM)
83
C o r o l l a r y 4.2.2 For F E D(g), 1
< 2g(F, F).
(4.2.1)
Note that the curvature constants which have appeared in the nondegenerate case do not appear here. This is because we use a different inner product on our spaces of admissible tangent vectors, and in this case it is easy to compare these inner products when V is metric for some metric with respect to which /) is bounded. However in the degenerate case we have no given Riemannian metric on M and so no canonical way of estimating curvatures, e,g, l~c # : T M --+ E and l~t does not preserve E. The definition of VH used here appears to be the most natural in the degenerate case, and so probably in the non-degenerate case.
4.3
Analysis on
Cid(DiffM)
A. It was pointed out in [ELJL96] that the integration by parts formula (4.1.2) was really derived from a 'mother formula' on the space of paths on the diffeomorphism group of M. Here we give that formula together with the resulting ' m o t h e r s ' for the Clark-Ocone formula and logarithmic Sobolev inequality. As observed in [ELJL96] the method and formulae are equally valid when the induced stochastic differential equation we use on DiffM is replaced by any right invariant systems on a Hilbert manifold with sufficiently regular group structure. We consider the Gaussian form of Proposition 1.1.3, w but use 7-/to denote the reproducing kernel Hilbert space H~ of sections of E. Recall {Wt : 0 <_ t < oo} is the Wiener process on F(E) which has law ? at time 1. We assume M is compact, possibly with smooth boundary. Since the connections used here are the right and left invariant connections on DiffM no conditions on our basic stochastic differential equation on M (i.e. on the Gaussian measure 3') are assumed apart from the smoothness of the fields in 7-/and their vanishing on OM. Let K . ( - ) E Lo2'l([0,T];7/). Let Z)~ be the Hilbert manifold of diffeomorphisms of M of Sobolev class H s, s > n/2 + 3 which are the identity on OM. Consider the random time dependent ordinary differential equation on l) 8, parameterized by r E R:
d r -~Ht
=
~-
ad ~-1 dKt ( t ) -~ ,
H~
=
id,
0
where ad(O) denotes the adjoint action of 0 E DiffM on F ( T M ) , i.e. ad(O)(V) = TO(V(O-I(.))), the push forward 0, (V). The solution exists and we can perturb
Application: Analysis on spaces of paths
84
our flow by it to obtain ~[ := ~t o H~t, 0 < t < T. This satisfies the analogue of (4.1.4): d~[ = TRy; o dWt + TRy; (A)dt + rTR~[ ([~[t)dt. As for x~ in w the flows of ~.~ on Cid([0, T]; DiffM) are equivalent a n d if F : Cid([O, T]; DiffM) ~ ]R is b o u n d e d and m e a s u r a b l e E F ( ~ . ) = EF(~T. ) M r where
{/:..
__
T 2
(4.3.1)
}
To stay flexible we will say t h a t F : Cid([O, T]; DiffM) --+ It( is C 1 with suitably bounded derivative D F if one of the following holds: Case (i), F is the restriction of a C 1 m a p F : Cid([O,T];D s) ~ I~ where the (Fr6chet) derivative d F : TCId([O,T]; 7:)8) -+ ~ is uniformly b o u n d e d using the Finsler metric on the tangent space TeCid([O, T]; D ~) at 8. given by
IlVllo
:=
sup IV lo, O
where ] 9 1o~ is the value at Ot of the left invariant R i e m a n n i a n s t r u c t u r e on 7:)s d e t e r m i n e d by a s t a n d a r d H s inner-product on F ( T M ) . Case (ii), T h e same as case (i) but with the Finsler metric given by O
where HS(Ot) is a s t a n d a r d H 8 n o r m on the space of H s vector fields over the diffeomorphism Or, i.e. on H s sections of 0~'(TM). Case (iii), T h e analogue of case (i) with CrDiffM replacing D s for I < r < co. Case (iv), T h e analogue of case (ii) with C r ( M , M ) replacing D s, 0 < r < oc or C~DiffM if r > 1. We need two lemmas. T h e proof of the first is by straightforward calculus, since 7-I is continuously included in C r F ( T M ) for each r. Lemma4.3.1 Let O : M ~ M be a C r diffeomorphism, some r > 1. Then the adjoint action ad(O) of O on 74 is continuous linear as a map ad(O) : 74 -+ C ~ - ] F ( T M ) with norm bounded by the C ~-1 norm of TO. L e m m a 4 . 3 . 2 For each of the Finsler norms described above there exists qP" : Cid(DiffM) -~ ]R(> O) with SUpo
IIT~ ~. / o a d ( ~ ) -x [(sdsll~ ns < O r ( ~ ) I K I L ] , I ( n ) for all K. G L2o'1 (74), almost surely.
(4.3.2)
Analysis on Cid( DiffM)
85
P r o o f . This follows from the previous l e m m a and Proposition 3.3.10 using the Sobolev embedding theorem to switch between C r and H~ norms. 9 If F is C 1 , bounded and with suitably bounded derivative, we can therefore differentiate (4.3.1) at T = 0 to obtain
]r~,dSQf~. fo'ad(~sl)gsds) ~- ~-,F(~.)fof (gs,dWslT_l which can immediately be written in terms of the law # 9 of ~. on DiffM, (since
Wt = f t TRy1 o des - f~ T R ~ I A ds)) fc,,(DiffM) dF (TO. fo ad(O: 1[(sds) d#~(O.) = fcle(DiffM)F(O.)f[
(4.3.3)
(Ks,dWs)nd#7)(O.).
As we saw for C~ o (M) this holds true when K is an adaps
process with sample
1+~
paths in L2'I([0, T ] ; ~ ) provided that E
IKsl~tds
< oo for some e > 0.
C. From (4.3.3) we see that the "tangent space" we obtained for # 9 at 0. is the Hilbert space Ho = H'~ 'A of V 9 ToCid(DiffM) with ad(Ot) d [(TOt)-lVt] 9 7-/for almost all 0 < t < T and having
I~
d 2 ad(Ot)--~ [(TOt)-lVt] 7{ dt < o~.
Note t h a t the first condition can be written as
z)
-~ Vt 9 TTio, (7-I) where now we define
~JD= TOtd [(TOt)-lvt] i.e. using the left invariant connection on DiffM, in complete analogy with the case of paths on M when M is a Lie group. For our bounded C 1 function F with suitably bounded derivative, by (4.3.2) we obtain VHF(O.) E Ho for each 0, satisfying
dF(V.) = (VHF(O.), V.>H~ , By (4.3.2),
all V. E
Ho.
[VHF(O)IH(O) <_O(O.)IldFllFins
for ff = fro. In particular
V H F lies in L p for 1 < p < o~.
D . Just as in Theorem4.1.2 there is the Clark-Ocone formula
Application: Analysis on spaces of paths
86
T h e o r e m 4.3.3 Let F : Cid(DiffM) --+ ~ be bounded and C 1 with suitably
bounded derivative. Then T
d
almost surely, where 3ct = a{Ws : 0 < s < t}. From this follows, as before, the logarithmic Sobolev inequality: T h e o r e m 4.3.4 For F bounded, C 1, and with suitably bounded derivative
fCi~(DiffM) F21~ F2d#v(o) - fc~d(DiffM) F21~ fc~d(DiffM) F2d#7~(O) <_2
fC
id(DiffM)
[VHF(O)I2H(~
