Lectures in Probability Theory and Statistics 1995

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg E Takens, Groningen B. Teissier, Paris

1690

Springer

Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Singapore Tokyo

M. T. Barlow D. Nualart

Lectures on Probability Theory and Statistics Ecole d'Et6 de Probabilit6s de Saint-Flour X X V - 1995 Editor: P. Bernard

Springer

Authors

Editor

Martin T. Barlow Department of Mathematics University of British Columbia # 121-1984 Mathematics Road Vancouver, B.C. Canada V6T 1Z2

Pierre Bernard Laboratoire de Mathdmatiques Appliqudes UMR CNRS 6620 Universitd Blaise Pascal Clermont-Ferrand F-63177 Aubibre Cedex, France

David Nualart Department d'Estadistica Universitat de Barcelona Facultat de Matem~ttiques Gran Via de les Corts Catalanes, 585 E-08007 Barcelona, Spain Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufilahme L e c t u r e s on p r o b a b i l i t y t h c o r y a n d statistics / Ecole d'Et~ de Probabilitds de Saint-Flour X X V - 1995. M. T. Barlow ; D. Nualart. Ed.: P. Bernard. - Berlin ; Heidelberg ; N e w York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ;

Singapore ; Tokyo : Springer, 1998 (Lecture notes in mathematics ; Vol. 1690) ISBN 3-540-64620-5 Mathematics Subject Classification (199 I): 60-01, 60-02, 60-06, 60D05, 60G57, 60H07, 60J15, 60J60, 60J65 ISSN 0075- 8434 ISBN 3-540-64620-5 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 for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1998 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 authors SPIN: 10649898 41/3143-543210 - Printed on acid-free paper

INTRODUCTION

This volume contains lectures givcn at the Saint-Flour Summer School of Probability Theory during the period 10th - 26th July, 1995. We thank the authors for all the hard work they accomplished. Their lectures are a work of refercncc in thcir domain. The school brought together i00 participants, 29 of whom gave a lecture concerning their research work. At the end of this volume you will find the list of participants and their papers. Finally, to facilitate research concerning previous schools we give here the number of the volume of "Lecture Notes" where they can be found :

L e c t u r e N o t e s in M a t h e m a t i c s 1971: n~ 1977:n~ 1982: n~ 1988: n~ 1993: n~

-

1973: n~ 1978 : n~ 1983:n~ 1989: n~ 1994: n~

L e c t u r e N o t e s in S t a t i s t i c s 1986 : n~

-

1974:n~ 1979: n~ 1984: n~ 1990: n~ 1996:n~

-

1975 : 1980: 19851991:

n~ n~ 1986 et n~

1976:n~ 1981:n~ 1987: n~ 1992:n~

-

TABLE OF CONTENTS

Martin 1

T. BARLOW

Introduction

: "DIFFUSIONS

ON FRACTALS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2

T h e Sierpinski Gasket . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3

Fractional Diffusions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

4

Dirichlet Forms, Markov P r o c e s s e s and Electrical Networks . . . . . . . . . .

46

5

G c o m e t r y of R e g u l a r Finitely Ramified Fractals

. . . . . . . . . . . . . . .

59

6

R e n o r m a l i z a t i o n on Finitely Ramified Fractals

. . . . . . . . . . . . . . . .

79

7

Diffusions on p.c.f.s.s, sets

. . . . . . . . . . . . . . . . . . . . . . . . . .

94

8

T r a n s i t i o n Density E s t i m a t e s References

David

. . . . . . . . . . . . . . . . . . . . . . . . .

106

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114

NUALART

: "ANALYSIS ON WIENER SPACE AND ANTICIPATING STOCHASTIC CALCULUS"

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

125

Derivative a n d divergence o p e r a t o r s on a G a u s s i a n space . . . . . . . . . . .

126

1.1 Derivative o p e r a t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . .

126

1.2 Divergence o p e r a t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . .

130

1.3 Local p r o p e r t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131

1.4 W i e n e r chaos e x p a n s i o n s

. . . . . . . . . . . . . . . . . . . . . . . . .

133

. . . . . . . . . . . . . . . . . . . . . . . . . . .

135

1.5 T h e w h i t e noise case

1.6 S t o c h a s t i c integral r e p r e s e n t a t i o n of r a n d o m variables 2

123

. . . . . . . . . .

139

O r n s t e i n - U h l e n b e c k semigroup a n d equivalence of n o r m s . . . . . . . . . . .

141

2.1 M e h l e r ' s f o r m u l a

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

142

2.2 H y p e r c o n t r a c t i v i t y

2.3 G e n e r a t o r of t h e O r n s t e i n - U h l e n b e c k s e m i g r o u p

. . . . . . . . . . . . .

145

2.4 M e y e r ' s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . .

146

VLLI

3

A p p l i c a t i o n of Malliavin calculus to s t u d y p r o b a b i l i t y laws . . . . . . . . . .

155

3.1 C o m p u t a t i o n of p r o b a b i l i t y densities

155

. . . . . . . . . . . . . . . . . . .

3.2 R e g u l a r i t y of densities a n d c o m p o s i t i o n of t e m p e r e d d i s t r i b u t i o n s w i t h e l e m e n t s of ~)-oo

. . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 T h e case of diffusion processes . . . . . . . . . . . . . . . . . . . . . . 3.4 Lp e s t i m a t e s of t h e density a n d a p p l i c a t i o n s . . . . . . . . . . . . . . . 4

5

160 163 164

Support theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

174

4.1 P r o p e r t i e s of t h e s u p p o r t

174

. . . . . . . . . . . . . . . . . . . . . . . . .

4.2 S t r i c t p o s i t i v i t y of t h e density a n d skeleton . . . . . . . . . . . . . . . .

177

4.3 Skeleton a n d s u p p o r t for diffusion processes

182

. . . . . . . . . . . . . . .

4.4 V a r a d h a n e s t i m a t e s . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183

A n t i c i p a t i n g s t o c h a s t i c calculus . . . . . . . . . . . . . . . . . . . . . . . .

188

5.1 S k o r o h o d i n t e g r a l processes . . . . . . . . . . . . . . . . . . . . . . . .

188

5.2 E x t e n d e d S t r a t o n o v i c h intcgral . . . . . . . . . . . . . . . . . . . . . .

197

5.3 S u b s t i t u t i o n f o r m u l a s . . . . . . . . . . . . . . . . . . . . . . . . . . .

201

A n t i c i p a t i n g s t o c h a s t i c differential e q u a t i o n s

210

. . . . . . . . . . . . . . . . .

6,1 S t o c h a s t i c differential e q u a t i o n s in t h e S t r a t o n o v i c h sense 6.2 S t o c h a s t i c differential e q u a t i o n s w i t h b o u n d a r y c o n d i t i o n s

........ ........

210 215

6.3 S t o c h a s t i c differential e q u a t i o n s in t h e S k o r o h o d sense . . . . . . . . . .

217

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221

DIFFUSIONS

ON FRACTALS

Martin T. BARLOW

1. I n t r o d u c t i o n . T h e notes are based on lectures given in St. Flour in 1995, and cover, in greater detail, most of the course given there. T h e word "fractal" was coined by M a n d e l b r o t [Man] in the 1970s, b u t of course sets of this t y p e have been familiar for a long time - their early history being as a collection of pathological examples in analysis. There is no generally agreed exact definition of the word "fractal", and a t t e m p t s so far to give a precise definition have been unsatisfactory, leading to classes of sets which are either too large, or too small, or both. This a m b i g u i t y is not a p r o b l e m for this course: a more precise title would be "Diffusions on some classes of regular self-similar sets". Initial interest in the properties of processes on fractals came from m a t h e m a t i c a l physicists working in t h e t h e o r y of disordered media. Certain m e d i a can be modelled by p e r c o l a t i o n clusters at criticality, which are expected to exhibit fractal-like properties. Following t h e initial p a p e r s [AO], [RT], [GAM1-GAM3] a very substantial physics l i t e r a t u r e has developed - see [HBA] for a survey and bibliography. Let G be an infinite s u b g r a p h of Z a. A simple r a n d o m walk (SRW) (X,~, n > 0) on G is j u s t the Markov chain which moves from x E G with equal p r o b a b i l i t y to each of the neighbours of x. Write p n ( x , y) = IP~(X,~ = y) for the n-step t r a n s i t i o n probabilities. If G is the whole of Z d then E(X,~) 2 = n with m a n y familiar consequences - the process moves roughly a distance of order v/n in time n, and the p r o b a b i l i t y law p,~(x, .) p u t s most of its mass on a ball of radius cdn. If G is not the whole of ~d then the movement of the process is on the average r e s t r i c t e d by the removal of p a r t s of the space. Probabilistically this is not obvious b u t see [DS] for an elegant argument, using electrical resistance, t h a t the removal of p a r t of the s t a t e space can only make the process X 'more recurrent'. So it is not unreasonable to expect t h a t for certain graphs G one m a y find t h a t the process X is sufficiently r e s t r i c t e d t h a t for some fl > 2 -

(1.1)

E ~ (X~ - x) ~ • n2/~.

(Here and elsewhere I use • to m e a n ' b o u n d e d above and below by positive constants', so t h a t (1.1) means t h a t there exist constants el, c2 such t h a t c l n 2/~ < E ~ ( X , - x) ~ < c2n~/a). In [AO] and [RT] it was shown t h a t if V is the Sierpinski gasket (or more precisely an infinite graph based on the Sierpinski gasket - see Fig. 1.1) t h e n (1.1) holds w i t h / 3 = log 5 / l o g 2.

Figure 1.1: T h e graphical Sierpinski gasket.

Physicists call b e h a v i o u r of this kind by a r a n d o m walk (or a diffusion - t h e y are not very i n t e r e s t e d in the distinction) subdiffusive - the process moves on average slower t h a n a s t a n d a r d r a n d o m walk on Z d. Kesten [Ke] proved t h a t the SRW on the 'incipient infinite cluster' C (a percolation cluster at p = Pc b u t conditioned to be infinite) is subdiffusive. T h e large scale structure of C is given by taking one infinite p a t h (the ' b a c k b o n e ' ) together with a collection of 'dangling ends', some of which are very large. K e s t e n a t t r i b u t e s the subdiffusive behaviour of SRW on C to the fact t h a t the process X spends a substantial amount of time in the dangling ends. However a g r a p h such as the Sierpinski gasket (SG) has no dangling ends, and one is forced to search for a different explanation for the subdiffusivity. This can be found in t e r m s of t h e existence of 'obstacles at all length scales'. W h i l s t this holds for the graphical Sierpinski gasket, the n o t a t i o n will be slightly simpler if we consider a n o t h e r example, the graphical Sierpinski c a r p e t (GSC). (Figure 1.2).

-t : H - H - H : H - H q

i i

,;fi fi:H:H: ; f:H-t [IIIII

q

t: : H F

. . . . . . . . .

f:H-H:H:H-H:

II

;H:F :H-F " I I I I I

III

Figure 1.2: T h e graphical Sierpinski carpet. This set can be defined precisely in the following fashion. Let H0 -- Z ~. For o~ x = (n, m) e H0 write n,m in t e r n a r y - so n -- ~i=o ni3", where ni 9 {0, 1,2}, and n~ -- 0 for all b u t finitely m a n y i. Set J~ -- { ( m , n ) : nk -- 1 and mk -- 1}, so t h a t Jk consists of a union of disjoint squares of side 3k: the square in Jk closest to t h e origin is { 3 ~ , . . . , 2 . 3 ~ - 1} • { 3 k , . . . , 2 . 3 k - 1}. Now set (1.2)

H,, = Ho- O Jk, k:l

H= 5 H,~. n=O

:~_

11111

IIII

IIII]

IIlI

-H:t

lIII

ilIi

itti

IIII iIII

IIlI

~HIIII

IlII

!

IIII

i

!~H~

IlIl

!HI::

Figure 1.3: T h e set H1 9

14' ; ~-

'.

.

.

.

!

~

!

T

F!

---

i

Li

i

i

i

Fi

.__

i

!i

i

i

i

i

i

ttt

r

ii

-L

!E:_I

t

i

1"

__.

---

!ii_I

H --~F-H

4~f-H--t:t--

.

Figure 1.4: T h e set/-/2 9 Note t h a t H A [0, 3n] 2 = Hn N [0, 3n] 2, so t h a t the difference between H and Hn will only be d e t e c t e d by a SRW after it has moved a distance of 3 n from the origin. Now let X (n) be a SRW on Hn, s t a r t e d at the origin, and let X be a SRW on H. T h e process X (~ is j u s t SRW on Z~_ and so we have (1.3)

E(X(~ 2 ~" n.

T h e process XO) is a r a n d o m walk on a the intersection of a t r a n s l a t i o n invariant subset of 7..2 w i t h 7.~_. So we expect 'homogenization': the processes n - 1 / 2 X [,~], O) t _> 0 should converge weakly to a constant multiple of Brownian motion in ~ 2 . So, for large n we should have E ( X O ) ) 2 ~ a l n , and we would expect t h a t al < 1, since the obstacles will on average t e n d to i m p e d e the motion of the process. Similar considerations suggest t h a t , writing ~on(t) = 1EO(X}n)) 2, we should have ~pn(t) " a n t

as t --* oo.

However, for small t we would expect t h a t ~n and ~0,~+1 should be a p p r o x i m a t e l y equal, since the process will not have moved far enough to detect the difference between Hn and H n + l . More precisely, if tn is such t h a t ~On(tn) = (3n) 2 then ~on

and ~o=+1 should b e a p p r o x i m a t e l y equal on [O, tn-bl ]. So we m a y guess t h a t the b e h a v i o u r of the family of functions ~o,~(t) should be roughly as follows: (1.4)

~on(t) = bn + an(t - in),

v.+l(s) = v.(s),

t >_ t,~,

0 < s < t~+l.

If we a d d the guess t h a t an = 3 - ~ for some ~ > 0 then solving the equations above we deduce t h a t tn ~ 3 (2+a)'~, bn ~- 3 2n. So ff ~o(t) = E ~ 2 t h e n as 4o(t) ~- lira= ~on(t) we deduce t h a t ~v is close to a piecewise linear function, and t h a t ~(t) • t 2/z where fl = 2 + a. T h u s the r a n d o m walk X on the graph H should satisfy (1.1) for some f~ > 2. T h e a r g u m e n t given here is not of course rigorous, b u t (1.1) does a c t u a l l y hold for t h e set H - see [BB6, BBT]. (See also [Jo] for the case of the graphical Sierpinski gasket. T h e proofs however run along r a t h e r different lines t h a n the heuristic a r g u m e n t sketched above). Given b e h a v i o u r of this t y p e it is n a t u r a l to ask if the r a n d o m walk X on H has a scaling limit. More precisely, does there exist a sequence of constants Tn such t h a t t h e processes (1.5)

( 3 - ' ~ X i t / , . ] , t > 0)

converge weakly to a non-degenerate limit as n --+ co? For the graphical Sierpinski c a r p e t the convergence is not known, though there exist m= such t h a t the family (1.5) is tight. However, for the graphical Sierpinski gasket the answer is 'yes'. Thus, for certain very regular fractal sets F C I~d we are able to define a limiting diffusion process X = ( X t , t > O, P~, x C F ) w h e r e ]?~ is for each x E F a p r o b a b i l i t y measure on fl = {w Z C([0, co), F ) : w(0) = x}. Writing T t S ( x ) = E ~ f ( X t ) for the semigroup of X we can define a 'differential' o p e r a t o r s defined on a class of functions 7:)(/:F) C C ( F ) . In m a n y cases it is reasonable to call s the Laplacian on F.

F r o m t h e process X one is able to o b t a i n information a b o u t the solutions to the Laplace a n d h e a t equations associated with L:F, the heat equation for example t a k i n g the form Ou

(1.6)

0-7 = LFu, u(0, x) = u0(x),

where u = u ( t , x ) , x E F , t > O. The wave equation is r a t h e r harder, since it is not very susceptible to probabilistic analysis. See, however [KZ2] for work on the wave equation on a some manifolds with a 'large scale fractal structure'.

T h e m a t h e m a t i c a l l i t e r a t u r e on diffusions on fractals and their associated infinitesimal generators can be divided into b r o a d l y three parts: 1. Diffusions on finitely ramified fractals. 2. Diffusions on generalized Sierpinski carpets, a family of infinitely ramified fractals. 3. S p e c t r a l p r o p e r t i e s of the ' L a p l a c i a n ' s These notes only deal with the first of these topics. On the whole, infinitely ramified fractals are significantly h a r d e r t h a n finitely ramified ones, and sometimes require a very different approach. See [Bas] for a recent survey. These notes also contain very little on spectral questions. For finitely ramified fractais a direct a p p r o a c h (see for example [FS1, Shl-Sh4, KL]), is simpler, and gives more precise information t h a n the heat kernel m e t h o d based on estimating

/F

p ( t , x , x ) d x = y ~ e-~' ~. i

In this course Section 2 introduces the simplest case, the Sierpinski gasket. In Section 3 I define a class of well-behaved diffusions on metric spaces, "Fractional Diffusions", which is wide enough to include m a n y of the processes discussed in this course. It is possible to develop their properties in a fairly general fashion, without using much of the special structure of the state space. Section 4 contains a brief i n t r o d u c t i o n to t h e theory of Dirichlet forms, a n d also its connection with electrical resistances. T h e remaining chapters, 5 to 8, give the construction and some p r o p e r t i e s of diffusions on a class of finitely ramified regular fractals. In this I have largely followed the analytic ' J a p a n e s e ' approach, developed by Kusuoka, Kigami, ~'klkushima a n d others. Many things can now be done more simply t h a n in the early probabilistic work - b u t there is loss as well as gain in a d d e d generality, and it is worth pointing out t h a t the early papers on the Sierpinski gasket ([Kusl, Go, BP]) contain a wealth of interesting direct calculations, which are not r e p r o d u c e d in these notes. A n y reader who is surprised by the a b r u p t end of these notes in Section 8 should recall t h a t some, at least, of the properties of these processes have a l r e a d y been o b t a i n e d in Section 3. c~ denotes a positive real constant whose value is fixed within each L e m m a , T h e o r e m etc. Occasionally it will b e necessary to use n o t a t i o n such as c3.5.4 - this is s i m p l y the constant c4 in Definition 3.5. c, c r, c ~ denote positive real constants whose values m a y change on each appearance. B(x, r) denotes the open ball with centre x a n d radius r, and if X is a process on a metric space F then

TA = inf{t > O : Xt E A } , Ty=inf{t>0:Xt=y}, ~-(x, r) = inf{t > 0 : Xt fd B(x, r)}. I have included in the references most of the m a t h e m a t i c a l papers in this area known to me, and so t h e y contain m a n y papers not mentioned in the text. I a m grateful to G e r a r d Ben Arous for a n u m b e r of interesting conversations on the physical conditions under which subdiffusive behaviour might arise, to Ben H a m b l y

for checking the final manuscript, and to Ann Artuso and Liz ttowley for their typing.

Acknowledgements. This research is supported by a NSERC (Canada) research grant, by a grant from the Killam Foundation, and by a EPSRC (UK) Visiting Fellowship.

2. T h e Sierpinski Gasket This is the simplest non-trivial connected symmetric fractal. The set was first defined by Sierpinski [Siel], as an example of a pathological curve; the name "Sierpinski gasket" is due to Mandelbrot [Man, p.142]. Let Go = {(0, 0), (1, 0), (1/2, ~/3/2)} = {a0, al, a2} be the vertices of the unit triangle in I~2, and let Tlu(Go) = Ho be the closed convex hull of Go. The construction of the Sierpinski gasket (SG for short) G is by the following Cantor-type subtraction procedure. Let b0, bl, b2 be the midpoints of the 3 sides of Go, and let A be the interior of the triangle with vertices {bo,bl,b2}. L e t / / 1 = H0 - A, so that HI consists of 3 closed upward facing triangles, each of side 2 -1. Now repeat the operation on each of these triangles to obtain a set H2, consisting of 9 upward facing triangles, each of side 2 -2 .

Figure 2.1: The sets//1 and H2. Continuing in this fashion, we obtain a decreasing sequence of closed non-empty ~o and set sets ( H n)==0, (2.1)

G = 5 n=0

/In.

Figure 2.2: The set H4. It is easy to see t h a t G is connected: just note that OHn C H,n for all m > n, so t h a t no point on the edge of a triangle is ever removed. Since IHnl = (3/4)nlH0 I, we clearly have t h a t IGI = 0. We begin by exploring some geometrical properties of G. Call an n-triangle a set of the form G N B, where B is one of the 3 '~ triangles of side 2 -'~ which make up Hn. Let #n be Lebesgue measure restricted to H,~, and normalized so that # n ( H n ) -- 1; that is #,~(dx) = 2- (4/3)'~1/./~ (x) dz. Let /~G = wlim#,~; this is the n a t u r a l "flat" measure on G. Note that # c is the u n i q u e measure on G which assigns inass 3 -'~ to each n-triangle. Set df=log3/log2~ 1.58... L e m m a 2.1. For x E G, O < r < l

3-1r d' <_ # a ( B ( x , r ) )

(2.2)

< lSr d'.

Proof. T h e result is clear if r = 0. If r > 0, choose n so that 2 -(~+1) < r < 2 -'~ we have n > 0. Since B(x, r) can intersect at most 6 n-triangles, it follows that

(B(x, r)) < 6.3

:

= 18(2-("+U) d, < 18r d'. As each (n + 1)-triangle has diameter 2 - ( n + U , B ( x , r ) must contain at least one (n + 1)-triangle a n d therefore

#r

> 3 -('~+1) -- 3-1(2-'~) d' > 3-1r d'.

[]

Of course the constants 3 -1 , 18 in (2.2) are not important; what is significant is t h a t the #G-mass of balls in G grow as r ds. Using terminology from the geometry of manifolds, we can say that G has volume growth given by r d~ .

Detour on Dimension. Let (F, p) be a metric space. There are a number of different definitions of dimension for F and subsets of F : here I just mention a few. The simplest of these is box-counting dimension. For e > 0, A C F , let N(A,e) be the smallest number of balls B(x, e) required to cover A. Then

(2.3)

d i m B c ( A ) = lim sup ~0

log N(A, e) log e-x

To see how this behaves, consider some examples. We take (F, p) to be ~a with the Euclidean metric.

Examples.

1. Let A -- [0,1] a C ~a. Then

N(A, e)

x e -a, and it is easy to verify

that lira log N([0, 1]d, e) = d. ~0 log e - 1 2. The Sierpinski gasket G. Since G C H~, and H,, is covered by 3 ~ triangles of side 2 - n , we have, after some calculations similar to those in Lemma 2.1, that N ( G , r ) y- (1/r)l~176 So,

dimBc(G) = 3. Let A = Q N [0,1]. Then N(A,e) dimBe({p}) = 0 for any p E A.

log3 log 2"

• ~-1, so dimBc(A)

= 1. On the other hand

We see t h a t box-counting gives reasonable answers in the first two cases, but a less useful number in the third. A more delicate, but more useful, definition is obtained if we allow the sizes of the covering balls to vary. This gives us Hausdorff dimension. I will only sketch some properties of this here - for more detail see for example the books by Falconer [Fal, Fa2]. Let h : ]~+ --* R+ be continuous, increasing, with h(0) = 0. For U C F write diam(U) -- sup{p(~, y) : z, y E U} for the diameter of V. For 6 > 0 let

~(a)--inf{E h(d(U,))

: A c gu,,

i

diam(U,) < 6}.

i

Clearly 7-/h(A) is decreasing in 6. Now let (2.4) we call T/h(-)

7/h(A) = lim 7/~(A); 610

Hausdorff h-measure. Let

B ( F ) be the Borel a-field of F.

10 Lemma

2.2. 7-/h is a measure on (F, B(F)).

For a proof see [Fal, Chapter 1]. We will be concerned only with the case h(x) = x~: we then write ~ for 7~h. Note that a ~ 7-/~(A) is decreasing; in fact it is not hard to see that 7 ~ ( A ) is either +o0 or 0 for all but at most one a. Definition

2.3. The Hausdorff dimension of A is defined by dimH(A) = inf{~ : ~ ( A )

Lemma

= 0} -- s u p { a : 7-/~(A) = + ~ } .

2.4. dimH(A) < dimBc(A).

Proof. Let a > dimBc(A). Then as A can be covered by N(A, e) sets of diameter 2e, we have 7-/~(A) < N(A,E)(2e) ~ whenever 2e < 6. Choose 0 so that d i m B c ( A ) < a - 0 < a; then (2.3) implies that for all sufficiently smMl e, N ( A , e) _< e-(~-0). So ? ~ ( A ) : 0, and thus 7/~(A) : 0, which implies that dimH(A) < a. [] Consider the set A = Q N [0, 1], and let A -- {Pa,P2,..-} be an enumeration of A. Let 5 > 0, and Ui be an open internal of length 2 - i A 6 containing Pi. Then (Ui) covers A, so that 7-/~(A) < ~ i ~ 1 (6 A 2-i) ~, and thus 7 ~ ( A ) -- 0. So dimH(A) = 0. We see therefore that dimH can be strictly smaller than dimBc, and that (in this case at least) dimH gives a more satisfactory measure of the size of A. For the other two examples considered above Lemma 2.4 gives the upper bounds dimH([0, 1] d) < d, dimH(G) < log 3 / l o g 2. In both cases equality holds, but a direct proof of this (which is possible) encounters the difficulty that to obtain a lower bound on 7 ~ ( A ) we need to consider all possible covers of A by sets of diameter less than 6. It is much easier to use a kind of dual approach using measures. 2.5. Let # be a measure on A such that #( A ) > 0 and there exist Cl < c~, ro > O, such that

Theorem

(2.5)

p ( B ( x , r ) ) < e a r ~,

xeA,

r
Then ~l"(A) >_ ci-a#(A), and d i m ~ ( A ) _> a. Proof. Let Ui be a covering of A by sets of diameter less than 6, where 25 < r0. If xi E Ui, then Ui C B(xi,diam(Ui)), so that #(Ui) _< ca diam (Ui)". So

"(Ui) ~ c11/d(A)"

diam (ui)" _> ei-1 E

E

i

i

Therefore 7-[~(A) _> c~ap(A), and it follows immediately that T/a(A) > 0, and dimH(A) _> a. [] C o r o l l a r y 2.6. dimH(G) = log 3 / l o g 2.

Proof. By Lemma 2.1 #G satisfies (2.5) with a = d I. So by Theorem 2.5 dimH(G) dl; the other bound has already been proved. [] Very frequently, when we wish to compute the dimension of a set, it is fairly easy to find directly a near-optimal covering, and so obtain an upper bound on d i m g directly. We can then use Theorem 2.5 to obtain a lower bound. However, we can also use measures to derive upper bounds on dimH.

11 T h e o r e m 2.7. L e t # be a finite measure on A such that # ( B ( x , r ) ) > C2r ~ lCor aH x 9 A, r < to. T h e n 7-l~(A) < cr and dimH(A) < a.

Proof. See [Fa2, p.61].

In particular we may note: C o r o l l a r y 2.8. I f # is a measure on A with #(A) 9 (0, ~ ) and (2.6)

c~ ~ < ~(B(x,r))

< c~r ~,

x 9 A,

r < r0

then 7-l~(A) 9 ( 0 , ~ ) and dimH(A) = a.

R e m a r k s . 1. If A is a k-dimensional subspace of R d then d i m g ( A ) = d i m B c ( A ) = k. 2. Unlike dimBc dimH is stable under countable unions: thus

i=1

i

3. In [Tri] Tricot defined "packing dimension" dimp(-), which is the largest reasonable definition of "dimension" for a set. One has dimp(A) > dimH(A); strict inequality can hold. The hypotheses of Corollary 2.8 also imply that dimp(A) = a. See [Fa2, p.48]. 4. The sets we consider in these notes will be quite regular, and will very often satisfy (2.6): that is they will be "a-dimensional" in every reasonable sense. 5. Questions concerning Hausdorff measure are frequently much more delicate than those relating just to dimension. However, the fractals considered in this notes will all be sufficiently regular so that there is a direct construction of the Hausdorff measure. For example, the measure # c on the Sierpinski gasket is a constant multiple of the Hausdorff x dj-measure on G. We note here how dimH changes under a change of metric. T h e o r e m 2.9. L e t Pl, P2 be metrics on F, and write 7-(~,i, dimH,i for the Hausdorff measure and dimension with respect to p,, i = 1, 2. (a) I f p ~ ( z , y ) < p 2 ( x , y ) for all x , y C A with p 2 ( z , y ) <_ 5o, then dimH,l(A) > dimH,2(A). (b) I f 1 A p l ( x , y ) • (1 A p 2 ( x , y ) ) ~ for some 0 > 0, then dimH,2 (A) = 0 dimH,1 (A). Proof. Write de(U) for the pj-diameter of U. If (U~) is a cover of A by sets with p2(Ui) < 5 < 50, then

E

dl(Ui) a < E i

so that ?-/~'I(A) < ~ ' 2 ( A ) . proving (a).

d2(Ui) a i

Then ?-/~'I(A) < 7/~,2(A) and dimH,l(A) > dimH,2(A),

12 (b) If Ui is a n y cover of A by sets of small d i a m e t e r , we have

Z dl(u,)~ • i

Zd2(U,) " i

Hence 7-/"'1 (A) = 0 if a n d only if 7-/e•'2 (A) = 0, a n d the conclusion follows.

[]

Metrics on the Sierpinski gasket. Since we will b e s t u d y i n g c o n t i n u o u s processes on G, it is n a t u r a l to consider the m e t r i c o n G given by t h e shortest p a t h in G b e t w e e n two points. We b e g i n w i t h a general definition. Definition

2 . 1 0 . Let A C I~d. For x, y E A set

d A ( x , y ) = inf{17 ] : 7 is a p a t h b e t w e e n x a n d y a n d 7 C A}. If dA(x, y) < oc for all x, y E A we call dA the geodesic metric on A. 2 . 1 1 . Suppose A is dosed, and that d A ( x , y ) < oc for all x, y E A. Then dA is a metric on A a n d (A, dA) has the geodesic property: Lemma

For each x, y E A there exists a map ~ ( t ) : [0, 1] --+ A such that dA(x,4~(t)) : t d A ( z , y ) ,

dA(4~(t),y) = (1 -- t)dA(x,y).

Proof. It is clear t h a t dA is a m e t r i c on A. To prove t h e geodesic property, let x , y E A, a n d D : d A ( x , y ) . T h e n for each n _> 1 there exists a p a t h %~(t), 0 < t < 1 + D such t h a t 7~ C A, [d'y~(t)l = dr, 7~(0) = x a n d 7 ~ ( t n ) : y for some D <_ tn <_ D q- n -1. I f p E [0, D] fq Q t h e n since Ix - "Yn(P)] -< P t h e sequence ('/n(p)) has a convergent subsequence. By a d i a g o n a l i z a t i o n a r g u m e n t there exists a s u b s e q u e n c e n~ such t h a t "/n, (P) converges for each p E [0, D] M Q; we c a n take = lim 7,~"

Lemma

[]

2 . 1 2 . For x , y E G, Ix -

Yl -< d a ( x , y ) _< ell x - YI'

Proof. T h e left h a n d i n e q u a l i t y is evident. It is clear from t h e s t r u c t u r e of H,~ t h a t if A, B are n - t r i a n g l e s a n d A fq B = 0, then la - b I _> (v/3/2)2 - ~

for a E A, b E B.

Let x, y E G a n d choose n so t h a t

(v /2)2 -("+a) < I x - yl <

2-".

So x, y are either in t h e s a m e n - t r i a n g l e , or in a d j a c e n t n-triangles. In either case choose z E G n so it is in t h e same n - t r i a n g l e as b o t h x a n d y.

13 Let z,, = z, a n d for k > n choose zk E G~ such t h a t x, zk are in the same ktriangle. T h e n since zk and zk+l are in the same k-triangle, and b o t h are contained in H k + l , we have da(zk,z~+l) = dH~+l(Zk,Zk+l ) < 2 - k . So,

da(z,x) < ~ da(zk,z~+l) <<21-~ < 4Ix -y]. k:n

Hence dG(x, y) < dG(x, z) + dv(z, y) < 8Ix

Y["

[]

Construction of a diffusion on the Sierpinski gasket. Let G,~ be the set of vertices of n-triangles. We can make Gn into a graph in a n a t u r a l way, by t a k i n g {x, y} to be an edge in G~ if x, y belong to the same n-triangle. (See Fig. 2.3). Write E,~ for the set of edges.

Figure 2.3: The graph G3. Let y ( n ) , k = 0 , 1 , . . . be a simple r a n d o m walk on G,~. Thus from x E G,~, the process Y('~) j u m p s to each of the neighbours of x with equal probability. ( A p a r t from the 3 points in Go, all the points in G,~ have 4 neighbours). The obvious way to construct a diffusion process (Xt,t > 0) on G is to use the graphs G,~, which provide a n a t u r a l a p p r o x i m a t i o n to G, and to t r y to define X as a weak limit of the processes Y('~). More precisely, we wish to find constants (an, n > 0) such t h a t

(2.7)

(y(n) [~.~1, t

_> 0)

(xt,~ _> 0).

We have two problems: (1) How do we find the right ( a n ) ? (2) How do we prove convergence? We need some more notation. D e f i n i t i o n 2.13. Let ,,q,~ be the collection of sets of the form G N A, where A is an n-triangle. We call the elements of ,S,~ n-complexes. For x C Gn let Dn(x) = [.J{S c

S~:zcS}. T h e key p r o p e r t i e s of the SG which we use are, first t h a t it is very symmetric, a n d secondly, t h a t it is finitely ramified. (In general, a set A in a metric space F is

14

finitely ramified if there exists a finite set B such t h a t A - B is not connected). For the SG, we see t h a t each n-complex A is disconnected from the rest of the set if we remove the set of its corners, t h a t is A f3 Gn. T h e following is the key observation. Suppose Y0( ' ) = y E G,~-I (take y r Go for simplicity), a n d let T = inf{k > 0 : y(n) e Gn-1 - {y}}. Then y ( n ) can only escape from D~-I(y) at one of the 4 points, { x l , . . . , x4} say, which are neighbours of y in the graph ( G n _ l , E n - ~ ) . Therefore Y('~) E { X l , . . . , x4 }. F u r t h e r the s y m m e t r y of the set G= (3 D=(y) means t h a t each of the events {YT(n) = xi} is equally likely. X2

X3

Xl

Y

X4

Figure 2.4: y and its neighbours. Thus

(Y(T~)=x~

y ( n ) = y ) 1= ~ ,

a n d this is also equal to ~(Y1(~-1) -- xiIY(~-1) = y). (Exactly the same argument applies if y C Go, except t h a t we t h e n have only 2 neighbours instead of 4). It follows t h a t Y(~) looked at at its visits to G ~ - I behaves exactly like y ( , ~ - l ) . To state this precisely, we first make a general definition. D e f i n i t i o n 2.14. Let T = ]~+ or Z+, let (Zt,t E 7~) be a cadlag process on a metric space F , a n d let A C F be a discrete set. T h e n successive disjoint hits by Z on A are t h e s t o p p i n g times To, T 1 , . . . defined by T0=inf{t>0:Zt

(2.8)

EA},

T~+I = i n f { t > T ~ : Z t c A - { Z T ~ } } ,

n>_O.

W i t h this notation, we can summarize the observations above. Lemma

2.15. Let (Ti)i>_o be successive disjoint hits by y(,O on G~-I.

Then

(YT(?), i _> O) is a simple random wMk on G,~-I and is therefore equal in law to

(y(,-1), i _> 0). Using this, it is clear t h a t we can build a sequence of "nested" r a n d o m walks on G,~. Let N _> 0, and let y(N),~ k _> 0 be a SRW on GN with y(N) = 0. Let

[TN,m~ 0 <_ m < N - 1 and ~ i )i>_o be successive disjoint hits by y(N) on Gin, and set

y(m) = y ( m ( T N , m ) = y (TuN,) m '

i > 0.

15 It follows from L e m m a 2.15 t h a t y(m) is a SRW on Gin, and for each 0 < n < m <_ N we have t h a t y ( m ) , s a m p l e d at its successive disjoint hits on G,~, equals y ( n ) . We now wish to construct a sequence of SRWs with this p r o p e r t y holding for 0 < n <_ m < co. This can be done, either by using the Kolmogorov extension theorem, or directly, b y building y ( g + l ) from y(N) with a sequence of i n d e p e n d e n t "excursions". T h e argument in either case is not hard, and I omit it. Thus we can construct a p r o b a b i l i t y space (fl, .T, ~), carrying r a n d o m variables (v(n) "k , n _ > 0 , k > 0 ) s u c h t h a t (a) For each n, ( y ( , 0 , k > 0) is a SRW on G= starting at 0. (b) Let T~ 'm be successive disjoint hits by Y('*) on G , , . (Here m < n). T h e n (2.9)

y(,,)(T:,m) = y(m),

i > O,

m < n.

If we j u s t consider the p a t h s of the processes y ( , 0 in G, we see t h a t we are viewing successive discrete approximations to a continuous path. However, to define a limiting process we need to rescale time, as was suggested by (2.7). W r i t e 7- = T~ '~ = min{k > 0 : [y(1)] = 1}, and set f(s) = E s ~, for s 6 [0,1]. L e m r n a 2.16. f ( , ) = s2/(4 - 3,), ET ----i f ( l ) = 5, a n d E r k < co for a / / k .

Proof. This is a simple exercise in finite state Markov chains. Let a l , a2 be the two 1 non-zero elements of Go, let b = 89 + a2), and ci -- ~ai. Writing fc(s) -- E c l s ~, a n d defining .fb, fa similarly, we have fa(S) = 1, f(8) = sfc(8),

fc(a) = 88

4- fc(S) 4- fb(s) 4- f a ( s ) ) ,

h(,) = 89

+ f,,(s)),

a n d solving these equations we o b t a i n f(s). T h e remaining assertions follow easily from this.

[]

a 1

0

b 2

a2

Figure 2.5: The graph G~.

~6 Now let Zn = Tff '~ n > 0. The nesting property of the random walks Y(~) implies that Zn is a simple branching process, with offspring distribution (p,~), where (2.10)

f(s) = ~

skpk.

k=2

To see this, note that y ( n + l ) , for T~ +1'~ _< k _< T ~ +1'~ is a SRW on G,~+I N Dn(Yi('0), and that therefore T~++l'n - T ~ +1'~ (d_)z. Also, by the Markov property, the r.v. ~ = T ~ +1''~ - T~ +1''~, i _> 0, are independent. Since Z~ --1

i=O (Z~) is a branching process. As E~-2 < o% and Ez -- 5, the convergence theorem for simple branching processes implies that 5-'~Z~ ~"% W for some strictly positive r . v . W . (See [Har, p. 13]). The convergence is easy using a martingale argument: proving that W > 0 a.s. takes a little more work. (See [Har, p. 15]). In addition, if

~o(u) = Be - ' w then ~o satisfies the functional equation (2.11)

~o(5u) = f(~oCu)),

~J(O) ----- 1 .

We have a similar result in general. Proposition

2.17. Fix m > O. The processes

r~m , Z(i) = T~'m - Ti_I

n > m

are branching processes with offspring distribution T, and Z (i) are independent. Thus there exist W(i m) such that for each m (W(im),i >_ O) are independent, 5-row, and n,m ) --+ W.(trn) a.$. 5--n ( T ? ' m - Ti_l Note in particular that E(T~ '~ ) ---- 5 n, that is that the mean time taken by y ( , 0 to cross G~ is 5 ~. In terms of the graph distance on G~ we have therefore that y(n) requires roughly 5 '~ steps to move a distance 2n; this may be compared with the corresponding result for a simple random walk on Z d, which requires roughly 4 n steps to move a distance 2 n. The slower movement of Y(n) is not surprising - - to leave Gn f) B(O, 1/2), for example, it has to find one of the two 'gateways' (1/2, 0) or (1/4, vf3/4). Thus the movement of y(n) is impeded by a succession of obstacles of different sizes, which act to slow down its diffusion.

17 Given the space-time scaling of y ( , 0 it is no surprise that we should take a s = 5 ~ in (2.7). Define X~'

= v(n)

"[5-t],

t >_ O.

I n view of the fact that we have built the y(n) with the nesting property, we can replace the weak convergence of (2.7) with a.s. convergence. T h e o r e m 2.18. The processes X '~ converge a.s., and uniformly on compact interwa/s, to a process Xt, t >__O. X is continuous, a n d X t E G for all t > O.

Proof. For simplicity we will use the fact that W has a non-atomic distribution function. Fix for now rn > 0. Let t > 0. Then, a.s., there exists i = i(w) such that i

i+1

Ew?

Ew?).

j=l

j----1

a s w j~) = l i m . _ ~ 5-" (T? '~ - T?:7) it follows that for n > n0(o~), n~m z ? '~ < 5'~t < Ti+x 9

(2.12)

Now Y ( - ) ( T ? ,~) = r, ('~) by (2.9). Since r~('~) ~ D . ( r , (~)) for T? '~ < k < T3;1, we have [l~:~]-- - Y/(m) I < 2 - ~

for all n >__no.

This implies that [X~ - X~' I < 2 -'~+1 for n, n ' > no, so that X~ is Cauchy, and converges to a r.v. Xt. Since X ~ 6 Gn, we have X t 6 G. W i t h a little extra work, one can prove that the convergence is uniform in t, on compact time intervals. I give here a sketch of the argument. Let a 6 N, a n d let

~m :

rain l
W: m) TM

T h e n ~,~ > 0 a.s. Choose no such that for n > no i

5-nTt'm-

E

W (m) < l~n ,

l
TM.

j=l n~m T h e n if i = i(t, w) is such that Wim < t < W i +m l , a n d i < a 5 m w e h a v e 5 _ ~ Ti_ 1 < --n n~m t <5 T~+2 for a l l n _ > n 0 . So, I X ~ - Yim[ < 2 -m+x for all n > no. This implies

t h a t if T,~ = ~ i = as~ 1 W (m), and S < Tin, then sup I X : O n O . If S < llm infm T m

X~'l <

2- " + 5

then the uniform a.s. convergence on the

(random) interval [0, S] follows. If s, t < g,~ and It - s I < (m, then we also have [X~ -X~n[ < 2 -m+~ for n _> no. Thus X is uniformly continuous on [0, S]. Varying a we also o b t a i n uniform a.s. convergence on fixed intervals [0, to]. []

18 Although the notation is a little cumbersome, the ideas underlying the construction of X given here are quite simple. The argument above is given in [BP], but Kusuoka [Kusl], and Goldstein [Go], who were the first to construct a diffusion on G, used a similar approach. It is also worth noting that Knight [Kn] uses similar methods in his construction of 1-dimensional Brownian motion. The natural next step is to ask about properties of the process X. But unfortunately the construction given above is not quite strong enough on its own to give us much. To see this, consider the questions (1) Is W = limn-.oo 5 - n T ~ '~ = inf{t _> 0 : X~ 9 G - {0}}? (2) Is X Markov or strong Markov? For (1), we certainly have X w 9 G - { 0 } . However, consider the possibility that each of the r a n d o m walks Y. moves from 0 to a2 on a path which does not include al, but includes an approach to a distance 2 - " . In this case we have al ~ {X~, 0 < t < W}, but X T = al for some T < W. Plainly, some estimation of hitting probabilities is needed to exclude possibilities like this. (2). The construction above does give a Markov property for X at stopping times of the form V.i z..,i= 1 w(-~) ,, j . But to obtain a good Markov process X = (Xt, t _> 0, ?~, x 9 G) we need to construct X at arbitrary starting points x 9 G, and to show that (in some appropriate sense) the processes started at close together points x and y are close. This can be done using the construction given above - - see [BP, Section 2]. However, the argument, although not really hard, is also not that simple. In the remainder of this section, I will describe some basic properties of the process X , for the most part without giving detailed proofs. Most of these theorems will follow from more general results given later in these notes. Although G is highly symmetric, the group of global isometries of G is quite small. We need to consider maps restricted to subsets. D e f i n i t i o n 2.19. Let (F, p) be a metric space. A local isometry of F is a triple (A, B, ~o) where A, B are subsets of F and ~ois an isometry (i.e. bijective and distance preserving) between A and B, and between OA and OB. Let ( X t , t > O,g~,x 9 F ) b e a M a r k o v process on F. For H C F , set TH = inf{t > 0 : X t 9 H } . X is invariant with respect to a local isometry (A, B, ~o) if

P= ( ~ ( x , ^ r o ~ ) 9

t > 0) = P~(=)

(x,^ro, 9 t > o).

X is locally isotropic if X is invariant with respect to the local isometries of F. T h e o r e m 2.20. (a) There exists a continuous strong Markov process X : (Xt, t > O, ? L x 9 G) on G.

(b) The semigroup on C(G) de~ed by Pal(x) = E= y ( x , ) is Fe//er, and is pG-symmetric:

/G f(x)P~g(x)~G(d~) = /

a(~)Pd(x)~G(dx).

19

(c) X is locally isotropic on the spaces ( a , I" - " I) and ( G, da). (d) For n > 0 let Tn,i, i > 0 be successive disjoint hits by X on Gn. Then ~('*) = ~(~) XT..~, i > 0 defines a S R W on Gn, and =[5*t] -+ X t uniformly on compacts, a.s. So, in particular (Xt, t > 0, I?~ is the process constructed in Theorem 2.18. This theorem will follow from our general results in Sections 6 and 7; a direct proof may be found in [BP, Sect. 21. The main labour is in proving (a); given this (b), (c), (d) all follow in a relatively straightforward fashion from the corresponding properties of the approximating random walks ~(n). The property of local isotropy on (G, da) characterizes X : T h e o r e m 2.21. (Uniqueness). Let (Zt, t > O, Qz, x C ~) be a non-constant locally isotropic diffusion on (G, da). Then there exists a > 0 such that (F(z,

~ .,t ___0) = ? = ( x o , ~ .,t _> 0).

(So Z is equal in law to a deterministic time change of X ) . The beginning of the proof of Theorem 2.21 runs roughly along the lines one would expect: for n > 0 let (~(=), i >_ 0) be Z sampled at its successive disjoint hits on Gn. The local isotropy of ~z implies that Y(=) is a SRW on G=. However some work (see [BP, Sect. 8]) is required to prove that the process Y does not have traps, i.e. points z such that Q* (Yt = z for all t) = 1. R e m a r k 2.22. The definition of invariance with respect to local isometries needs some care. Note the following examples. 1. Let x , y E Gn be such that D , ( z ) n Go = ao, Dn(y) C3Go = 0. Then while there exists an isometry ~ from D,~(z) n G to D=(y) n G, ~v does not map OnD=(z) ;3 G to ORDn(y) n G. (OR denotes here the relative boundary in the set G). 2. Recall the definition of Hn, the n-th stage in the construction of G, and let B , = OH,~. We have G = cl(UB,). Consider the process Zt on G, whose local motion is as follows. If Zt E H,~ - H=-I, then Zt runs like a standard 1-dimensional Brownian motion on H , , until it hits Hn-1. After this it repeats the same procedure on Hn-1 (or H,~-k if it has also hit H,,-k at that time). This process is also invariant with respect to local isometries (A, B, ~o) of the metric space (G, I" - " I). See [He] for more on this and similar processes. To discuss scale invariant properties of the process X it is useful to extend G to an unbounded set G with the same structure. Set = U 2'~G, rt=0

and let G~ be the set of vertices of n-triangles in G,~, for n > 0. We have G~ = U 2kGn+k, k=0

20 and if we define Gm = {0} for m < 0, this definition also makes sense for n < 0. We can, almost exactly as above, define a limiting diffusion .~ = (_~,, t _> 0, ~x, x E G) Oil G: =

t _> o, a s

where (Y~('~),n > 0, k _> 0) are a sequence of nested simple random walks on G=, and the convergence is uniform on compact time intervals. The process X satisfies an analogous result to Theorem 2.20, and in addition satisfies the scaling relation (2.13)

F~(22~ 9 -, t > 0) : ~2~(25t 9 -, t > 0).

Note that (2.13) implies that _~ moves a distance of roughly t l~176 Set d,~ = d~(G) -- l o g 5 / l o g 2 .

in time t.

We now turn to the question: "What does this process look like?" The construction of X, and Theorem 2.20(d), tells us that the 'crossing time' of a 0-triangle is equal in law to the limiting random variable W of a branching process with offspring p.g.f, given by f ( s ) = s 2 / ( 4 - 3 s ) . From the functional equation (2.11) we can extract information about the behaviour of ~o(u) = E e x p ( - u W ) as u --* oo, and from this (by a suitable Tauberian theorem) we obtain bounds on P ( W < t) for small t. These translate into bounds on Px(1Xt - x I > A) for large A. (One uses scaling and the fact that to move a distance in G greater than 2, X has to cross at least one 0-triangle). These bounds give us many properties of X. However, rather than following the development in [BP], it seems clearer to first present the more delicate bounds on the transition densities of .Y and X obtained there, and derive all the properties of the p r o c ~ s from them. Write ~ c for the analogue o f # a for G, and Pt for the semigroup of X. Let ~. be the infinitesimal generator of Pt. 2.23. fit and Pt have densities ~(t, x, y) and p(t, x, y) respectively. ~(t, ~, y) i~ continuous on (0, ~ ) • ~ • ~. ~(t,~,y) = ~(t,y,~) for ~ t,~,y. t -~ ~ ( t , . , y) i~ C ~ on (0, ~ ) for each (~, y). For each t, y

Theorem

(a) (b) (c) (d)

I~(t,x,y) - ~(t,x',y)] <_ Clt-Xlx - x'l d~-d',

x,x' 9 G.

(e) Fort E (O, cc), x , y C (2.14)

c2t - d ~ / ~ exp

-c3

--ty-

~ c4t -d2/d~ exp

<_~ ( t , x , y ) --c5

21 (f) For each Yo 9 G, p(t,x, yo) is the fundamental solution o f the heat equation on G with pole at Yo:

,yo) = Z (t, x, yo),

y0) =

(g) p(t, x, y) satisfies (a)-(f) above (with G replaced by G and t 9 (0, c~] replaced by t 9 (0, 1]). R e m a r k s . 1. T h e p r o o f of this in [BP] is now largely obsolete - - simpler m e t h o d s are now available, t h o u g h these axe to some extent still based on the ideas in [BP]. 2. If d f -- d and d~, = 2 we have in (2.14) the form of the transition density of Brownian m o t i o n in R a. Since d~ -- log 5 / l o g 2 > 2, the tail of the d i s t r i b u t i o n of [Xt - x[ u n d e r I?~ decays more r a p i d l y t h a n an exponential, b u t more slowly t h a n a Gaussian. It is fairly straightforward to integrate the b o u n d s (2.14) to obtain information a b o u t X . At this point we just present a few simple calculations; we will give some further p r o p e r t i e s of this process in Section 3. D e f i n i t i o n 2.24. For x 9 G, n 9 Z, let x,~ be the point in Gn closest to x in E u c l i d e a n distance. (Use some p r o c e d u r e to b r e a k ties). Let D , ( x ) = D,(x,~). Note t h a t "fiG(D,(x,~)) is either 3 - " or 2.3 - " , t h a t (2.15)

Ix - Yl -< 2 ' 2 - ~

if y 9 D , ( x ) ,

and that (2.16)

Ix - Yl >- ~-~2-('~+1)

if y C G N D , ~ ( x ) c .

T h e sets D,~(x) form a convenient collection of neighbourhoods of points in G. Note t h a t U , + z D , ( x ) = G. C o r o l l a r y 2.25. F o r x E G,

clt 2/d= <_ E ~ IX~ - x} 2 <_ c~t 2/d= ,

t > O.

Proof. We have E: [Xt - xl 2 = ~ ( y - x)2~(t, x, y)~to(dy ). Set Am = Dm(x) - Dm+l(x). T h e n

(2.17)

/A (y - x)2P(t'x'Y) c(dY)

=

e:p

22 Choose n such t h a t 5 -'~ _< t < 5 -~+1, and write am(t) for the final t e r m in (2.17). Then

E~(x, - x) ~ <_ ~

am(t) +

m:--oo

am(t) m:n

For m < n, 5 - r " / t > 1 a n d the exponential t e r m in (2.17) is dominant. After a few calculations we o b t a i n n--1

a . ( t ) _< c ( 2 - " ) ' + ' , t - ' , / ~ ~rt~-- oo

at (2h-df)/g~-d2/4~ ~ Ct (2+4f)/g~-df/d~ ~

ct 2/4~,

where we used t h e fact t h a t ( 2 - n ) d~ x t. For m > n we neglect the exponential term, a n d have

am(t) < ~ t - ~ , / ~

Z

(2-")~+~,

< ct-~lld~ (2---)2+d~ < c't21 d~. Similar calculations give the lower bound. Remarks

[]

2.26. 1. Since 2 / d ~ = log 4 / l o g 5 < 1 this implies t h a t X is subdiffusive.

2. Since ~G (B(x, r)) • r d,, for x E G, it is t e m p t i n g to t r y and prove Corollary 2.25 by the following calculation:

(2.18)

E~12~ - ~1~

=

•

/{ /o

= t 2/d~

f

r2dr

~(t,x,y)~tc(dy)

JOB(x,r)

dr r 2 rds-lt-ds/d'~ exp --c(rd~/t) l[d~'-I

(

/o

s l+a~ exp - c ( s d~)l/a~-I

(

)

)

ds = ct 2/d~.

Of course this calculation, as it stands, is not valid: the e s t i m a t e

~(B(~,~

+ ~ ) - B(x,~)) • ~ ' - ' ~

is c e r t a i n l y not valid for all r. But it does hold on average over length scales of 2 '~ < r < 2 n+l, a n d so splitting G into suitable shells, a rigorous version of this calculation m a y be o b t a i n e d - a n d this is what we did in the proof of Corollary 2.25. T h e / k - p o t e n t i a l kernel density of )~ is defined by

~,(~, y) =

e-~'~(t, ~, y) dr.

From (2.14) it follows t h a t ux is continuous, t h a t u ~ ( x , z ) <_ r dl/d~-l, and t h a t ux ~ ~ as A --* 0. Thus the process X (and also X ) "hits points" - t h a t is if

23 Ty -- inf{t > 0 : )f~ = y} then

(2.19)

< oo) > 0.

It is of course clear that X must be able to hit points in Gn - otherwise it could not move, but (2.19) shows that the remaining points in G have a similar status. The continuity of u x ( x , y) in a neighbourhood of x implies that F ' ( T : = 0) = 1, that is t h a t x is regular for {x} for all x 6 G. The following estimate on the distribution of I)ft - x[ can be obtained easily from (2.14) by integration, but since this bound is actually one of the ingredients in the proof, such an argument would be circular. Proposition

2.27. For x 6 G, A > 0, t > 0,

X[ >

),)

_< c3 e x p From this, it follows that the paths of _~ are HSlder continuous of order l/d,,, for each e > 0. In fact we can (up to constants) obtain the precise modulus of continuity of X. Set h(t) = t l/d~ ( l o g t - 1 ) (d~-l)/d~ . T h e o r e m 2.28. (a) For x E G

cl _< lim

sup

8J.0 0<s
[)(, - Xt[ _< c2, h(s - t)

I?~ - a.s.

I~-~l<S (b) T h e p a t h s of .~ are of infinite quadratic variation, a.s., and so in particalar is n o t a semimartingale. The proof of (a) is very similar to that of the equivalent result for Brownian motion in I~d. For (b), Proposition 2.23 implies that [Xt+h - Xt[ is of order h 1/d~ ; as d,o > 2 this suggests that X should have infinite quadratic variation. For a proof which fills in the details, see [BP, Theorem 4.5]. [] So far in this section we have looked at the Sierpinski gasket, and the construction and properties of a symmetric diffusion X on G (or G). The following three questions, or avenues for further research, arise naturally at this point.

24 1. Are there other n a t u r a l diffusions on the SG? 2. Can we do a similar construction on other fractals? 3. W h a t finer p r o p e r t i e s does the process X on G have? (More precisely: what a b o u t p r o p e r t i e s which the b o u n d s in (2.17) are not strong enough to give information on?) T h e bulk of research effort in the years since [Kusl, Go, BP] has been devoted to (2). O n l y a few p a p e r s have looked at (1), and (apart from a number of works on s p e c t r a l properties), the same holds for (3). Before discussing (1) or (2) in greater detail, it is worth extracting one p r o p e r t y of the SRW y ( D which was used in the construction. Let V = (V,~,n > O,I?a,a E Go) be a Markov chain on Go: clearly V is specified by the t r a n s i t i o n probabilities

p(al,aj) = Fa'(V1 = hi),

0 _< i,j <_ 2.

We take p(a,a) = 0 for a E G0, so V is d e t e r m i n e d by the three probabilities p(ai,aj), where j -- i + 1 (mod 3). Given V we can define a Markov Chain V ~ on G1 by a process we call replication. Let {b01, b02, b12} be the 3 points in G1 - Go, where bij -- 89 + aj). We consider G1 to consist of three 1-cells {hi, bij,j ~ i}, 0 < i < 2, which intersect at the points {bij}. T h e law of V I m a y be described as follows: V I moves inside each 1-cell in t h e way same as V does; if V~ lies in two 1-cells then it first chooses a 1-cell to move in, and chooses each 1-cell with equal probability. More precisely, writing Y' = (V~n,n >_ o,~a,a E G1), and p'(a, b) =

= b),

we have (2.20)

p'(ai, biy) ----p(ai, a j ) , Pl(bij,bik) = lp(aj, ak),

hi). pl(bij,ai) = ~p(aj, 1

Now let T~, k > 0 be successive disjoint hits by V I on Go, and let Uk = V~-k , k > 0. T h e n U is a Markov Chain on Go; we say t h a t V is decimation invariant if U is equal in law to V. We saw above t h a t the SRW y(0) on Go was decimation invariant. A n a t u r a l question is: W h a t other d e c i m a t i o n invariant Markov chains are there on Go? Two classes have been found: 1. (See [Go]). Let p(ao,al) = p(al,ao) = 1, p(a2 ao) - 1 2. "p-stream r a n d o m walks" ([Kuml]). Let p E (0, 1) and

p(ao,al) = p ( a l , a 2 ) =p(a2,ao) =p. From each of these processes we can construct a limiting diffusion in the same way as in T h e o r e m 2.18. T h e first process is reasonably easy to understand: essentially its p a t h s consist of a downward drift (when this is possible), and a behaviour

25 like 1-dimensional Brownian motion on the portions on G which consist of line segments parallel to the x-axis. For p > 89 Kumagai's p-stream diffusions tend to rotate in an anti-clockwise direction, so are quite non-symmetric. Apart from the results in [Kuml] nothing is known about this process. Two other classes of diffusions on G, which are not decimation invariant, have also been studied. The first are the "asymptotically 1-dimensional diffusions" of [HHW4], the second the diffusions, similar to that described in Remark 2.22, which are (G, [ . - . [)-isotropic but not (G, d~)-isotropic - s e e [He]. See also [HH1, HK1, HHK] for work on the self-avoiding random walk on the SG. Diffusions on o~her /racta/ sets. Of the three questions above, the one which has of making similar constructions on other fractals. which can arise, consider the following two fractals, by a Cantor type procedure, based on squares rather the figure gives the construction after two stages.

mml

mm

=%

received most attention is that To see the kind of difficulties both of which are constructed than triangles. For each curve

mmm

9

9

=%

Figure 2.6: The Vicsek set and the Sierpinski carpet. The first of these we will call thc "Vicsek set" (VS for short). We use similar notation as for the SG, and write Go, G 1 , . . . for the succession of sets of vertices of corners of squares. We denote the limiting set by F = Fvs. One difficulty arises immediately. Let Y~ be the SRW on Go which moves from any point z E Go to each of its neighbours with equal probability. (The neighbours of x are the 2 points y in Go with Ix - y[ = 1). Then y(0) is not decimation invariant. This is easy to see: y(0) cannot move in one step from (0, 0) to (1,1), but y(1) can move from (0, 0) to (1, 1) without hitting any other point in Go. However it is not hard to find a decimation invariant random walk on Go. Let p E [O, 1], and consider the random walk (Y~,r > 0, E~,x E Go) on Go which moves diagonally with probability p, and horizontally or vertically with probability 89( l - p ) . Let (Y~',r _> 0, E~,x 9 G1) be the Markov chain on G1 obtained by replication, and let T~, k > 0 be successive disjoint hits by Y' on Go.

26 0 ! T h e n writing f ( p ) = ~p(Y~'x = (1,1)) we have (after several minutes calcula-

tion) f(P)=

1 4-3p"

T h e equation f(p) = p therefore has two solutions: p = ~ and p = 1, each of which corresponds to a d e c i m a t i o n i n w r i a n t walk on Go. (The number ~ here has no general significance: if we h a d looked at the fractal similar to the Vicsek set, b u t b a s e d on a 5 x 5 square r a t h e r t h a n a 3 x 3 square, t h e n we would have o b t a i n e d a different number). One m a y now c a r r y through, in each of these cases, the construction of a diffusion on the Vicsek set F , very much as for the Sierpinski gasket. For p = 1 one gets a r a t h e r uninteresting process, which, if s t a r t e d from (0, 0), is (up to a constant t i m e change) 1-dimensionM Brownian motion on the diagonal {(t, t), 0 < t < 1}. It is w o r t h r e m a r k i n g t h a t this process is not strong Markov: for each x E F one can take I?* to be t h e law of a Brownian motion moving on a diagonal line including x, b u t the strong Markov p r o p e r t y will fail at points where two diagonals intersect, such as the point (89 89 For p = ] one obtains a process (Xt,t > 0) with much the same behaviour as t h e Brownian m o t i o n on the SG. We have for the Vicsek set (with p = 89 df(Fvs) = log 5 / l o g 3, dw(Fvs) = log 1 5 / l o g 3. This process was studied in some d e t a i l b y K r e b s [Krl, Kr2]. T h e Vicsek set was mentioned in [Go], and is one of the "nested fractals" of L i n d s t r 0 m [L1]. This e x a m p l e shows t h a t one m a y have to work to find a decimation invariant r a n d o m walk, a n d also t h a t this m a y not be unique. For the VS, one of the decimation invariant r a n d o m walks was degenerate, in the sense t h a t P~(Y hits y) = 0 for some z, y E Go, a n d we found t h e associated diffusion to be of little interest. But it raises t h e possibility t h a t there could exist regular fractals carrying more t h a n one " n a t u r a l " diffusion. T h e second example is the Sierpinski carpet (SC). For this set a more serious difficulty arises. T h e VS was finitely ramified, so t h a t if Yt is a diffusion on Fvs, a n d (Tk, k > 0) are successive disjoint hits on G,~, for some n > 0, then (YTk, k >_O) is a Markov chain on Gn. However the SC is not finitely ramified: if (Zt, t > 0) is a diffusion on Fsc, t h e n the first exit of Z from [0, 89 could occur anywhere on t h e line segments {($,y),01 _< y _< ~},1 {(z, 5),10 < x < ~}. It is not even clear t h a t a diffusion on Fsc will hit points in Gn. Thus to construct a diffusion on Fsc one will need very different m e t h o d s from those outlined above. It is possible, a n d has been done: see [BBI-BB6], a n d [Bas] for a survey. On the t h i r d question mentioned above, disappointingly little has been done: most known results on the processes on t h e Sierpinski gasket, or o t h e r fractals, are of roughly the same d e p t h as the bounds in T h e o r e m 2.23. Note however the results on t h e s p e c t r u m of/~ in [FS1, FS2, S h l - S h 4 ] , and the large deviation results in [Kumh]. Also, K u s u o k a [Kus2] has very interesting results on the behaviour of h a r m o n i c functions, which i m p l y t h a t the measure defined formally on G by

v(dx) --- I v / I z (~)~(dz) is singular with respect to p. There are m a n y open problems here.

27 3. F r a c t i o n a l D i f f u s i o n s . In this section I will introduce a class of processes, defined on metric spaces, which will include m a n y of the processes on fractals mentioned in these lectures. I have chosen an axiomatic approach, as it seems easier, and enables us to neglect (for t h e t i m e being!) much of fine detail in the geometry of the space. A m e t r i c space (F, p) has t h e midpoint property if for each x, y E F there exists 1 X , y). Recall t h a t the geodesic metric dG in z E F such t h a t p(z, z) = p ( z , y ) = ~p( Section 2 h a d this property. The following result is a straightforward exercise: L e m m a 3.1. (See [Blu]). Let (F, p) be a complete metric space with the midpoint property. Then for each x, y E F there ex/sts a geodesic path (7(t), 0 < t < 1) such that 7(0) = z, 3'(1) = y a n d p(~/(s),'r(t)) = ]t - s[d(z,y), 0 < s < t < 1. For this reason we will frequently refer to a metric p with the m i d p o i n t p r o p e r t y as a geodesic metric. See [Stul] for a d d i t i o n a l remarks a n d references on spaces of this type. D e f i n i t i o n 3.2. Let (F, p) be a complete metric space, and # be a Borel measure on ( F , B ( F ) ) . We call ( F , p , p ) a fractional metric space (FMS for short) if (3.1a)

(F, p) has the m i d p o i n t p r o p e r t y ,

a n d there exist d f > 0, and constants Cl,C2 such t h a t if r0 = s u p { p ( z , y ) : x , y E F } E (0, oo] is t h e d i a m e t e r of F t h e n (3.1b)

clr d' < # ( B ( x , r ) ) <_ c2 rdf

for

x E F, 0 < r < r0.

Here B ( z , r ) = {y 9 F : p ( z , y ) < r}. R e m a r k s 3.3. 1. I~d, with Euclidean distance and Lebesgue measure, is a FMS, with d I = d and r0 = cr 2. If G is the Sierpinski gasket, dG is the geodesic metric on G, and # = P c is t h e measure c o n s t r u c t e d in Section 2, then L e m m a 2.1 shows t h a t (G, dG, #) is a F M S , w i t h d I = dr(G) = l o g 3 / l o g 2 and r0 = 1. Similarly ( G , d ~ , ~ ) is a F M S with r 0 ~(~2.

3. If ( F k , d k , p k ) , k = 1,2 are F M S with the same d i a m e t e r r0 and p 9 [1, oo], then setting F = F1 x F2, d((Xl, z2), (Yl, Y2)) = (dl(Xl, y l ) P + d 2 ( z 2 , yz)P) l/p, # = Pl x #~, it is easily verified t h a t ( F , d , # ) is also a F M S with d r ( f ) = dr(F1 ) + dr(F2 ). 4. For simplicity we will from now on take either r0 = ~ or r0 = 1. We will write r 9 (0, r0] to m e a n r 9 (0, r0] n (0, ~ ) , and define r~' = ~ if a > 0 and r0 = co. A n u m b e r of p r o p e r t i e s of (F, p, #) follow easily from the definition. L e r n m a 3.4. (a) d i m H ( F ) = d i m p ( F ) = dr.

(b) F is locally compact. (c) df > 1. Proof. (a) is i m m e d i a t e from Corollary 2.8. (b) Let x 9 F, A = B ( x , 1), a n d consider a m a x i m a l packing of disjoint balls B ( x i , e ) , zi 9 A, 1 < i < m. As #(A) <_ c2, and # ( B ( z i , e ) ) >_ Cle d~, we have m < c2(cled~) -1 < cr Also A = U'~=lB(Xi,2e ). Thus any b o u n d e d set in F can be

28 covered by a finite n u m b e r of balls radius s; this, with completeness, implies t h a t F is locally compact. (c) Take x, y E F with p(x, y) = D > 0. A p p l y i n g the m i d p o i n t p r o p e r t y r e p e a t e d l y we obtain, for m = 2 k, k > 1, a sequence x -- zo,zl,... ,zm -- y with p(zi,zi+l) = D i m . Set r = D/2m: the balls B(zi,r) must be disjoint, or, using the triangle inequality, we would have p(x, y) < D. But then m--1

U B(z,, r) C B(x,D), i=0

so t h a t

rn--1

c2D d' > # ( B ( x , D ) ) > E

#(B(z,,r))

i=0

> mclD d~( 2 m ) - d s = cml--d~. If d f < 1 a c o n t r a d i c t i o n arises on letting m --* ~ .

[]

3.5. Let (F,p,#) be a fractional metric space. X = (~x, x E F, Xt, t ~ 0) is a fractional diffusion on F if Definition

A Markov process

(3.2a) X is a conservative Feller diffusion with state space F . (3.2b) X is # - s y m m e t r i c . (3.2c) X has a s y m m e t r i c t r a n s i t i o n density p(t, x, y) = p(t, y, x), t > 0, x, y E F , which satisfies, the C h a p m a n - K o l m o g o r o v equations a n d is, for each t > 0, j o i n t l y continuous. (3.2d) T h e r e exist constants (~,fl, 7, cl - c4, to = v0~, such t h a t

(3.3)

Clt

exp

<

~_ c3t -~ exp (-c4p(x, y)flTt-7) , x, y E F, 0

< t ~ to.

E x a m p l e s 3.6. 1. If F is ~d, a n d a(x) = aij(x), 1 < i, j < d, x C R d is bounded, s y m m e t r i c , measurable and uniformly elliptic, let s be the divergence form o p e r a t o r

~Oa,

x

0

zj

T h e n Aronsen's b o u n d s JAr] i m p l y t h a t the diffusion with infinitesimal generator s isaFD, witha=d/2, fl=2,7=l. 2. By T h e o r e m 2.23, the Brownian motion on the Sierpinski gasket described in Section 2 is a F D , with a = dl(SV)/dw(SG), fl = dw(SG) and 7 = 1/(fl - 1). T h e hypotheses in Definition 3.5 are quite strong ones, and (as the examples suggest) t h e assertion t h a t a p a r t i c u l a r process is an F D will usually be a substantial theorem. One could of course consider more general bounds t h a n those in (3.3) (with a correspondingly larger class of processes), b u t the form (3.3) is reasonably natural, a n d a l r e a d y contains some interesting examples. In an interesting recent series of papers S t u r m [Stul-Stu4] has studied diffusions on general metric spaces. However, the processes considered there t u r n out to have an essentially Gaussian long range behaviour, and so do not include any F D s with

29 In the rest of this section we will s t u d y the general properties of FDs. In the course of our work we will find some slightly easier sufficient conditions for a process to be a F D t h a n the b o u n d s (3.3), and this will be useful in Section 8 when we prove t h a t certain diffusions on fractais are FDs. We begin by obtaining two relations between t h e indices dl, a , fl, 7, so reducing the p a r a m e t e r space of F D s to a two-dimensional one. We will say t h a t F is a F M S ( d f ) if F is a F M S and satisfies (3.1b) with p a r a m e t e r df (and constants el, c2). Similarly, we say X is a F D ' ( d f , a, f l , 7 ) if X is a F D on a F M S ( d f ) , and X satisfies (3.3) with constants a , /3, 7. (This is t e m p o r a r y n o t a t i o n - - hence the '). It w h a t follows we fix a F M S (F, p, #), with p a r a m e t e r s r0 and d I. Lemma

3.7. L e t a , 7 , x > 0 and set I ( % x) =

~1 ~176 e -xt~ dt,

S(a,7 ,x) = E

n=0

a~e-=~'"

Then (3.4)

( a - 1 ) S ( ( ~ , 7 , a T z ) < I ( % x ) <_ (a - 1 ) S ( a , 7 , z),

and (3.5)

I ( 7 , z ) • z -1/~

(3.6)

I ( 7 , z) • z - l e - x

for z <_ 1, for x > 1,

Proof. We have

,(%,): and e s t i m a t i n g each t e r m in the sum (3.4) is evident. If 0 < x < 1 t h e n since x 1 / ~ I ( 7 , x) =

1/~ e - e ' d s --* c(7) as x ~ O,

(3.5) follows. If x _> 1 t h e n (3.6) follows from the fact t h a t =

Lemma drift.

+

--> 7 - '

3.8. ("Scaling relation"). L e t X be a F D ' ( d l , a , f l , 7 )

-->

on F . T h e n a =

Proof. From (3.1) we have p ( t , x , y ) >_ e l t - ~ e -c~ = c3t - ~

[]

for p ( x , y ) < t 1/p.

30 Set to = To~. So i r A = B(x, tl/~), and t < to

1 > ~*(p(z, xt) <_t 1/~) = fAP(t,z,y)g(dy) >_c3t-~#(A) > ct -~+d'/a. If ro = cr t h e n since this holds for all t > 0 we must have a = dl/ft. If T0 = 1 t h e n we only deduce t h a t a < dr~ft. Let now T0 = 1, let A > 0, t < 1, and A = B(x, Atl/~). We have t t ( F ) < c3.1.2, a n d therefore I=~(Xt 9

~)

< g(A)supp(t,x,y) + # ( f - A) sup p(t,x,y) yEA < c4t--ct+d~/~Ad~/~ if-

yEA ~

c~t-ae-CO x~.

Let )~ = ((df/fl)c61 log(1/t))l/~'r; then we have for all t < 1 t h a t

1 < ct-"+aH#(1 + (log(lit)) 1/#'t, which gives a contradiction unless a _> drift.

[]

T h e next relation is somewhat deeper: essentially it will follow from the fact t h a t t h e l o n g - r a n g e b e h a v i o u r o f p ( t , z, y) is fixed by the exponents d i a n d fl governing its s h o r t - r a n g e behaviour. Since 7 only plays a role in (3.3) when p(z, y)# >> t, we will be able to o b t a i n 7 in terms of df and ~ (in fact, it turns out, of/3 only). We begin by deriving some consequences of the bounds (3.3). L e m m a 3.9. Let X be a FD'(df,dl/~,13,7 ). Then (a) For t E (0, to], r > 0

lP~(p(x,X,) > T) _< Cl e x p ( - c 2 r B ' r t - ' r ) . (b) There exists c3 > 0 such that c4exp (--chT#Tt -7) < ~=(p(x, Xt) > r) forr < C3T0, t < r ~.

(c) F o r x 9 F , 0 < T < car9, /YT(X,T) = inf{s > 0 : X , • B ( z , r ) } then

(3.7)

COT~~ E~(x,r) _< C~T~.

Proof. F i x z E F , a n d set D(a, b) = {y E F : a < p(x, y) <_b}. T h e n by (3.15) c3.1.zbdt > p(D(a,b)) > c3.1.1bd~ - c3.1.2adl. Choose 8 > 2 so t h a t C3.1.1od! > 2C3.1.2: t h e n we have

(3.8)

csa ~' < ~(D(a,0a))< cga~,.

Therefore, writing Dn = D(Snr, 8 n+1 r), we have # ( D n ) x 8 nd~ provided r8 n+1 _~ r0. Now

31 /.

P~(p(x,X,) > r) =

(3.9)

p(t,x,y)it(dy)

/

B(z,r) ~ = ~n~: <- ~

p(t,z,y)it(dy)

c(re')~'t-~'/€ exp(-c~ot-"(,-O")")

n=O

= c(r~/t)d'/~s(o,p'r, clo(r~/t)D. If clo r~ > t then using (3.6) we deduce that this sum is bounded by c11exp (--c12(r~/t)'r) , while ff Cxor~ <_ t then (as P ~ (p(x, Xt) > r) < 1) we obtain the same bound, on adjusting the constant c11. For the lower b o u n d (b), choose c3 > 0 so that c3O < 1. Then It(D0) _> cr ~, and taking only the first term in (3.9) we deduce that, since r ~ > t,

~(p(~, x,) > ~) > c(~lt)~,/" exp(-c~3(~/t)~) _> c e~p(-c~ (~'/t)~). (c) Note first that

(3.10)

~(~(~,~) > t) < ~ ( x ,

e B(x,~))

= _J~(~,r)p(t, y, z)it(dz) So, for a suitable c14

_~,

yEF.

Applying the Markov property of X we have for each k _> 1

P~(T(x,r) > kel4r ~) <_2 -k,

y E F,

which proves the upper bound in (3.7). For the lower bound, note first that

]

\O<s
< P~(p(x, Xt) > r) +P~(T(x, 2r) < t,p(x, Xt) < r) Writing S = ~-(x, 2r), the second term above equals

E~l(s<,)~xs(p(x,x,_s) < ~) <

sup

supP~(p(y,x,_.) > ~),

yEOB(z,2v) s<:t

32 so that, using (a), F~(~-(x,2r) < t) < 2supsupFU(p(y, Xs) > r) s<_t yEF

(3.11)

2Cl exp(-c2(r"

/t)7).

So if 4cle -c2a~ = 1 then I?~(T(x, 2r) < ar m) _< 89 which proves the left hand side of

(3.7).

[]

R e m a r k 3.10. Note that the bounds in (c) only used the upper bound on p(t, x, y). The following result gives sufficient conditions for a diffusion on F to be a fractional diffusion: these conditions are a little easier to verify than (3.3). T h e o r e m 3.11. Let (F,p,#) be a F M S ( d I ) . Let (Yt, t > O,P*,x C F) be a #symmetric diffusion on F which has a transition density q(t, x, y) with respect to # which is jointly continuous in x, y /'or each t > O. Suppose that there exists a constant/3 > O, such that

q(t, x, y) < clt -d'/~ for all x, y E F, t E (0, to], q(t,x,y) >_ c2t-a,/~ i~p(x,y) < c3t 1/~, t e (O, to], c4r/3 < E~T(x, r) < chr ~, for x E F, 0 < r < coro,

(3.12) (3.13) (3.14)

where T(x,r) = inf{t > 0 : Yt ~ B(x,r)}. parameters dl, dl//3 ,/3 and 1/(/3 1).

Then /3 > 1 a n d Y is a F D with

-

C o r o l l a r y 3.12. Let X be a FD'(dl,df//3,/3,7) on a FMS(df) F. Then fl > 1 and 7 = 1/(/3 - 1).

Proof. By L e m m a 3.8, and the bounds (3.3), the transition density p(t, x, y) of X satisfies (3.12) and (3.13). By Lemma 3.9(c) X satisfies (3.14). So, by Theorem 3.11 /3 > 1, and X is a FD'(dl,df//3,fl,(/3 - 1)-1). Since p(t,x,y) cannot satisfy (3.3) for two distinct values of 7, we must have 7 = (/3 - 1) -1. [] R e m a r k 3.13. Since two of the four parameters axe now seen to be redundant, we will shorten our notation and say that X is a FD(df,/3) i f X is a FD'(df, dr~~3,~3,7). The proof of Theorem 3.11 is based on the derivation of transition density bounds for diffusions on the Sierpinski carpet in [BB4]: most of the techniques there generalize easily to fractional metric spaces. The essential idea is "chaining": in its classical form (see e.g. [FaS]) for the lower bound, and in a slightly different more probabilistic form for the upper bound. We begin with a some lemmas. Lemma

3.14. [BB1, L e m m a 1.1] Let ~1,~2,''" ,~n, V be non-negative r.v. such ~i. Suppose that for some p E (0, 1), a > 0,

that V > ~ (3.15)

P(~{ < t[a(~l,...,~{_l)) < P + at,

t>O.

Then (3.16)

l o g P ( V < t) <_ 2 ( p t ) -

-

1/2

-

n

log -1. P

33 Proof.

If ~? is a r.v. w i t h d i s t r i b u t i o n function PO1 <- t) = (p + at) A 1, t h e n E ( e - ~ & ] a ( ~ l , . . . , ~ , - 1 ) ) < Ee -~''7

P + f ( 1 - p ) / ~ e-~tadt

_

--

JO

< p + aA -1. So

m ( v < t) = P (e -~'v > r

<_ d'~Er - ~ v

n

G e : ~ t E e x p A E ~ i < e)'t(p + h A - l ) " 1

T h e r e s u l t follows on s e t t i n g A -- (an/pt) 1/2.

[]

R e m a r k 3.15. T h e e s t i m a t e (3.16) a p p e a r s slightly odd, since it t e n d s to + c r as P I 0. However if p = 0 t h e n from the last b u t one line of the p r o o f above we o b t a i n log P ( V <_ t) ~_ At + n log ~, a n d s e t t i n g A = n / t we d e d u c e t h a t

ate

(3.17) Lemma

log P ( V <_ t) _< n l o g ( ~ - ) . 3 . 1 6 . Let (Yt,t > O) be a diffusion on a metric space (F,p) such that,/'or

x C F~ r > O, clr ~ ~_ EX T(x,r) ~ c2r f~. Then for x C F, t > 0 ,

~(~(x,r)

< t) < (1 - el/(2'c~)) + c3r-~t.

Proof. Let x e F , a n d A = B ( x , r ) , T ----T(x,r). Since ~- _< t + (~" - t)l(~>t) we have EZ'r <_ t q- E a l ( r > t ) E Y` (T -- t)

~_ t ~-Pz(T > t) supEYT. Y

As ~- ~ ~-(y, 2r) IFU-a.s. for any y C F , we d e d u c e

clr~ < E ~ < t + ~ ( ~ > t)c2(2r)~, so t h a t C22~(T

<_ t) < (2Zc2 - el) -~- t r -f~.

[]

T h e n e x t couple of results are n e e d e d to show t h a t the diffusion Y in T h e o r e m 3.11 can r e a c h d i s t a n t p a r t s of the space F in an a r b i t r a r i l y short time.

34

L e m m a 3 . 1 7 . Let Yt be a it-symmetric diffusion with semigroup Tt on a complete metric space (F, p). If f, g >_ 0 and there exist a < b such that

/ f(x)E~ g(Yt)it(dx) = 0 for t e (a,b),

(3.18)

then f f(x)E~ g(Yt)#(dx) = 0 for all t > O.

Proof. Let (Ex, s > 0) be t h e s p e c t r a l family associated with Tt. Thus (see [FOT, p. 17]) Tt = e-XtdEx, and

I~

(f, T,g) =

/o

e-:"d(f, E ,g) =

/o

where ~, is of finite variation. (3.18) and the uniqueness of the Laplace transform i m p l y t h a t t, = O, a n d so (f, Ttg) = 0 for all t. [] 3.18. Let F a n d Y satisfy the hypotheses of Theorem 3.11. If p(x,y) < c3r 0 then ~z(Yt E B(y,r)) > 0 for all r > 0 and t > O.

Lemma

R e m a r k . T h e restriction p(z,y) < c3ro is of course unnecessary, b u t it is all we need now. T h e conclusion of T h e o r e m 3.11 implies t h a t ~ ( Y t E B(y, r)) > 0 for all r > 0 a n d t > 0, for M i x , y E F .

Proof. Suppose t h e conclusion of the L e m m a fails for x, y, r, t. Choose g E C(F, N+) such t h a t fmgdit = 1 a n d g = 0 outside B(y,r). Let tl = t/2, rl = c3(tl) ~, a n d choose f 9 C ( F , I ~ + ) so t h a t f F f d # = 1, f ( x ) > 0 and f = 0 outside A = B(x, rl). If 0 < s < t t h e n t h e construction of g implies t h a t 0

E~ g(Yt) = f~F q(s, x, x' )E x' g(Yt-,)it(dx ' ).

Since b y (3.13) q(s,x,x') > 0 for t/2 < s < t, x' 9 B(x, rl), we deduce t h a t E*'g(Yu) = 0 for x' 9 S ( x , rl), u 9 (0, t/2). Thus as s u p p ( f ) C B(x, rl)

F f (X')EX'g(Yu)dit

= 0

for all u 9 ( 1 , t / 2 ) , a n d hence, by L e m m a 3.17, for all u > 0. But by (3.13) if u = (p(x,y)/c3) ~ t h e n q(u,x,y) > 0, and by the continuity of f , g and q it follows t h a t f fE*g(Y=)eit > 0, a contradiction. []

Proof of Theorem 3.11. For simplicity we give full details of the p r o o f only in the case r o = co; the a r g u m e n t in the case of b o u n d e d 17 is essentially the same. We begin by o b t a i n i n g a b o u n d on

<

t).

Let n > 1, b = r/n, a n d define stopping times Si, i > 0, by So = 0,

Si+~ = inf{t _> Si : p(Ys,,Yt) _> b}.

35 Let ~i = Si - Si-1, i > 1. Let (~'t) be the filtration of Yt, and let ~i = J=s,. We have by L e m m a 3.16

~(~i+1 <_ tlGi) = IPYs' (T(Ys,, b) < t) < p + c6b-~t, w h e r e p 6 (0,1). As Ys,,Ys,§ = b, we have p(Yo,Ys~) <_ r, so that S , = ~ 1 ~i _< 7(Y0,r). So, by L e m m a 3.14, with a = c6(r/n) -~, < t) < 2p

-

(3.19)

.log 1P

1

= cT(r-t~nl+t~t) ~ - csn. If fl < 1 then taking t small enough the right hand side of (3.17) is negative, and letting n ~ oo we deduce IP~('r(x,r) _< t) = 0, which contradicts the fact that F~(Yt 9 B ( y , r ) ) > 0 for all t. So we have fl > 1. (If r0 = 1 then we take r small enough so that r < c3). If we neglect for the moment the fact that n 9 bt, and take n = no in (3.19) so that

then

n~o-1 = (c2s/4c~)r~t -1,

(3.20) and

_< t) _< -}csn0. So if r~t -1 >_ 1, we can choose n 6 l~ so that 1 < n < no V 1, and we obtain (3.21)

I?~(T(x,r) < t) _< c9 exp

-clo

9

Adjusting the constant c9 if necessary, this bound also clearly holds if r~t -1 < 1. Now let z , y 6 F , write r = p(x,y), choose e < r/4, and set C~ = B(z,e), z = x,y. Set Ax = {z 6 F : p(z,x) < p(z,y)}, Ay = { z : p(z,x) >_ p(z,y)}. Let u~, uy be the restriction of # to C~, Cy respectively. We now derive the upper bound on q(t, x, y) by combining the bounds (3.12) and (3.21): the idea is to split the journey of Y from C~ to C~ into two pieces, and use one of the bounds on each piece. We have (3.22)

F~'(Yt 6 C,) = [ [ q(t,x',y')#(dx')#(dy') C~C.

<_ IF~" (Yt 6 Cu,Yt/2 6 A~) + F " ( Y t 9 Cu,Yt/2 9 Ay). We begin with second term in (3.22): (3.23) P~'(Yt 9 Cu,Yt/2 9 Au) = IP~" (T(ro,r/4) < t/2,Yt/2 9 Au,Yt 9 Cu) -< F~" (7(Y0,r/4) < t/2) sup F y' (Yt/2 9 y'6A.

Cy)

36

_< ,,.(c~

exp k-c,o

)

=

exp

where we used (3.21) and (3.12) in the last b u t one line. To handle the first t e r m in (3.22) we use symmetry:

~'.(~ c c y , v u 2 ~ A~) = P'~(Y~ c C~,Yu2 e A~), and this can now be b o u n d e d in exactly the same way. We therefore have

/ / q(t,x',y')#(dx')#(dy') C~ C~ < It(C, )#(Cy)2Cll t -d'/~ exp ( - c 1 2 (r ~ It) 1/(f~-l)) , so t h a t as (3.24)

q(t,., .)

is continuous

q(t,x,y) < 2c, lt-a'/f~ exp (-c,2(rf~/t)l/(~-l)) .

T h e p r o o f of the lower b o u n d on q uses the technique of "chaining" the C h a p m a n - K o l m o g o r o v equations. This is quite classical, except for the different scaling. Fix x,y,t, and write r -- p(x,y). If r <_c3tl/~ then by (3.13)

q(t, x, y) > c2t -dr~z, a n d as exp(-(rfl/t) U(~-U) >_ e x p ( - c ~ / ( ~ - l ) ) , we have a lower b o u n d of the form (3.3). So now let r > c3t x/~. Let n _> 1. By the mid-point hypothesis on the metric p, we can find a chain x = xo,xl,...,xn = y in F such t h a t p(xi-l,xi) = r/n, 1 < i < n. Let Bi = B(xi,r/2n); note t h a t i f y i E Bi then P(yi-x,Yi) <_ 2r/n. We have by the C h a p m a n - K o l m o g o r o v equation, writing Y0 = x0, y , = y,

q(t,x,y)>__/#(dyl)... #(dy,.-1)Hq(t/n,y,-1,y,). n

(3.25)

/

Ba

B~_a

i----1

We wish to choose n so t h a t we can use the b o u n d (3.13) to estimate the terms q(t/n, Yi-1, Yi) from below. We therefore need: (3.26)

n

which holds provided (3.27)

n ~-1 _> 2 Z c [ ~ .

37 As 17 > 1 it is c e r t a i n l y possible to choose n satisfying (3.27). By (3.25) we then obtain, since # ( B i ) > c(rl2n) ds,

(3.28)

q(t, x, y) > c(rl2n) d,(',-1) (c2(tln) -dsi~) --

= c(,'/2n) -d'

= c'(~,/n) -'~,

n

(~(tln) '/~(~/2~)

((tln)-liZ(,,In ))~

d

')

n

Recall t h a t n satisfies (3.27): as r > c3t l l j we can also ensure t h a t for some cl3 > 0 (3.29)

r >_ c13(tln)l//3 '

u

so t h a t u ~-1 < 2~c~3~r~lt. So, by (3.28)

q(t,~,y) > c(tln)-~'i~c74 _> cl 5t -ds IZ exp ( n log cl4 )

>_ eist -d' i' exp

lt )'ir

) .

[]

Remarks 3.19. 1. Note t h a t the only point at which we used the "midpoint" p r o p e r t y of p is in the derivation of the lower b o u n d for q. 2. T h e essential i d e a of t h e proof of T h e o r e m 3.11 is t h a t we can obtain b o u n d s on t h e long range b e h a v i o u r of Y provided we have good enough information a b o u t the b e h a v i o u r of Y over distances of order t ll2. Note t h a t in each case, if r = p(x, y), the e s t i m a t e of q ( t , x , y ) involves splitting the j o u r n e y from x to y into n steps, where n x (r2lt) 1/(~-l). 3. B o t h the arguments for the u p p e r and lower bounds a p p e a r quite crude: the fact t h a t t h e y yield the same b o u n d s (except for constants) indicates t h a t less is thrown away t h a n might a p p e a r at first sight. T h e explanation, very loosely, is given by "large deviations". T h e off-diagonal bounds are relevant only when r 13 >> t - otherwise t h e t e r m in t h e exponential is of order 1. If r E >> t then it is difficult for Y to move from x to y by time t and it is likely to do so along more or less the shortest p a t h . T h e proof of the lower b o u n d suggests t h a t the process moves in a 'sausage' of radius t i n • t i t ~1-1. T h e following two theorems give additional bounds and restrictions on the p a r a m e t e r s d f a n d / 3 . Unlike the proofs above the results use the s y m m e t r y of the process very strongly. T h e proofs should a p p e a r in a forthcoming paper. Theorem (3.30)

3.20. Let F be a F M S ( d l ) , and X be a F D ( d f , / 3 ) on F. Then 2
T h e o r e m 3.21. Let F be a F M S ( d l ) . i = 1,2. Then/31 = /32.

l + d s.

Suppose X i are FD(df,/3i ) on F, for

R e m a r k s 3.22. 1. T h e o r e m 3.21 implies t h a t the constant 13 is a p r o p e r t y of the metric space F , a n d not just of the F D X. In p a r t i c u l a r any F D on ~d, with the

38 usual metric and Lebesgue measure, will have/3 = 2. It is very unlikely that every FMS F carries a FD. 2. I expect that (3.30) is the only general relation between/3 and d I. More precisely, set A = {(dr,/3) : there exists a F D ( d l , / 3 ) } , and F = {(df,fl) : 2 < /3 < 1 + d I } . Theorem 3.20 implies that A C I', and I conjecture that int I' C A. Since B M ( ~ a) is a F D ( d , 2), the points (d, 2) 9 A for d > 1. I also suspect that { d r : (dr,2) 9 A} = N, that is that if F is an FMS of dimension dr, and df is not an integer, then any FD on F will not have Brownian scaling. Properties of Fractional Diffusions. In the remainder of this section I will give some basic analytic and probabilistic properties of FDs. I will not give detailed proofs, since for the most part these are essentially the same as for standard Brownian motion. In some cases a more detailed argument is given in [BP] for the Sierpinski gasket. Let F be a F M S ( d l ) , and X be a F D ( d l , f l ) on F. Write T, = E ~ f ( X t ) for the semigroup of X , a n d / 2 for the infinitesimal generator of T~. D e f i n i t i o n 3.23. Set d",=fl,

2dr d e = d"" .

This notation follows the physics literature where (for reasons we will see below) d", is called the "walk dimension" and d~ the "spectral dimension". Note that (3.3) implies that p ( t , x , x ) • t -d~ 0 < t < to, so that the on-diagonal bounds on p can be expressed purely in terms of d~. Since m a n y important properties of a process relate solely to the on-diagonal behaviour of its density, d, is the most significant single parameter of a F D . Integrating (3.3), as in Corollary 2.25, we obtain:

Lemma3.24.

E~p(Xt, x) p • t p/d~ , x C F, t > O, p > O.

Since by Theorem 3.20 d", > 2 this shows that FDs are diffusive or subdiffusive. L e m m a 3.25. (Modulus of continuity). Let ~(t) = t 1/d~ (log(1/t)) (d~-D/d~. Then (3.31)

Cl < lira sup p(X~,X~) <_ c2. ~10 0<~<~<1 ~ ( t - s)

So, in the metric p, the paths of X just fail to be Hhlder (1/d,,). The example of divergence form diffusions in I~d shows that one cannot hope to have ci =- c~ in general.

39 L e m m a 3.26. (Law of the iterated logarithm - see [BP, Thin. 4.7]). Let r = tl/d~(loglog(1/t))(d~--l)/d~. There exist Cl, c2 and constants c(x) E [Cl,C2] such that lim0u p p ( ~ X o ) _ c(x) ~'-a.s. Of course, the 01 law implies that the limit above is non-random. L e m m a 3.27. (Dimension of range). dimH ({X~: 0 < t < 1}) = d I A d~.

(3.32)

This result helps to explain the terminology "walk dimension" for d,~. Provided the space the diffusion X moves in is large enough, the dimension of range of the process (called the "dimension of the walk" by physicists) is d~.

Potential Theory of ~ractional Diffusions. Let A >_ 0 and set

O ~

Then ff

v~f(x) =

E~

/?

e-~V(x,)

de

is the A-resolvent of X, ux is the density of UA:

Uxf(x) = f uA(x, V)p(dy). t] F

Write u for u0. P r o p o s i t i o n 3.28. Let A0 : l/r0. (If to : oo take Ao = 0).

(a) If d~ < 2 then u~(x,y) is jointly continuous on F x F and for )~ > Ao (3.33)

c1)~d'/2-1 exp(--c2A1/d~p(x,y)) <_u~(x,y) < c3z./2-, exp

(b) If d, = 2 and A > ~o then writing R = p(x,y)A 1/d~ (3.34)

e5 (log+(1/R) + e - ~ ' ' ) < un(x,y) <_e7 (log+(1/R) + e - * " ) .

( c ) / f de > 2 then

(3.35)

~p(~,Y)~-d' < u~o(~,V) _< ~,0p(~,V) ~~

These bounds are obtained by integrating (3.3): for (a) and (b) one uses Laplace's method. (The continuity in (b) follows from the continuity of p and the uniform bounds on p in (3.3)). Note in particular that:

4O (i) i f d s < 2 t h e n u ~ ( x , x ) < +oo and lim u x ( x , y ) = +oo. A---~0

(ii) if ds > 2 t h e n u(x, x) -- +c~, while u(x, y) < c~ for x ~ y Since the p o l a r i t y or n o n - p o l a r i t y of points relates to the on-diagonal behaviour of u, we deduce from P r o p o s i t i o n 3.28 C o r o l l a r y 3.29. (a) Ifd~ < 2 then for each x , y E F

P ~ ( X hits y) = 1. (b) I f d~ > 2 then points are polar for X . (c) I f d~ < 2 then X is set-recurrent: for ~ > 0 I?y ({t : X , 9 B(y, E)} is non-empty and unbounded) = 1.

(d) I f d~ > 2 and r0 = c~ then X is transient. In short, X behaves like a Brownian motion of dimension ds; b u t in this context a continuous p a r a m e t e r range is possible. L e m m a 3.30. (Polar and non-polar sets). Let A be a Borel set in F. (a) ]~(TA < oo) > 0 if d i m H ( A ) > df - dw, (b) A is polar for X f f d i m g ( A ) < df - dw. Since X is s y m m e t r i c any semipolar set is polar. As in the Brownian case, a more precise condition in terms of capacity is true, and is needed to resolve the critical case dim/~(A) = df - d~o. If X , X ' are i n d e p e n d e n t F D ( d y , • ) on F , and Zt = (Xt, X~), then it follows easily from the definition t h a t Z is a F D on F x F , with p a r a m e t e r s 2dr and ft. If D -~ { ( x , x ) : x 9 F } C F x F is the diagonal in F • F , then d i m H ( D ) -- df, and so Z hits D (with positive p r o b a b i l i t y ) if

df > 2 d f - d w , t h a t is if ds < 2. So (3.36)

lP~(Xt = X~ for some t > 0) > 0

if d8 < 2,

~(Xt

if d~ > 2.

and (3.37)

= X~ for some t > 0) = O

No d o u b t , as in the Brownian case, X and X ' do not collide if d, = 2. Lemma

3.31. X has k-multiple points i f a n d only if d, < 2 k / ( k - 1).

Proof. By [Rog] X has k-multiple points if and only if

fB (~,1) ui(x'Y)k'(dY) < (XD;

41

the integral above converges or diverges with

o I rkd~--(k--1)d~r -1 dr, by a calculation similar to that in Corollary 2.25.

[]

T h e b o u n d s on the potential kernel density u ~ ( x , y ) lead immediately to the existence of local times for X - see [Sha, p. 325]. T h e o r e m 3.32. I f d~ < 2 then X has jointly measurable local times ( L t9 , x E F , t >_

O) which satisfy the density of occupation formula with respect to #: (3.38)

f(X~)ds =

f(a)L~#(da),

f bounded a n d measurable.

I n the low-dimensional case (that is when d~ < 2, or equivalently d I < d,~) we can o b t a i n more precise estimates on the HSlder continuity of ux(x, y), and hence on the local times L~'. The m a i n lines of the argument follow that of [BB4, Section 4], b u t on the whole the arguments here are easier, as we begin with stronger hypotheses. We work only in the case r0 = oo: the same results hold in the case r0 = 1, with essentially the same prooofs. For the next few results we fix F , a F M S ( d f )

with r0 = 0r

and X , a

F D ( d l , d~) on F . For A C F write TA = TAo = inf{t _> 0 : X~ C A~}. Let Rx be an i n d e p e n d e n t exponential time with mean A-1. Set for A > 0

u A ( x , y ) -: E ~

f0

TM

e - ~ d L y -- E ~ LrA^R~ y

u2f(~) : f ~(~, y),(dy). JF

Let

p~(x, y) = ~ ( T ~ < ~A A R~); note t h a t

(3.39) Write uA(x, y) = u A ( x , y ) , U A = UoA, and note that u ~ ( x , y ) = u F ( x , y ) , Ux = UA. As in the case of u we write pA, px for pA, pF. As (I?~, Xt) is #-symmetric we have uA(x, y) = uA(y, X) for all x, y e f . The following L e m m a enables us to pass between bounds on u~ a n d UA. L e m m a 3.33. Suppose A C F, A is bounded, For x, y E F we have

~A(x,y) = ~ f ( x , y) + ~ (I(R~<~A)uA(XR~,y)) - E ~ (I(R~>~A)~f(x~A,y)).

42 P r o o f . From the definition of u A, u A ( x , y ) = E (L~A ;Rx < ~'A)+ E~(L~A ;Rx > ~'A)

= E=(L~ ;R~ < ~A) + E=(IIR~<,A)EXR~ L~A) y y , --L~A ;R~ > rA) + E~e(LR~ ;R~ > ~-A)- Ea~(LR~^~ =ux(x,y)+E

z

(I(R~_<~A)UA ( X , x , y ) ) - E ~ ( I ( R ~ > ~ A ) U X ( X ~ A , y ) ) .

[]

C o r o l l a r y 3.34. Let x E F, and r > 0. Then

clrd~-dl <_ uB(~,~)(x,z) < e2rd~--d~. Proof. Write A = B ( x , r ) , and let A = Or -d~, where /9 is to be chosen. We have from L e m m a 3.33, writing 9 = ~ ( x , r ) ,

uA(~, y) _< ~ ( ~ , Y) + E~ I(,~<~)~A(XR~, Y). So if ~ = sup~ ~A(~, y) then using (3.33)

(3.40)

v ~_ c3/~d'/2-1 -[-]]~Z(RA < T)V.

Let to > 0. Then by (3.10)

P~(/r < ~-) = P~(R:~ < T,~- < to) + IP~(R~ < T,~" > to) _< ~ ( R ~ < to) + P~(~- > to) < (1 - e -xt~ + ctodl/d~r dl. Choose first to so that the second term is less than 88 and then A so that the first term is also less than 88 We have to ~ r d~ ~ A-1, and the upper bound now follows from (3.40). The lower bound is proved in the same way, using the bounds on the lower tail of T given in (3.11). [] L e m . m a 3.35. There exist constants cl > 1, c2 such that ff x, y E F, r = p ( x , y ) , t 0 = r d~ then ~ ( T ~ < to < ~(x,c~r))> c~.

Proof. Set ~ -- (e/r) d~ ; we have px(x, y) > c3 exp(-c4e) by (3.33). So since p x ( z , y ) = E~ e -xT, <_ P~(T~ < t) + e - ~ , we deduce that P~(Tv < t) > c3 exp(-c4O) - exp(--0d~). As dw > 1 we can choose 0 (depending only on c3, c~ and d,o) such that P~(Tv < t) >_ ~31c exp(--c4O) ----C5. By (3.11) for a > 0

P~('r(z, an) < ~'~) <_c~ exp(--c~ad~/~d'-~)), so there exists c~ > 1 such that IP):e(T(X, Cl r ) < tO ) _<( ~.C5 . 1 So 1 l~=(Tu < to < T(X, c~r))> II~=(Tu < to) - ~=(T(X, clr) < to) >_ ~c5.

[]

43 D e f i n i t i o n 3.36. We call a function h harmonic (with respect to X ) in a n open subset A C F if s = 0 on A, or equivalently, h(Xt^TAo ) is a local martingale. 3 . 3 7 . (Harnack inequality). There exist constants cl > 1, c2 > O, such that if xo E F, and h >_ 0 is harmonic in B(xo,clr), then

Proposition

h(x) > c~h(y),

x,y e B(~o,r).

Proof. Let cl = 1 + ca.35.1, so t h a t B(x, c3.ah.lr) C B(xo,clr) if p(x, xo) < r. Fix x, y, write r -----p(x, y), and set S -- Tu A~(x, c3.ah.lr). As h(X.hs) is a supermartingaie, we have by L e m m a 3.35, h(x) > E= h ( X s ) > h(y)P~(Ty < T(X,e3.35.1r)) ~_ ca.35.2h(y).

[]

C o r o l l a r y 3.38. T h e r e exists Cl > 0 such that ff xo C F, and h > 0 is harmonic

in B(xo,r), then

h(~) >_ clh(y),

~,y e B(xo,88

Proof. This follows by covering B(xo, 88 by balls of the form B(y, c2r), where c2 is small enough so t h a t P r o p o s i t i o n 3.37 can be applied in each ball. (Note we use the geodesic p r o p e r t y of the metric p here, since we need to connect each ball to a fixed reference point by a chain of overlapping balls). [] Lemma

3 . 3 9 . Let x, y e F, r : p(x, y). IT R > r and B(y, R) c A then

Proof. We have, writing T = ~-(y,r), T -- TAo,

uA(y, y) = ~ L ~ + EYEX, L~ = u'(y, y) + E~uA(X~, y), so b y Corollary 3.34

(3.41)

E~(~(y,y)-

~(x~,y))

-- ~B(y,y) < c ~ - . , .

Set ~ ( x ' ) = uA(y,y)--uA(x',y); ~ is harmonic on A - {y}. As p(x,y) = r and p has t h e geodesic p r o p e r t y there exists z with p(y, z) = 88 p(x, z) = ~r. ~ By CoroUary 3.38, since ~ is harmonic in B(x, r),

~(~) _>c3 3~ ~ ( ~ ) . Now set r = E =' ~(X~) for x' E B. T h e n r is harmonic in B and ~ < r on B. A p p l y i n g Corollary 3.38 to r in B we deduce

r

__~r

__~C3.38A~P(Z) __~(C3.38.1)2~0(X).

Since r

-- E y (uA(y, y) -- uA(Xr, y)) the conclusion follows from (3.41).

Theorem

3.40. (a) Let ~ > 0. Then t'or x, x', y E F , a n d f E L I ( F ) , g E L ~ ( F ) ,

(3.42) (3.43) (3.44)

I ~ ( ~ , y ) - uA(x',y)l ~ clp(x,x') d~-d', IV~f(~) - v~f(~')l < ~,p(z,~')~o-d'llfll,. IV~g(~) - v~g(~')l _< c~-d./~p(~, ~,)d~-d, Ilgll~.

[]

44

Proof. Let x, x' C F, write r = p(x,x') and let R > r, A = B(x,R). uA(y, x') > pA(y, x)uA(x, x'), we have using the s y m m e t r y of X that (3.45)

=2(~, y)

-

Since

u2(~', y) < ~2(y, ~) p~(y, x)~2(~, ~') = p~(y, ~)(u~(~, x) u~(~, ~')). -

-

Thus

lu~(~,y) - ~(z',y)l -< 1~2(~,~) - u2(~,~')l. Setting ~ = O and using L e m m a 3.39 we deduce (3.46)

luA(x,y) _ ~.4(x,,y)l _< ~:d~-,~,.

So f

Iu~/(~) - UAf(~')E <_ ./~ IuA(~,Y) -- uA(~', Y)I If(Y)I.(dY) <_ c: "d~-d' llflAll~To obtain estimates for A > 0 we apply the resolvent equation in the form

u~(x, y) -- uA(x, y) - ~v%(~), where

v(x) = uA(x,y). (Note that Ilvlll = ;~-1). Thus < c : d ~ - d , + ~clrd~-~'tlvll,

2c3rd~--d:. Letting /~ --* ~ we deduce (3.42), and (3.43) then follows, exactly as above, by integration. To prove (3.46) note first that p~(y,x) = ux(y,x)/ux(x,x). So by (3.33) (3.47)

f p2(y,~)lf(y)l.(dy)<_ I l f l l ~ u ~ ( z , z ) - 1 / A u x ( y , x ) t t ( d Y ) _-I If II,,,,u~,(z, z)-~.~, -~ <__c411.fll~,\

-d'/2.

From (3.45) and (3.46) we have

1~2(~,y) - u~(x',y)l < c~ (p2(y, ~) + V2(y,~'))r~-~', and (3.44) then follows by intergation, using (3.47).

[]

T h e following modulus of continuity for the local times of X then follows from the results in [MR].

45 T h e o r e m 3.41. If d, < 2 then X has jointly continuous local times (L~,x E F,t >

0). Let ~o(u) = u(d~-d')/2(log(I/u) ) I/2. The modulus of continuity in space of L" is given by: l i m sup ~Oo<s
sup 0<s<~

IL~ - Z~l < c ( s u p ~(p(z,y)) ~eF

L~)1/2.

I~-yl<6 It follows t h a t X is space-filling: for each x, y E F there exists a r.v. T such t h a t I?~(T < cr = i a n d B ( y , 1) C {Xt,O < t < T}. T h e following P r o p o s i t i o n helps to explain why in early work m a t h e m a t i c a l physicists found t h a t for simple examples of fractal sets one has d~ < 2. (See also [HHW]). 3.42. Let F be a FMS, and suppose F is finitely ram/fled. Then if X is a FD(dI, d~) on F, d~(X) < 2. Proposition

Proof. Let F1, F2 be two connected components of F , such t h a t D = F1 ~ F2 is finite. I f D = {Yl,-..,Y,~}, fix ~ > 0 and set Ms = e T M ~

u),(Xt,yi).

i=1

T h e n M is a supermartingale. Let To = inf{t > 0 : Xt E D}, and let x0 C F1 - D. Since I?~~ E F2) > 0, we have IP~~ < 1) > 0. So oo > EX~

> EZ~

a n d thus MTD < ar a.S. So u~(XTD,Yi) < Or for each Yl E D, and thus we must have u~(yi, Yi) < ~ for some y~ E D. So, by Proposition 3.25, ds < 2. [] R e m a r k 3.43. For k = 1,2 let (Fk,dk,#k) be F M S with dimension d r ( k ) , a n d c o m m o n d i a m e t e r r0. Let F = F1 • F2, let p _> 1 and set d((xl,x2),(yl,Y2)) = (dl(xl,Yl) p + d2(x2,Y2)P) Up, # = #1 • #2. T h e n (F,d,#) is a F M S with dimension d f = d f ( 1 ) + d f ( 2 ) . Suppose t h a t for k = 1,2 X k is a FD(dl(k),dw(k)) on Fk. T h e n if X = ( X ~ , X 2) it is clear from t h e definition of F D s t h a t if dw(1) = d,~(2) = / 3 t h e n X is a F D ( d f , f l ) on F . However, if dw(1) r dw(2) then X is not a F D on F . (Note from (3.3) t h a t the metric p can, up to constants, be e x t r a c t e d from the t r a n s i t i o n density p(t, x, y) by looking at limits as t ~ 0). So the class of F D s is not stable u n d e r products. This suggests t h a t it might be desirable to consider a wider class of diffusions with densities of the form: Tt

1

where Pi are a p p r o p r i a t e non-negative functions on F • F . Such processes would have different space-time scalings in the different 'directions' in the set F given by t h e functions pi. A recent p a p e r of H a m b l y and K u m a g a i [HK2] suggests t h a t

46 diffusions on p.c.f.s.s, sets (the most general type of regular fractal which has been studied in detail) have a behaviour a little like this, though it is not likely that the transition density is precisely of the form (3.48).

Spectral properties. Let X be a FD on a FMS F with diameter r0 = 1. The bounds on the density

p(t, x, y) imply that p(t,., .) has an eigenvalue expansion (see [DaSi, Lemma 2.1]). T h e o r e m 3.44. There exist continuous functions ~ol, and Ai with 0 <_ ~o <- ~1 <_ ...

such that/or each t > 0 p(t,x,y) = ~

(3.49)

e-~'t~n(x)~n(y),

n-~O

where the sum in (3.49) is ,miformly convergent on F • F. R e m a r k 3.45. The assumption that X is conservative implies that A0 -- 0, while the fact that p(t, x, y) > 0 for all t > 0 implies that X is irreducible, so that A1 > 0. A well known argument of Kac (see [Ka, Section 10], and [HS] for the necessary Tauberian theorem) can now be employed to prove that ff N(A) = #{),i : Ai < )t} then there exists ci such that (3.50)

ClAd~ < N(A) <_c2)ta~

for ~ > ca.

So the number of eigenvalues of Z: grows roughly as )~d./2. This explains the term spectral dimension for ds.

4. D i r i c h l e t F o r m s , M a r k o v P r o c e s s e s , a n d E l e c t r i c a l N e t w o r k s . In this chapter I will give an outline of those parts of the theory of Dirichlet forms, and associated concepts, which will be needed later. For a more detailed account of these, see the book [FOT]. I begin with some general introductory remarks. Let X = ( X t , t >_0,]?~,x E F ) be a Maxkov process on a metric space F. (For simplicity let us assume X is a Hunt process). Associated with X are its semigroup (Tt, t > 0) defined by

Ttf(x) = E~ f ( X t ) ,

(4.1)

and its resolvent (U~, A > 0), given by (4.2)

Uj, f ( z ) =

/o

T t f ( z ) e -~t dt -- E ~

/o

e - J " f ( X , ) ds.

While (4.1) and (4.2) make sense for all functions f on F such that the random variables f ( X t ) , or f e-X~f(Xs)ds, are integrable, to employ the semigroup or resolvent usefully we need to find a suitable Banaeh space (B, II 9 118) of functions on F such that Tt : B ~ B, or Ux : B --, B. The two examples of importance here are

47

Co(F) and L2(F,#), where # is a Borel measure on F. Suppose this holds for one of these spaces; we then have that (Tt) satisfies the semigroup property Tt+, = TtT,,

s,t > O,

and (Ux) satisfies the resolvent equation

U~-V~=(fl-~)UoU~,

a , ~ > O.

We say (Tt) is strongly continuous ff IITtf - f]]s ~ 0 as t ~ O. If Tt is strongly continuous then the infinitesimal generator (Z:, 7)(L)) of (T,) is defined by L ; / = lirnt-l(Ttf - f), tl0

(4.3)

/ e 7)(/2),

where 7)(L) is the set of f E B for which the limit in (4.3) exists (in the space B). The Hille-Yoshida theorem enables one to pass between descriptions of X through its generator s and its semigroup or resolvent. Roughly speaking, if we take the analogy between X and a classical mechanical system, s corresponds to the equation of motion, and Tt or Ux to the integrated solutions. For a mechanical system, however, there is another formulation, in terms of conservation of energy. The energy equation is often more convenient to handle than the equation of motion, since it involves one fewer differentiation. For general Markov processes, an "energy" description is not very intuitive. However, for reversible, or symmetric processes, it provides a very useful and powerful collection of techniques. Let # be a Radon measure on F : that is a Borel measure which is finite on every compact set. We will also assume # charges every open set. We say that Tt is #-symmetric if for every bounded and compactly supported f, g, (4.4)

/ T,.f(x)g(x)#(dx) = f Ttg(x)f(x)#(dx).

Suppose now (Tt) is the semigroup of a Hunt process and satisfies (4.4). Since Ttl _< 1, we have, writing (.,-) for the inner product on L2(F, #), that

ITtf(x)I <_ (Ttf2(x)) 1/2 (Ttl(x)) 1/2 <_(TtfZ(x)) 1/2 by HSlder's inequality. Therefore

IITtflll <

IlTtf2111 = (Ttf2,

1) = (f2,Ttl) < (f2, 1) = Ilfll~,

so that Tt is a contraction on L 2 (F, #). The definition of the Dirichlet (energy) form associated with (Tt) is less direct than that of the infinitesimal generator: its less intuitive description may be one reason why this approach has until recently received less attention than those based on the resolvent or infinitesimal generator. (Another reason, of course, is the more restrictive nature of the theory: many important Markov processes are not symmetric. I remark here that it is possible to define a Dirichlet form for non-symmetric Markov processes - - see [MR]. However, a weaker symmetry condition, the "sector condition", is still required before this yields very much.)

48 Let F be a metric space, with a locally compact and countable base, and let # be a R a d o n measure on F . Set H = L2(F,#). D e f i n i t i o n 4.1. Let 7) be a linear subspace of H. A symmetric form (E, 7)) is a m a p E : 7) • 7) -~ • such t h a t (1) s is bilineax (2) s f ) _> O, f C 79.

allfl[ ~, and

For a > 0 define Ea on 7) by E a ( f , f ) -~ e ( f , f ) +

llfll~ = llfll~+ c,E(f,f)

=

write

E,~(I,f).

D e f i n i t i o n 4.2. Let (s 7)) be a symmetric form. (a) E is closed if (79, II-Ile~) is complete (b) (s is Markov if for f E 7), i f g = ( 0 V f ) A 1 t h e n g E 79 and s < E(f,f). (c) (s 7)) is a Dirichlet form if 7) is dense in LZ(F, #) a n d (s 7)) is a closed, Markov s y m m e t r i c form. Some further p r o p e r t i e s of a Dirichlet form will be of importance: D e f i n i t i o n 4.3. (E, 7)) is regular if (4.5) (4.6)

79 N Co(F) is dense in 7) in II" I1~1, a n d 7) N Co(F) is dense in Co(F) in [[. I1~.

s is local if C(f, g) = 0 whenever f, g have disjoint support. E is conservative if 1 E 7) and s = 0. s is irreducible if s is conservative and g ( f , f ) = 0 implies t h a t f is constant. T h e classical example of a Dirichlet form is t h a t of Brownian motion on Ra:

s

f) - -

1 ~

f IVfl2 d~, I e H1'2(R% ~a

L a t e r in this section we will look at the Dirichlet forms associated with finite state Markov chains. J u s t as the Hille-Yoshida theorem gives a 1 - 1 correspondence between semigroups a n d their generators, so we have a 1 - 1 correspondence between Dirichlet forms a n d semigroups. Given a semigroup (Tt) the associated Dirichlet form is o b t a i n e d in a fairly straightforward fashion. D e f i n i t i o n 4.4. (a) T h e semigroup (Tt) is iarkovian if f E L2(F,#), 0 <_ f < 1 implies t h a t 0 _< Ttf _< 1 #-a.e. (b) A Markov process X on F is reducible if there exists a decomposition F = A1UA2 with Ai disjoint a n d of positive measure such t h a t P~(Xt E Ai for all t) = 1 for x E Ai. X is irreducible if X is not reducible.

49 4.5. ([FOT, p. 23]) Let (Tt, t _> 0) be a strongly continuous It-symmetric contraction semigroup on L2(F, It), which is Markovian. For f C L 2 ( F , # ) the function ~l(t) defined by Theorem

q o f ( t ) = t - x ( f --Ttf, f),

t>0

is non-negative and non-increasing. Let V = {f 9 L 2 ( F , # ) : lim~oy(t) < oo}, tlo

C(Lf) =limbos(t), tlo

f e 7).

Then (s 7)) is a Dirichlet form. If (Z:, 79(/:)) is the infinitesimal generator of (Tt), then 7)(1:) C V, V(C) is dense in L2(F, It), and s

(4.7)

= ( - E l , g),

f E 7)(s

g e 7).

As one might expect, by analogy with the infinitesimal generator, passing from a Dirichlet form (s 7)) to the associated semigroup is less straightforward. Since formally we have Us = (a - 12)-1, the relation (4.7) suggests that (4.8)

( f , g ) = ((a - E)U~f,g) -- a(U~,f,g) + E(Usf, g) = C~(UsI, g).

Using (4.8), given the Dirichlet form E, one can use the Riesz representation theorem to define U~f. One can verify that Us satisfies the resolvent equation, and is strongly continuous, and hence by the Hille-Yoshida theorem (Us) is the resolvent of a semigroup (Tt). T h e o r e m 4.6. ([FOT, p.18]) Let (e,7)) be a Oirichlet form on L2(F,#). Then there exists a strongly continuous #-symmetric Maxkovian contraction semigroup (Tt) on L2(F,#), with infinitesimal generator (L:,7)(E)) and resolvent (U~,a > O)

such that 1: and ~ satisfy (4.7) and also (4.9)

s

+ a(f,g) -- (f,g),

f e L2(F, It), g e 7).

Of course the operations in Theorem 4.5 and Theorem 4.6 are inverses of each other. Using, for a moment, the ugly but clear notation C -- T h m 4.5((Tt)) to denote the Dirichlet form given by Theorem 4.5, we have T h m 4.6(Tam 4.5((Tt))) = (Tt), and similarly T h m 4.5(Whm 4.6 (s

= E.

R e m a r k 4.7. The relation (4.7) provides a useful computational tool to identify the process corresponding to a given Dirichlet form - at least for those who find it more natural to think of generators of processes than their Dirichlet forms. For example, given the Dirichlet form s f) = f IV f[ 2, we have, by the Gauss-Green formula, for f , g e C2(~d), ( - s =s = f v f.Vg = -- f g A f , so that s = A. We see therefore that a Dirichlet form (~,7)) give us a semigroup (Tt) on L2(F, It). But does this semigroup correspond to a 'nice' Markov process? In general it need not, but if C is regular then one obtains a Hunt process. (Recall that

50 a Hunt process X = (X,, t > 0, ~ , x E F ) is a strong Markov process with cadlag sample paths, which is quasi-left-continuous.) T h e o r e m 4.8. ([FOT, Thin. 7.2.1.]) (a) Let (~, 9 ) be a regu/ar Dirichlet form on L 2 ( F , # ) . Then there exists a #-symmetric Hunt process X = ( X t , t > O , ~ , x E F ) on F with Dirichlet form C. (b) In addition, X is a diffusion if and only i r e is local. R e m a r k 4.9. Let X = (X~,t >_ 0,I?~,x E I~2) be Brownian motion on R 2. Let A C I~2 be a polar set, so that P~(TA < oo) = 0 for each x. T h e n we can obtain a new Hunt process Y -- (X~ >_ 0, Q=, x E ]~') by "freezing" XonA. SetQ= =~*,xEA c,andforzEAletQ~(xt=x, a l l t E [0, oo)) = 1 . Then the semigroups (TX), (TY), viewed as acting on L2(~2), are identical, and so X and Y have the same Dirichlet form. This example shows that the Hunt process obtained in Theorem 4.8 will not, in general, be unique, and also makes it clear that a semigroup on L 2 is a less precise object t h a n a Markov process. However, the kind of difficulty indicated by this example is the only problem - - see [FOT, Thm. 4.2.7.]. In addition, if, as will be the case for the processes considered in these notes, all points are non-polar, then the Hunt process is uniquely specified by the Dirichlet form g. We now interpret the conditions that ~ is conservative or irreducible in terms of the process X. L e m m a 4.10. I_fs is conservative then Ttl = 1 and the associated Markov process X has infinite lifetime.

Proof. If f e 9 ( s then 0 <_ e(1 + 1+ for any e R, and so e(1, f ) = 0. Thus ( - s f ) --- 0, which implies that s --- 0 a.e., and hence that Tel = 1. [] L e m m a 4.11. / f s is irreducible then X is irreducible. Proof. Suppose that X is reducible, and that F -- A1 U A2 is the associated decomposition of the state space. Then TtlA~ = 1A~, and hence f(1A~, 1A~) = 0. As 1 # 1A~ in L2(F, #) this implies that E is not irreducible. [] A remarkable property of the Dirichlet form E is that there is an equivalence between certain Sobolev type inequalities involving E, and bounds on the transition density of the associated process X. The fundamental connections of this kind were found by Varopoulos [V1]; [CKS] provides a good account of this, and there is a very substantial subsequent literature. (See for instance [Co] and the references therein). We say (C, 7)) satisfies a Nash inequality if

(4.10)

llflll~/~(~lifll~ +E(f,/)) _>cllfl[~ +~/~, f e v.

This inequality appears awkward at first sight, and also hard to verify. However, in classical situations, such as when the Dirichlet form E is the one connected with the Laplacian on ]~d or a manifold, it can often be obtained from an isoperimetric inequality.

51 In w h a t follows we fix a regular conservative Dirichlet form (E, V). Let (Tt) be the associated semigroup on L 2 (F, #), and X = (Xt, t > 0, P~) be the Hunt process associated with E. 4.12. ([CKS, Theorem 2.1]) (a) Suppose E satisfies a Nash inequality ~ t h constants c, 6, 8. Then there exists c' = c'(c, 8) such that

Theorem

(4.11)

[[Ttlh-~ < c ' d q - ~

t > O.

(b) If (Tt) satisfies (4.11) with constants c', 6, 0 then C satisfies a Nash inequality with constants c" = c"(c', 8), 6, and O. Proof. I sketch here only (a). Let f Z :D(s

T h e n writing ft = Ttf, and

gth : h-l(/t+h - ft) - T t s

we have llg~hll~ < IIg0hll2 ~ 0 as h ~ 0. It follows that (d/dt)ft exists in L 2 ( F , ~ ) and t h a t

d

-~ft = T t s

= s

Set ~(t) = (ft, ft). T h e n

h -1 (T(t + h) - T(t)) - 2(TtEf, T t f )

= (gth, ft + ]t+h) + (TtEf, ~t+h -- ft),

a n d therefore T is differentiable, a n d for t > 0 (4.12)

~o'(t) = 2(/25, ft) = - 2 E ( f t , ft).

If f C L2(F,t~), T t f E :D(s for each t > 0. So (4.12) extends from f E :P(s

to all

f e L2(F,#). Now let f > 0, a n d Hf]]l = 1: we have ]1s

(4.13)

~'(t) -- -2c(1,, ft)) < 2~11/~11~ cllI~ll~ +4/~ = 2~i~o(t)2 -

Thus ~ satisfies a differential inequality. Set r r If r

= 1. T h e n by (4.10), for t > 0,

is the solution of r

< -2cr176

-

c~~ 1+2/~

= e-2~t~o(t). T h e n

e46t/~ < - 2 c r

1+2/~

.1+~./0 = -c~v 0 then for some a E ]~ we have, for co = co(c, 0),

Co(t) =- co(t -4- a) -0/2. If r

is defined on (0, oc), t h e n a > 0, so t h a t < cot -~

r

t > o.

It is easy to verify t h a t r satisfies the same b o u n d - so we deduce t h a t

(4.14)

IlTdll~ ~-- e2~t~b(t) <_ coe26tt-~

Now let f , g E L2+(F,#) with

Ilfll~

--

Ilgll~

f e I2+,

Ilfllx = 1.

-- 1. T h e n

(T~.tf, g) = (Ttf, Ttg) < IIT~III~IIT~glI~ <_c~S2q -~

52 Taking the s u p r e m u m over g, it follows that replacing 2t by t, t h a t

IlT2tfll~

~ c~ e~2tt-~

that is,

IlTtll1-~ _~ c2 e~tt -e/z.

[]

R e m a r k 4.13. In the sequel we will be concerned with only two cases: either 5 = 0, or 5 = I and we are only interested in bounds for t E (0, 1]. In the latter case we can of course absorb the constant e 5t into the constant c. This theorem gives b o u n d s in terms of contractivity properties of the semigroup

(Tt). If Tt has a 'nice' density p ( t , x , y ) , then IlTtlll --- sup~,yp(t,x,y), so that (4.11) gives global upper b o u n d s on p(t,., .), of the kind we used in Chapter 3. To derive these, however, we need to know that the density of Tt has the necessary regularity properties. So let F, $, Tt be as above, and suppose that (Tt) satisfies (4.11). Write Pt(x, ") for the t r a n s i t i o n probabilities of the process X. By (4.11) we have, for A E B(F), a n d writing ct = eeStt-el2,

P t ( x , A ) < cry(A)

for # - a . a . x .

Since F has a countable base (An), we can employ the arguments of [FOT, p.67] to see that

Pt(x,A,~) < ct#(A,~),

(4.15)

x E F - Nt,

where the set Nt is "properly exceptional". In particular we have #(N~) = 0 and

~z~(Xs E Nt or X s - E Nt for some s > 0) = 0 for x E F - Nt. From (4.15) we deduce that Pt(x, .) << p for each x E F - Nt. If s > 0 a n d y ( B ) = 0 t h e n P , ( y , B ) = 0 for #-a.a. y, and so

Pt+~(x,B) =

f P~(z, dy)Pt(y,B) = O, z E F - N~.

So Pt+s(x, .) << # for all s > O, x E F - Nt. So taking a a single properly exceptional set N = U,~Nt. such that x E F - N. Write F ~ -- F - N: we can reduce the state T h u s we have for each t, x a density p(t, x, .) of Pt(x, can be regularised by integration.

sequence tn + 0, we obtain Pt(x, ") << tt for all t > 0, space of X to F ~. .) with respect to #. These

P r o p o s i t i o n 4.14. (See [Y, Thm. 2]) There exists a jointly measurable transition

density p ( t , x , y ) , t > O, x , y E F' • F ~, such that P t ( x , A ) = f p ( t , x , y ) y ( d y ) for x E F',

p(t,x,u)

t > O, A E B ( F ) ,

A = p ( t , u , x ) rot all x , u , t ,

p(t + s,~,z) = [ p ( s , z , y ) v ( t , y , z ) , ( d y ) J

for all

x,z,t,s.

53 C o r o l l a r y 4.15. Suppose (E,/3) satisfies a Nash inequality with constants c, 6, O. Then, for all x , y E F ~, t > O,

p(t, x , y ) <_ c'e~t -~ We also obtain some regularity properties of the transition functions p(t, x, .). Write qt,~(Y) = p(t, x, y). P r o p o s i t i o n 4.16. Suppose (g, ~)) satisfies a Nash inequality with constants c, 6, O. Then for x E F ~, t > O, qt,~ E l)([:), and [[qt,~l[~ -< Cl J ~ t t - ~

(4.16)

s

(4.17)

qt,~) <_ c~e~tt-l-~

Proof. Since qt,~ = Tt/2qt/2,x, and qt/2: E L 1, we have qt,~ E / ) ( L ) , and the bound (4.16) follows from (4.14). Fix x, write ft = q,,~, and let ~(t) = llftll]. Then

~o"(t)= ~t (2Lf~,ft) = 4(/:ft, s

>_ O.

So, W~ is increasing and hence o < ~,(t) = ~ ( t / 2 ) +

/i

~'(s) ds _ ~ ( t / 2 ) + (t/2)~'(t). 2

Therefore using (4.13), e(It, lt)=

- ~1 , (t) ~ t - l ~ ( t / 2 ) ~ ce~tt -1-~

[]

Traces of Dirichlet forms and Markov Processes. Let X be a #-symmetric Hunt process on a LCCB metric space (F,#), with semigroup (Tt) and regular Dirichlet form (E, :D). To simplify things, and because this is the only case we need, we assume (4.18)

Cap({x}) > 0 for all x e F.

It follows that x is regular for {x}, for each x C F, that is, that

I?~(T~ = O) --1,

x C F.

Hence ([GK]) X has jointly measurable local times (L~, x e F, t > 0) such that

/0

](X~)ds =

f(x)L~#(dx),

f C L2(F,#).

54 Now let v be a a-finite measure on F . (In general one has to assume v charges no set of zero capacity, b u t in view of (4.18) this condition is vacuous here). Let At be the continuous additive functional associated with v:

At = / L~u(da), a n d let Tt = inf{s : A , > t} be the inverse of A. Let G be the closed s u p p o r t of v. Let -~t = X~,: t h e n by [BG, p. 212], .~ = (Xt,I? ~, x E G) is also a Hunt process. We call ) ( the trace of X on G. Now consider the following o p e r a t i o n on the Dirichlet form s For g E L 2 (G, v) set

~(g,g) = i n f { s

(4.19)

f ) : f i g = g}-

4.17. ("Trace theorem": [FOT, Thm. 6.2.1]). (a) (s is a regular Dirichlet form on L2(G,u). (b) .~ is u-symmetric, and has Dirichlet form (g, ~ ) .

Theorem

Thus g is t h e Dirichlet form associated with X : we call g the

trace of s (on G).

R e m a r k s 4 . 1 8 . 1. T h e d o m a i n ~ on ~ is of course the set of g such t h a t the infimum in (4.19) is finite. If g E :D then, as g is closed, the infimum in (4.19) is a t t a i n e d , by f say. If h is any function which vanishes on G c, then since ( f + Ah)l v = g, we have

s which implies s

<s

f+)~h),

AEII~

h) = 9. So, if f ~ :D(L), and we choose h C Z), then (-h, s

= O,

so t h a t s = 0 a.e. on G c. This calculation suggests t h a t the minimizing function f in (4.19) should be the h a r m o n i c extension of g to F ; t h a t is, the solution to the Dirichlet p r o b l e m

.f= g s

= 0

on G on G ~.

2. We shall sometimes write

= Tr(EIG) to denote the trace of the Dirichlet form s on G. 3. Note t h a t t a k i n g traces has t h e "tower p r o p e r t y " ; if H C G C F , then

Tr(el H) = Tr(Tr(elG) [ H). We now look at continuous time Markov chains on a finite state space. Let F be a finite set.

55 Definition

4.19.

A conductance matrix on F is a m a t r i x A = (asv), x, y E F ,

which satisfies as~>_0,

x#y,

a s y ---- a y s ,

EY asy ~- O. Set as = ~

axy -- - a z s . Let EA = ({x,y} : a~y > 0}. We say t h a t A is irreducible

if the g r a p h (F, EA) is connected. We can i n t e r p r e t the pair (F, A) as an electrical network: asu is the conductance of t h e wire connecting t h e nodes x a n d y. T h e intuition from electrical circuit t h e o r y is on occasion very useful in Markov Chain t h e o r y - - f o r more on this see [DS]. Given (F, A) as above, define the Dirichlet form E -- s with domain C(F) = { f : F --, ~ } by (4.20)

E ( f , g) ---- ~1 E ~ , y asy (f(x) - f(y))(g(x) - g(y)).

Note t h a t , writing fs = f ( x ) etc.,

e ( f , g ) = 89

Z

=~

as,(f

- :,)(gs - a N )

y~s

x

~

asyfsgs- Z

y~s

s

= - Z assf go x

s

~

azyfsgy

y~

Z Z as fxg s

y~s

y

In electricai terms, (4.20) gives the energy dissipation in the circuit (F, A) if the nodes are held at potential f. (A current l~y = as~ (f(y) - f(z)) flows in the wire connecting x a n d y, which has energy dissipation Isy (f(y) - f ( z ) )

asy(f(y) - f ( x ) ) 2. T h e sum in (4.20) also use this i n t e r p r e t a t i o n of Dirichlet (4.20) gives a 1-1 correspondence vative Dirichlet forms on C(F). Let # point.

=

counts each edge twice). We can of course forms in more general contexts. between conductance matrices a n d conserbe any measure on F which charges every

4 . 2 0 . (a) I r A is a conductance matrix, then gA is a regu/ar conservative Dirichlet form. (b) IrE is a conservative Dirichlet form on L2(F, #) then s = s for a conductance

Proposition

m a t r i x A. (c) A is irreducible ff and only/rE is irreducible.

Proof. (a) It is clear from (4.20) t h a t e is a bilinear form, and t h a t s f ) > 0. If g = 0 V (1 h f ) t h e n lax - gyl -< Ifs - fyl for all x,y, so since asy _> 0 for x # y, s is Markov. Since E ( f , f ) < c(A,p)llfl[~, II.lle, is equivalent to ll.ll2, and so E is closed. It is clear from this t h a t s is regular.

56 (b) As s is a s y m m e t r i c bilinear form there exists a symmetric m a t r i x A such t h a t F_.(f,g) = --fTAg. Let f = fo,f~ ----a l x + flly; then E(f, f ) = - ~ 2 a ~ x - 2o~a~.y _/32avv. Taking ~ = 1,/3 -- 0 it follows t h a t a ~ < 0. The Markov p r o p e r t y of s implies t h a t E(f01,f01) _~ E ( f ~ l , f ~ l ) i f a < 0. So 0 ~_ --o~2a~ -- 2oLa~y,

which implies t h a t a~y >_ 0 for x # y. Since s is conservative we have 0 -- E(f, 1) = --fTA1 for all f . So A1 = 0, and therefore ~-~ya~y = 0 for all x. (c) is now evident. [] E x a m p l e 4.21. Let # be a measure on F , w i t h # ( { x } ) = # ~ > 0 for x E F . Let us find the generator L of the Markov process associated with s = ~A on L2(F, #). Let z 9 F , g -- 1~, a n d f 9 L2(F,#). Then

E(s,g) = - : A S

=

az S(y)= Z azy(S(z)- S(Y))y

y

and using (4.7) we have, writing (., . ) , for the inner p r o d u c t on L2(F, #),

~(f,g) ---(-L f,g), = -#~L f(z).

So,

LS(z) =

(4.21)

Z(a.z/.z)(S(.)

-

:(z))

Note from (4.21) that (as we would expect from the trace theorem), changing the measure # changes the jump rates of the process, but not the jump probabilities.

ElectricalEqa/va/ence. Definition 4.22. Let (F, A) be an electrical network, and G C F. conductance m a t r i x on G, and

If B is a

E . = T (EAIG) we will say t h a t the networks (F, A) and (G, B) are (electrically) equivalent on G. In intuitive terms, this means t h a t an electrician who is able only to access the nodes in G (imposing potentials, or feeding in currents etc.) would be unable to distinguish from the response of the system between the networks (F, A) and (G, B). D e f i n i t i o n 4.23. (Effective resistance). Let Go, G1 be disjoint subsets of F . The effective resistance between G0 and G1, R(Go, G1) is defined by (4.22)

R(Go,G1) -1 = i n f { E ( f , f ) : fIBo = 0, f l - 1 = 1}.

This is finite if (F, A) is irreducible.

57 If G = {x, y}, then from these definitions we see that (F, A) is equivalent to the network (G, B), where B = (b~y) is given by

b~y = by~ -- - b ~ = -byy -- R(x, y ) - l . Let (F, A) be an irreducible network; and G C F be a proper subset. Let H -- G c, and for .f E C(F) write f -- (IH, ]G) where fH, fV are the restrictions of f to H and G respectively. If g e C(G), then if g = Tr(s

~(g,g):inf {(,T,gT)A(f;),

fg e C(H)}.

We have, using obvious notation (4.23)

(fH,g )A

-= f T A H H f H + 2 f T A H G g + g T A G G g .

g

The function fH which minimizes (4.23) is given by f g ----A~IHAHVg. (Note that as A is irreducible, 0 cannot be an eigenvalue of AHH , so AH~/ exists). Hence (4.24)

~(g, g) = gT(AGG -- AGHAHI AHc)g,

so that ~ -- CB, where B is the conductivity matrix --1

B = AGG -- AGHAHHAHG.

(4.25) E x a m p l e 4.24. ( A - y matrix defined by,

transform). Let G -- {x0, Xl, x2} and B be the conductance

b~o~1

---

a2,

b~lx 2 =

ao,

b~:~ o

=

O~1.

Let F = G U {y}, and A be the conductance matrix defined by a,,,j =0, a,~y = H i ,

i#j, 0
If the ai and Hi are strictly positive, and we look just at the edges with positive conductance the network (G,B) is a triangle, while (F,A) is a Y with y at the centre. The A - y transform is that (F, A) and (G, B) are equivalent if and only if O~0

(4.26)

al

fllfl2 -

-

80 + 81 + 82'

8~Zo -

-

Ot 2 - -

130 + H1 + H~' Z0H1 H0 + 131 + H2"

Equivalently, if S =- aoal + ala2 -4- a2a0, then (4.27)

S Hi=--,

0
58 This can be proved by elementary, but slightly tedious, calculations. The A - y transform can be of great use in reducing a complicated network to a more simple one, though there are of course networks for which it is not effective.

Proposition 4.25. (See [Kib]). Let (F,A) be an irreducible electric network, and R(z, y) = R({z}, {y}) be the 2-point effective resistances. Then R is a metric on F. Proof. We define R(z, x) = 0. Replacing f by 1 - f in (4.22), it is clear that R(z, y) = R(y, z), so it just remains to verify the triangle inequality. Let z0, zl, z2 be distinct points in F , and G = {x0, zl, z2 }. Using the tower property of traces mentioned above, it is enough to consider the network ( G , B ) , where B is defined by (4.25). Let a0 = b. . . . . and define a~, a2 similarly. Let/3o, ill,/32 be given by (4.27); using the A - Y transform it is easy to see that

R(zi, zj) =/3:,-1 + /3~-1,

i # j.

The triangle inequality is now immediate.

[]

R e m a r k 4.26. There are other ways of viewing this, and numerous connections here with linear algebra, potential theory, etc. I will not go into this, except to mention that (4.25) is an example of a Schur complement (see [Car]), and that an alternative viewpoint on the resistance metric is given in [Me6]. The following result gives a connection between resistance and crossing times. 4.27. Let (F, A) be an electrical network, let /Z be a measure on F which charges every point, and let (Xt, t > O) be the continuous time Markov chain associated with s on L2(F,/Z). Write T~ = inf{t > 0 : X t = z}. Then if z # y, Theorem

(4.28)

E~T~ + EYT~ = R(x, y)/Z(F).

R e m a r k . In view of the simplicity of this result, it is rather remarkable that its first appearance (which was in a discrete time context) seems to have been in 1989, in [CRRST]. See [Tet] for a proof in a more accessible publication.

Proof. A direct proof is not hard, but here I will derive the result from the trace theorem. Fix x,y, let G = {x,y}, and let g = s = Tr(E[G). If R = R(x,y), then we have, from the definitions of trace and effective resistance, B =

R_ 1

_R_ 1

.

Let v = /zig; the process -~t associated with (E, L2(G,v)) therefore has generator given by f(z) =

- f(z)). w~z

Writing T~, Ty for the hitting times associated with X we therefore have +

= R(/Z

59 We now use the trace theorem. If f ( x ) : l~(x) then the occupation density formula implies that

#zL zt =

1,(X,)

ds =

[{s _< t: X , = z}[.

So

At =

I'

l a ( X , ) ds,

and thus if S = inf{t >__Ty : Xt = x} and S is d e l e d

~ =

f

similarly, we have

1G(xs) ds.

However by Doeblin's theorem for the stationary measure of a Markov Chain

(4.29/

.(a) = (E~S/-1E 9

f 1G(X.lds~(F).

Rearranging, we deduce that

= (#(F)/#(G))E~S

= (.(F)/~(G))(E~ + E~.) = R.(F).

[]

C o r o l l a r y 4.28. Let H C F, z ~ H. Then

E ' T H <_R(x, H)#(F). Proof. If H is a singleton, this is immediate from Theorem 4.27. Otherwise, it follows by considering the network ( F ' , H ' ) obtained by collapsing all points in H into one point, h, say. (So f ' = ( f - H) U {h}, and a'~h = ~yEH a~y). [] Remark.

This result is actually older than Theorem 4.27 - see [Tell.

5. G e o m e t r y

of Regular Finitely Ramified Fractals.

In Section 2 I introduced the Sierpinski gasket, and gave a direct "hands on" construction of a diffusion on it. Two properties of the SG played a crucial role: its s y m m e t r y and scale invariance, and the fact that it is finitely ramified. In this section we will introduce some classes of sets which preserve some of these properties, and such that a similar construction has a chance of working. (It will not always do so, as we will see). There are two approaches to the construction of a family of well behaved regular finitely ramified fractals. The first, adopted by LindstrCm [L1], and most of the mathematical physics literature, is to look at fractal subsets of R d obtained by generalizations of the construction of the Cantor set. However when we come to study processes on F the particular embedding of F in •d plays only a small role,

60 and some quite n a t u r a l sets (such as the "cut square" described below) have no simple embedding. So one m a y also choose to a d a p t an a b s t r a c t approach, defining a collection of well behaved fractal metric spaces. This is the approach of Kigami [Ki2], and is followed in much of the subsequent m a t h e m a t i c a l literature on general fractal spaces. ( " A b s t r a c t " fractals may also be defined as quotient spaces of p r o d u c t spaces - see [Kus2]). T h e question of e m b e d d i n g has lead to confusion between m a t h e m a t i c i a n s and physicists on at least one (celebrated) occasion. If G is a graph then the n a t u r a l metric on G for a m a t h e m a t i c i a n is the s t a n d a r d graph distance d(x, y), which gives the length of the shortest p a t h in G between z and y. Physicists call this the chemical distance. However, physicists, thinking in terms of the graph G being a m o d e l of a polymer, in which the individual strands are tangled up, are interested in the Euclidean distance between x and y in some embedding of G in ~ a . Since t h e y regard each p a t h in G as being a random walk p a t h in Z a, they generally use the metric d'(x, y) = d(x, y)U2. In this section, after some initial remarks on self-similar sets in R d, I will introduce the largest class of regular finitely ramified fractals which have been studied in detail. These are t h e pc.f.s.s, sets of Kigami [Ki2], and in what follows I will follow the a p p r o a c h of [Ki2] quite closely. D e f i n i t i o n 5.1. A m a p r : R d --* ~ d is a similitude if there exists a E (0,1) such t h a t [r - r = a]x - y[ for all x, y 9 I~a. We call a the contraction factor of

r Let M _> 1, a n d let r A C ~ d set

CM be similitudes with contraction factors a i . For

(5.1)

ffJ(A) = U r

M i=1

Let ~(n) denote the n-fold composition of 9 . D e f i n i t i o n 5.2. Let 35 be the set of non-empty compact subsets of R d. For A C ~ d set 6E(A) = {x : [ x - - a ] < E for some a 9 A}. The g a u s d o r f f m e t r i c d on 35 is defined by d ( A , B ) -- inf{E > O: A C 6~(B) and B C 6e(A)}. 5.3. ( S e e / F e , 2.10.21]). (a) d is a m e t r i c on 35. (b) (35, d) is complete. ( c ) / f E N = { K 9 35: g C B(0, N)} then K N is compact in g .

Lemma

CM) be as above, with ai C (0, 1) for each 1 < i < M . Then there exists a unique F 9 35 such that F = ~ ( F ) . b-hrther, if G E 35 then @'~(G) -~ F in d. l_f G 9 35 satisfies ~ ( G ) C G then F = A~=0~('~)(G). T h e o r e m 5.4. Let ( r

Proof. Note t h a t ~ : 35 --, 35. Set a = m a x i a i < 1. If Ai, Bi E 35, 1 < i < M note that

M M -- ma~xd(Ai,Bi). d(Ui=lAi,ui=lBi )<

61 (This is clear since if ~ > 0 and Bi C 5,(A~) for each i, then 0B~ C 5e(OA~)). Thus

d(q(A), q(B)) <_m~x d(~b,(A), r = max ai d(A, B) = ad(A, B). z

So 9 is a contraction on/C, and therefore has a unique fixed point. For the final assertion, note that if ff2(G) C G, then ~ ( ' 0 ( G ) is decreasing. So N,~q?(")(G) is non-empty, and must equal F. [] E x a m p l e s 5.5. The fractal sets described in Section 2 can all be defined as the fixed point of a map ~ of this kind. 1. The Sierpinski gasket. Let {al, a2, a3 } be the 3 corners of the unit triangle, and set (5.2)

r

= ~ + ~(x - ~),

x c R 2,

1 < i < 3.

2. The Vicsek Set. Let {al . . . . ,ad) be the 4 corners of the unit square, let M -- 5, let a5 = (21-,89 and let (5.3)

r

= ~, + ~1 ( ~ - ~,),

1 < i < 5.

It is possible to calculate the dimension of the limiting set F from ( r However an "non-overlap" condition is necessary.

CM).

D e f i n i t i o n 5.6. ( r CM) satisfies the open set condition if there exists an open set U such that r 1 < i < M, are disjoint, and q ( U ) C U. Note that, since q ( U ) C U, then the fixed point F of 9 satisfies F = Nq('0(U). For the Sierpinski gasket, if H is the convex hull of {al, a2, a3}, then one can take U = int(H). T h e o r e m 5.7, Let ( e l , . . . ,r satisfy the open set condition, and let F be the fixed point of 9. Let/3 be the unique real such that M

(5.4)

Eaf

= 1.

i=l

Then d i m H ( F ) --/3, and 0 < ?/f~(F) < oo. Proof. See [Fa2, p. 119]. R e m a r k . If ai -- a, 1 < i < M, then (5.4) simplifies to M a ~ = 1, so that (5.5)

~ =

log M log a -1 "

We now wish to set up an abstract version of this, so that we can treat fractals without necessarily needing to consider their embeddings in R 4. Let (F, d) be a compact metric space, let I -- IM = { 1 , . . . , M}, and let

r

I
62 be continuous injections. We wish the copies r to be strictly smaller than F, and we therefore assume that there exists 5 > 0 such that (5.6)

d(r162

~_ (1 - 5 ) d ( x , y ) ,

x , y E F,

i E IM.

D e f i n i t i o n 5.8. (F, r 1 < i < M ) is a self-similar structure if (F, d) is a compact metric space, r are continuous injections satisfying (5.6) and M

(5.7)

f = U r i=1

Let (F, r 1 < i < M ) be a self-similar structure. We can use iterations of the maps r to give the 'address' of a point in F. Introduce the word spaces W , = I s,

W =- I N.

We endow W with the usual product topology. For w E W , , v in W , or W, let w 9 v = ( w l , . . . ,w,~,vl . . . . ), and define the left shift a on W (or W , ) by

= For w -- (wl, ...,wn) E Wn define

(5.8) It is clear from (5.7) that for each n > 1,

F= U wEW~

If a = ( a l , . . . , a M ) is a vector indexed by I, we write (5.9)

a~ = l ~ a~,,,

w E Wn.

i~l

Write A~ = r m > n) write (5.10)

for w E UnW,~, A C F. If n_> 1, a n d w E W (or W,n with w i n -= ( w l , . . . ,wn) E Wn.

L e m m a 5.9. For each w E W, there ex/sts a x ~ E F such that (5.11)

fi r

= {x~}.

n----1

Since r = Cw]~(r C r the sequence of sets (5.11) is decreasing. As r are continuous, r are compact, and therefore A N~F~I n is non-empty. But as diam(F~ln) _< (1 - 5)~diam(f), we have diam(A) = so that A consists of a single point. Proof.

in = 0, []

63 Lemma

5.10. There exists a unique map Ir : W ~ F such that

(5.12)

~(i.w) : r

w ~ W,

i ~ I.

lr is continuous and surjective. Proof. Define ~r(w) : zw, where z~, is defined by (5.11). Let w E W. T h e n for any n,

r ( i - w) E F(i.w)l, = Fi.(~l,~_i ) --- r

).

So 7r(i. w) E Amr = {r proving (5.12). If r' also satisfies (5.12) t h e n ~d(v. w) = r (Tr'(w)) for v E Wn, w E W, n > 1. T h e n r ' ( w ) E F~,in for any n > 1, SO

71" ! ~

"ft.

To prove t h a t 7r is surjective, let x E F . By (5.7) there exists Wl E IM such that x E F= 1 = r M = U~o~=zF~I~ 2. So there exists w2 such t h a t x E F ~ 2, and continuing in this way we o b t a i n a sequence w -- ( w l , w 2 , . . . ) E W such t h a t z E F,~ln for each n. It follows t h a t x = ~r(w). Let U be open in F , a n d w E 7r-l(U). T h e n F=ln M U c is a decreasing sequence of c o m p a c t sets w i t h e m p t y intersection, so there exists m with F~,[m C U. Hence V -- {v E W : v[m --- w[m} C 71"-l(U), and since V is open in W, r - l ( U ) is open. Thus lr is continuous. [] Remark (5.13) Lemma

5.11. It is easy to see t h a t (5.12) implies t h a t r(v-w) = r

v E W,,

w ~ W.

5.12. For x E F, n >_ O set

N,(x) = U{F= : w E W,,z

E F~,}.

Then { N , ( x ), n > 1} form a base of neighbourhoods of x. Proof. Fix x and n. If v E W~ and x r F+ then, since F . is compact, d(x,F~) = i n f { d ( x , y ) : y E F~} > 0. So, as Wn is finite, d(x, N n ( x ) c) = min{d(x,F~) : x r F~,v E W~} > 0. So x E i n t ( N n ( x ) ) . Since d i a m F w < ( 1 - 6 ) n d i a m ( F ) f o r w E W,~ we have d i a m N , ( x ) < 2(1 - 6 ) ' ~ d i a m ( f ) . So if g ~ x is open, N~(x) C U for all sufficiently large n. [] T h e definition of a self-similar structure does not contain any condition to prevent overlaps between the sets r i E IM. (One could even have r -- r for example). For sets in I~d t h e open set condition prevents overlaps, b u t relies on t h e existence of a space in which the fractal F is embedded. A general, a b s t r a c t , non-overlap condition, in terms of dimension, is given in [KZl]. However, for finitely ramified sets the s i t u a t i o n is somewhat simpler. For a self-similar s t r u c t u r e S = (F, r i E IM) set

B = B(S) =

U i,j,i~ j

n Fj.

64 As one might expect, we will require B ( S ) to be finite. However, this on its own is not sufficient: we will require a stronger condition, in terms of the word space W. Set Y = rc-1 (B(S)), P = 0

~r~(F)"

n=l

D e f i n i t i o n 5.13. A self-similar structure (F, r is post critically finite, or p.c.f., if P is finite. A metric space ( F , d ) is a p.c.f.s.s, set if there exists a p.c.f, self-similar structure (r 1 < i < M ) on F. R e m a r k s 5.14. 1. As this definition is a little impenetrable, we will give several examples below. The definition is due to Kigami [Ki2], who called F the critical set of 8, and P the post critical set. 2. The definition of a self-similar structure given here is slightly less general than that given in [Ki2]. Kigami did not impose the constraint (5.6) on the maps r but made the existence and continuity of 7r an axiom. 3. The initial metric d on F does not play a major role. On the whole, we will work with the natural structure of neighbourhoods of points provided by the self-similar structure and the sets F,~, w E W ~ , n >_ O. E x a m p l e s 5.15. 1. T h e Sierpinski gasket. Let al, a2, a3 be the corners of the unit triangle in IRd, and let r

= ai + 89 - a,),

x e R 2,

1
Write G for the Sierpinski gasket; it is ctear that (G, r structure. Writing ~ = (s, s , . . . ) , we have r(~)=a~,

r

r

is a self-similar

l<s<3.

So B(S)

1

1

i a

= { (a3 + i(al + a2), i( 2 + a3)}, r = {(1i), (3i), (12), (2i), (2i), (32)},

and P=

o(r)={(i),(~),(i)}.

2. T h e cut square. This is an example of a p.c.f.s.s, set which has no convenient embedding in Euclidean space. (Though of course such an embedding can certainly be found). Start with the unit square Co = [0, 1]2 . Now make 'cuts' along the line L1 = 1 1 {(if,y) : 0 < y < if}, and the 3 similar lines (L2, L3, L4 say) obtained from L1 by rotation. So the set C1 consists of Co, but with the points in the line segment ( 1 , y _ ) , ( 89 viewed as distinct, for 0 < y < i" 1 (And similarly for the 3 similar sets obtained by rotation). Alternatively, C1 is the closure of A = Co - U,4=ILI in the geodesic metric dA defined in Section 2. One now repeats this construction on each of the 4 squares of side 89which make up C1 to obtain successively C2, C3,...; the cut square C is the limit.

65 This is a p.c.f.s.s, set; one has M = 4, and if a l , . . . , a 4 are the 4 corners of [0, 1]2, then the maps r agree at all points with irrational coordinates with the

maps ~,(x) = a, + 89 - a,). We have 1 ( 1 , ~), 1 (~, 1 0), (~, 1 1)} ~), (~,1 ~), r = {(12), (2i), (22), (32), (3~), (42), (4i), (14), (12), (3i), (24), (42)},

B=((0,1

so that P = {(i), (2), (3), (4)}. Note also that ~r(12) = 7r(2i), and r(13) = ~r(3i) = ~-(2~1) --- r(42) = z, the centre of the square. In b o t h the examples above we had P = {(h),s e IM}, and P = amP for all n > 1. However P can take a more complicated form if the sets e l ( F ) , C j ( F ) overlap at points which are sited at different relative positions in the two sets.

3. Sierpinski gasket with added triangle. (See [Kum2]). We describe this set as a subset of I~2. Let {ax,a2,a3} be the corners of the unit triangle in ]r and let r = 89 hi) + h i , 1 <_ i _< 3. Let a4 = 1(al + a s + a3) be the centre of the triangle, and let r = a4 + ~(x - a4). Of course ( r 1 6 2 1 6 2 gives the Sierpinski gasket, but q / = (r r r r still satisfies the open set condition, and if F -- F(iI~) is the fixed point of qt then ( F , r ,r is a self-similar structure. Writing bl, b2, b3 for the mid-points of (as, a3), (a3, hi), (al, a2) respectively, and c~ = ~(ai 1 + bi), 1 < i < 3, we have

B = {bl,b2,b3,cl,c2,c3}, ~r-l(bl) -- {(23), (32)}, while ~r-l(cl) -- {(123), (139.), (4i)}, with similar expressions for ~r-l(bj), ~r-l(ci), j -- 2,3. So # ( F ) -- 15, and

~(r) = {(i), (2), (2), (22), (32), ~(r) = {(i), (2), (2)}.

(3i), (13), (12), (2i)},

Then P = a(F) consists of 9 points in W, and #(~r(P)) = 6.

Fig. 5.1 : Sierpinski gasket with added triangle.

4. (Rotated triangle). Let hi, bi, r 1 < i < 3, be as above. Let A E (0,1), and let Pl = )~b2 + (1 - ;~)b3, with P2, P3 defined similarly. Evidently {Px,P2,P3} is an

66

equilateral triangle; let r be the similitude such that r -- Pi. Let F = F(@) be the fixed point of ~P. If H is the convex hull of {a~,a~,a3}, then ~ ( H ) C H, so clearly F is finitely ramified, and

B -= {bl,b2,b3,pl,p2,p3}.

Fig. 5.2 : Rotated triangle with A -- 2/3. As before, 7r-l(bl) -- {(23), (32)}. Let Yl -- r then Yl lies on the line segment connecting a2 and a3. If A = r -1 (Yl) then A consists of one or two points, according to whether )~ is a dyadic rational or not. Let A = ~v, w}, where v = w if ~ D. Note that for each element u E A, we have, writing u -- ( u l , u 2 , . . . ) , that uk 9 {2,3}, k > 1. Then I r - l ( p l ) = { ( 4 i ) , ( 1 . v),(1 .w)}. I f S : W --* W is defined by 8(w) -- w', where w~ -= w, + 1 (mod 3), and

A . = {(i), ~ v , o"w}, then ~ " ( i ~) = An U 8(A~) U 02(A~). (a) A -- 89gives Example 3 above. (b) If A is irrational, then P = u~>la~(r) is infinite. This example therefore shows that the "p.c.f." condition in Definition 5.13 is strictly stronger than the requirement that the set F be finitely ramified and self-similar. (c) Let )~ = ~. Then v = w = (23). Therefore B consists of Pl and bl, with their rotations, and a(L) consists of (23), (32), (4i), (12323) and their "rotations" by 8. Hence P = {(i), (2), (3), (23), (32), (3i), (i3), (i2), (2i)}. So A ---- 32-does give a p.c.f.s.s, set. (d) In general, as is clear from the examples above, while F is finitely ramified for any A 9 (0, 1), F is a p.c.f.s.s, set if and only if A 9 Q n (0,1).

67

Fig. 5.3 : R o t a t e d t r i a n g l e w i t h )~ -- 0.721. We n o w i n t r o d u c e some m o r e n o t a t i o n . Definition

5.16. Let ( F , r

,r

be a p.c.f.s.s, set. Set for n _> 0,

p ( n ) = {w E W : ~ w

E P},

v(") = ~(P(~)). A n y set of t h e form Fw, w E W=, we call a n n-complex, a n d a n y set of t h e form Cw(V(0)) _- V(~ we call a n-cell. Lemma

5.17.

(a) L e t x E V (n). T h e n x -- Cw(Y), where y E V (~ a n d w E W,~.

(b) v(n) = U ~ w o v 2 ) Proof. (a) F r o m t h e definition, x = ~r(w- v), for w E W,,, v E W. T h e n if y - ~r(v), y E V (~ a n d b y (5.13), x = r ( w . v) = r (b) Let x E V(~

Then x = r

( g ( v ) ) , where v E P . Hence x = ~ ( w . v), a n d since

w 9 v E P(~), x E V (~). T h e other i n c l u s i o n follows from (a).

[]

We t h i n k of V (~ as b e i n g t h e " b o u n d a r y " of t h e set F . T h e set F consists of the u n i o n of M '~ n - c o m p l e x e s F~o (where w E W ~ ) , which intersect o n l y at their b o u n d a r y points. Lemma

5.18.

(a) I f w, v E W,~, w • v, then F~ M F , = V (~ M V (~

(b) ~ n > 0, ~-1 (~(p(~))) = ~-I(V(~) ) : p(n). Proof. (a) Let n ~ 1, v, w E W n , a n d x E Fw M Fv. So x = ~ ( w . u) ~ ~r(v. u ' ) for u, u ' E W. Suppose first t h a t wl ~ Vl. T h e n as F ~ C Fw,, we have x E F~,, M Fv~ C B. So w . u , v . u ' E F, a n d t h u s u = a ' ~ - l a ( w . u ) E P. Therefore ~r(u) E V (~

68 and x = r E V(~ If wl -- vl t h e n let k be the largest integer such t h a t w l k = vik. A p p l y i n g r we can t h e n use the a r g u m e n t above. (b) It is e l e m e n t a r y T h e n t h e r e exists v m > 1. H e n c e t h e r e r = ~r(u. v) So

t h a t p ( , 0 C 7r-1 ( l r ( p ( ' 0 ) ) . Let n = 0 a n d w C ~r-1 (~r(P)). e P such t h a t lr(w) = r ( v ) . As v e P , v e a ' ~ ( r ) for some exists u E Wm such t h a t u . v C ~ r - l ( B ) . However 7r(u. w) -9 B , a n d t h u s u . v 9 a. Hence v 9 P .

If n _> 1, a n d 7r(w) 9 7r(P(~)) = Y (~), t h e n 7r(w) e V(~ for some v e W~. (0) a n d t h u s 9 ~ n Fwlo __ (~ n V (~ by (a). Therefore 7r(w) 9 Vwl"'

r(w) = r where v 9 P . So r ( w ) = r ( w i n , v), a n d t h u s r ( a ~ w ) = 7r(v). By t h e case n = 0 above a ~ w 9 P , a n d hence w 9 p ( , 0 . [] R e m a r k 5 . 1 9 . Note we used the fact t h a t ~r(v.w) = ~r(v.w') implies lr(w) = r ( w ' ) , which follows from t h e fact t h a t Cv is injective. Lemma n>l.

5 . 2 0 . L e t s 9 { 1 , . . . , M } . T h e n r(~) is in e x a c t l y one n-complex, for each

Proof. Let n -- 1, a n d write x, = 7r(~). P l a i n l y x, 9 F , ; suppose x, E Fi where i # s. T h e n x~ -- r for some w 9 W. S i n c e x , -- r for a n y k_> 1, x~ = r (1r(i.w)) = 7r(s k . i . w ) , where s k = (s, s . . . . . s) 9 Wk. Since x , 9 F, MF, C B , z c - l ( x , ) 9 C. B u t therefore s k 9 i . w 9 C for each k > 1, a n d since i # s, C is infinite, a c o n t r a d i c t i o n . Now let n > 2, a n d s u p p o s e x , = ~r(~) 9 F ~ , where w 9 W,~ a n d w # s n. Let 0 < k < n - 1 b e such t h a t w = s k . a k w , a n d Wk+l # s. T h e n a p p l y i n g r to F,~ we have t h a t x, 9 F~,~w M F , ~ - ~ , which c o n t r a d i c t s the case n = 1 above. [] Let ( F , r

,r

be a p.c.f.s.s, set. For x 9 F , let

ran(x) = # {w e Wn : x e Fw} be t h e n - m u l t i p l i c i t y of x, t h a t is t h e n u m b e r of d i s t i n c t n - c o m p l e x e s c o n t a i n i n g x. P l a i n l y , i f x r U,~V (n), t h e n m n ( x ) = 1 for all n. Note also t h a t m . ( x ) is increasing.

Proposition

5 . 2 1 . For a11 x E F , n >_ 1, m,(x) < M#(P).

Proof. S u p p o s e x E Fwl M . . . M Fwh, where w i, 1 < i < k are d i s t i n c t e l e m e n t s of W,). S u p p o s e first t h a t w~ # w~ for some i # j . T h e n x E B, a n d therefore there exist v l , . . . , v k C W such t h a t ~r(wl . v t) -- x, 1 < l < k. Hence wt . v I E r for each l, a n d so # ( r ) _> k. B u t # ( P ) > M - I # ( F ) , a n d thus k <_ M # ( P ) . If all t h e w z c o n t a i n a c o m m o n i n i t i a l s t r i n g v, t h e n a p p l y i n g r we can use the a r g u m e n t above. []

69

Nested Fractals and AfIine Nested fractals. Nested fraetats were introduced by Lindstrem [L1], and affine nested fractals (ANF) by [FHK]. These are of p.c.f.s.s, sets, but have two significant additional properties: (1) T h e y are embedded in Euclidean space, (2) They have a large s y m m e t r y group. I will first present the definition of an ANF, and then relate it to that for p.c.f.s.s, sets. Let r 1 6 2 be similitudes in I~d, and let F be the associated compact set. Writing r also for the restrictions of r to F, (F, r . . . . , eM) is a self similar structure. Let W, ~r, V (~ etc. be as above. For x, y E V (~ let g~u : R d --* I~d be reflection in the hyperplane which bisects the line segment connecting x and y. As each r is a contraction, it has a unique fixed point, z, say. Let V = {zl,..., ZM } be the set of fixed points. Call x E V an essential fixed point if there exists y E V, and i # j such that r = e j ( y ) . Write V (~ for the set of essential fixed points. Set also V ( " ) = [,.J V (~ wEW.

D e f i n i t i o n 5.22.

(F,r

. . . . ,r

is an affine nested fractal if r

,r

the open set condition, # ( V (~ > 2, and (A1) (Connectivity) For any i, j there exists a sequence of 1-cells V_-{~ such t h a t i o = i ,

ik=jandV

satisfy V_-(~

!~ N V I r~ Zv--1

(h2) (Symmetry) For each x, y E V (~ n _> 0, g~y maps n cells to n ceils. (Aa) (Nesting) If w, v E W,~ and w # v then F~ n F, = --(0)Vwn V~~ In addition (F, r factor. If r

eM) is a nested ffactal if tile r all have the same contraction

has contraction factor a~, then by (5.4) d i m H ( F ) =/~, where f~ solves M

(5.14)

E

a t = 1.

i=1

If ai ----a, so that F is a nested fractal, then (5.15)

log M dimH(f) = log(l/a)"

Following Lindstrem we will call M the mass scale factors and 1 / a the length scale factor, of the nested fractM F.

70 L e m m a 5.23. L e t ( F , r ,r be an arlene nested [ractM. fixed p o i n t o f r T h e n zi r F j t'or any j ~ i. Proof. S u p p o s e t h a t zx 9 F2. T h e n by

Write zi for the

Definition 5.22(A3) F1 n F~ = VI~ N ~0),

so zl 9 ~ 0 ) , a n d Zl = r for some zi C ~(0). We cannot have i = 2, as r = z2 ~ zl. Also, if i = 1 then r would fix b o t h zl and z2, so could not be a contraction. So let i = 3. Therefore for any k _> 0, i _> 0,

~1

=

r

o

r

o

ri

Z

9 FI~ 2 3,

W r i t e Dn = {w 9 Wn : zl 9 F~}: by the above # ( D n ) >_ n. Let U be the open set given by t h e open set condition. Since F C U we have zi 9 U for each i. So zl 9 U~ for each w 9 Dn, while the open set condition implies t h a t the sets {Uw, w 9 Dn} are disjoint. So zl is on the b o u n d a r y of at least n disjoint open sets. If (as is t r u e for n e s t e d fractals) all these sets are congruent then a contradiction is almost immediate. For t h e general case of affine nested fractals we need to work a little h a r d e r to o b t a i n t h e same conclusion. Let a > 0 be such t h a t

IB(z. 1) n U[ > a

for each i.

Let a i , 1 < i < M be the contraction factors of the r Recall the n o t a t i o n n a,, : IIi=la~, , w E Wn. Set 5 = minwED, a ~ , and let /3 : m i n i a i . For each w 9 Dn let w' : w 9 1...1 be chosen so t h a t / 3 6 < a~, < 6. Then za 9 Fw, C Uw,, for each w 9 On, a n d the sets { U w , , w 9 Dn} are still disjoint. (Since @(U) C V we have U~, C Uw for each w 9 Dn). Now i f w E Dn t h e n i f j is such t h a t Zl : r

IB(~,~) n V~'I = ~ , l B ( z j , ~ / ~ , ) n Vl >_ (/3~)~IB(~j, 1) n UI >

~(Z~)~.

So

Ca#= IB(zl,5)] __ ~

IB(zl,~)n Uw,I _>ha(~36)a.

wED~ Choosing n large enough this gives a contradiction. P r o p o s i t i o n 5.24. L e t ( F , r . . . . . CM) be an atIine nested [ractal. the fixed p o i n t o f r T h e n (F, C a , . . . , CM) is a p.c.f.s.s, set, a n d

(a) V ~~ = Vr176 P : -{(~): ~, ~

(b)

[] Write zi for

v~~

(c) I[ z 9 V (0) then z is in exactIy one n - c o m p l e x for each n >_ 1. (d) Each 1-complex contains at most one element o f V (~ Proof. It is clear t h a t (F, r

CM) is a self-similar structure. Relabelling the r

we can assume ~(0) = { z l , . . . ,zk} where 2 < k < M . We begin by calculating B, F a n d P . It is clear from (A3) t h a t B = lJ(v~ ~ n s~t

V~o)).

71 Let w 9 I'. r ( a w ) 9 V-/~ 5.23 we must

Then 7r(w) 9 B, so (as ~r(w) 9 F~,) 7r(w) 9 V ~ ,

and therefore

Say ~r(aw) = z~, where s 9 {1, .., k}. Then since z, 9 F,~, by L e m m a have

= 8. S o

=

=

= z,, and therefore

r ( a 2 w ) ---- z~. So w3 = 8, and repeating we deduce that aw -- (~). { a w , w 9 I'} -- {(~), 1 < s < k}. This proves (b); as P is finite ( F , r

Therefore ,r is a

p.c.f.s.s, set. (a) is immediate, since r ( P ) = Y (~ = {lr(~)} = V (~ (c) This is now immediate from (a), (b) and Lemma 5.23. (d) Suppose Fi contains z~ and zt, where s ~ t. Then one of s,t is distinct from i suppose it is s. T h e n z~ 9 F~ M Fi, which contradicts (c). [] R e m a r k s 5.25. 1. Of the examples considered above, the SG is a nested fractal and the SG with added triangle is an ANF. The cut square is not an ANF, since if it were, the maps r : lI~d --* R a would preserve the plane containing its 4 corners, and then the nesting axiom fails. The rotated triangle fails the symmetry axiom unless A -- 1/2. The Vicsek set defined in Section 2 is a nested fractal, but the Sierpinski carpet fails the nesting axiom. 2. The simplest examples of p.c.f.s.s, sets, and nested fractals can be a little misleading. Note the following points: (a) Proposition 5.24(c) fails for p.c.f.s.s, sets. See for example the SG with added triangle, where V (~ contains the points {bl, b2, b3 } as well as the corners {a~, a~, a3 }, and each of the points bl lies in 2 distinct 1-cells. (b) This example also shows that for a general p.c.f.s.s, set it is possible to have F - V(~ disconnected even if F is connected. (c) Let Vi(~ and V(~ be two distinct 1-cells in a p.c.f.s.s, set. Then one can have # ( V i (~ M V(~ > 2. (The cut square is an example of this). For nested fractals, I do not know whether it is true that (5.16)

#(Vi (~ M V(~

_< 1

if i ~ j.

In [FHK, Prop. 2.2(4)] it is asserted that (5.16) holds for affine nested fractals, quoting a result of J. Murai: however, the result of Murai was proved under stronger hypotheses. While much of the work on nested fractals has assumed that (5.16) holds, this difficulty is not a serious one, since only minor modifications to the definitions and proofs in the literature are needed to handle the general case. 3. The s y m m e t r y hypothesis (A2) is very strong. We have (5.17)

g ~ y : V (~ ~ V (~

for all

z#y,

z,y 9

(~

The question of which sets V (~ satisfy (5.17) leads one into questions concerning reflection groups in R ~. It is easy to see that V (~ satisfies (5.17) if V (~ is a regular planar polygon, a d-dimensional tetrahedron or a d-dimensional simplex. (That is, the set V (~ = { e i , - e i , 1 < i < d} C ~d where ei = (fili,-.-,6di). I have been assured by two experts in this area that these are the only possibilities, and my web page see ( h t t p : / / w w w . m a t h . u b r c a / ) contains a letter from G. Maxwell with a sketch of a proof of this fact. Note that the cube in ~3 fails to satisfy (5.17).

72 4. Note also t h a t if F is a nested fractal in ~d, and V (~ C H where H is a kdimensional subspace, one does not necessarily have F C H. This is the case of the Koch curve, for example. (See ILl, p. 39]). E x a m p l e 5.26. (LindstrCm snowflake). This nested fractal is the "classical example", used in [L1] as an illustration of the axioms. It m a y be defined briefly as follows. Let z~, 1 < i < 6 be the vertices of a regular hexagon in R 2, and let 1 z~ = ~(Zl + . . . z6) be the centre. Set r

= z, + 89 - zi),

1 < i < 7.

It is easy to verify t h a t this set satisfies the axioms ( A 1 ) - ( A 3 ) above.

Fig. 5.4. Lindstr0m snowflake.

Measures on p.c.s

sets.

T h e s t r u c t u r e of these sets makes it easy to define measures which have good p r o p e r t i e s relative to t h e maps r We begin by considering measures on W. Let 0 = ( 0 1 , . . . ,OM) satisfy M

Z

Oi-=l,

0 <0i < 1

for each

iCIM.

i=1 n

Recall the n o t a t i o n 0~ = 1-Ii=l 0~,~ for w C Wn. We define the measure/~e on W to be the n a t u r a l p r o d u c t measure associated with the vector 0. More precisely, let ~,~ : W -~ IM be defined by ~ ( w ) = w~; then/2o is the measure which makes (~,~) i.i.d, r a n d o m variables with d i s t r i b u t i o n given by ~(~= = r) = 0~. Note t h a t for a n y n _ > 1, w E Wn, n

(5.18)

({v c w :

=

= II i=l

D e f i n i t i o n 5.27. Let /~(F) be the a-field of subsets of F generated by the sets {F~, w E W,~,n _> 1}. (By L e m m a 5.12 this is the Borel e-field). For A E B ( F ) , set =

73 T h e n for w E W~ ~t

(5.19)

#o(F~)=f~o(r-~(F~))

=~o((,:~l~=~})--o~=IIo~, i----1

In contexts when 0 is fixed we will write # for #e. R e m a r k . If ( F , r 1 6 2 is a nested fractal, then the sets r 1 < i < M are congruent, a n d it is n a t u r a l to take 0, -- M -1. More generally, for an ANF, the ' n a t u r a l ' 0 is given by where/3 is defined by (5.4). The following L e m m a summarizes the self-similarity of # in terms of the space LI(F,#). L e m m a 5.28. Let f E L 1(F, #). Then t'or n > 1

wGW~

Proof. It is sufficient to prove (5.20) in the case n : 1: the general case then follows by iteration. Write G -- F - V (~ Note that G~ M G~ = ~ if v, w E Wn a n d v ~ w. As # is non-atomic we have # ( F ~ ) = # ( G ~ ) for any w E Wn. Let f = l a ~ for some w E W~. T h e n f o r = 0 if i ~ Wl, and f o r = 1 G ~ . Thus

fIfo

0=:f fd ,

proving (5.20) for this particular f . The equality then extends to L 1 by a s t a n d a r d argument. [] We will also need related measures on the sets V (n). Let No -- # V (~ and set (5.21)

#~(x) = N0-1 ~ 0~lv(o,(X), wcw~

Fix

x E Y (~).

L e m m a 5.29. /z,~ is a probability measure on V ('~) a n d w l i m , ~ o o # n -- #e.

Proof. Since # V (0) -- No we have

~(v(~)) : Z No' Z e=~,o~(~): Z e~:~, zEV(~)

wGW~

wEW~

proving the first assertion. We may regard #n as being derived from # by shifting the mass on each ncomplex F ~ to the b o u n d a r y V(~ with an equal amount of mass being moved to

74 each point. (So a point x E V(~ obtains a contribution of 0,~ from each n-complex it belongs to). So if f : F --~ R then (5.22)

f fd#JF

f fd#n JR

_< max sup I f ( x ) - f(Y)l wEW~ x,yEF~

It follows t h a t #n--*#0.~ Symmetries of p.c.s

[] sets.

D e f i n i t i o n 5.30. Let g be a group of continuous bijections from F to F. We call G a symmetry group of F if (1) g : V (~ ~ V (~ for all g E G. (2) For each i E I, g E G there exists j E I, g~ E G such that (5.23)

g

o

r

= e j o g'.

Note that if g, h satisfy (5.23) then (goh) or162162 =

(r

o

g') o h'

=

ek o g",

for some j , k E I, g',h*,g" E ~. The calculation above also shows that if G1 and G2 are s y m m e t r y groups then the group generated by G1 and G2 is also a s y m m e t r y group. Write G(F) for the largest symmetry group of F. If G is a symmetry group, and g E G write ~(i) for the unique element j E I such that (5.23) holds. L e m m a 5.31. Let g E G. Then for each n _> 0, w E W,~, there exist v E W~, g' E G such that g o ev~ = ev o g'. In particular g : V ('q --* V (n). Proof. The first assertion is just (5.23) if n = 1. If n _> 1, and the assertion holds for all v E W,~ then if w = i 9v E W,,+I then

gor for

j E I, gl,g. E ~.

= gor

or

= ~ og' o r

= ~j o r

og",

[]

Proposition

5.32. Let (F, r . . , ~)M) be an A N F . Let G1 be the set of isometries of R a generated by reflections in the hyperplanes bisecting the line segments [zi, zj ], i ~ j, z,, zj E V (~ Let Go be the group generated by G1. Then GR = {giF : g E GO} is a s y m m e t r y group of F. Proof. I f g E G1 t h e n g : V (n) --~ V (n) for each n and hence a l s o g : F - - ~ F. Let i E I: by the s y m m e t r y axiom (A2) g(Vi(~ = V(~ for some j E I. For each of the possible forms of V (~ given in Remark 5.25(3), the symmetry group of V (~ is generated by the reflections in G1. So, there exists g' E G0 such that g o r = r o g'. Thus (5.23) is verified for each g E G~, and it follows that (5.23) holds for all g E G0. [] R e m a r k 5.33. In [BK] the collection of 'p.e.f. morphisms' of a p.c.f.s.s, set was introduced. These are rather different from the symmetries defined here since the definition in [BK] involved 'analytic' as well as 'geometric' conditions.

75 Connectivity Properties. D e f i n i t i o n 5.34. Let F be a p.c.f.s.s, set. For n > 0, define a graph structure on V (n) by taking {x, y} E En if x # y, and x, y E V~(~ for some w E W=. P r o p o s i t i o n 5.35. Suppose that (V (1), E1 ) is connected. Then (V(n), En) is connected for each n > 2, and F is pathwise connected. Proof. Suppose that ( v ( n ) , E n ) is connected, where n > 1. Let x , g E V ('~+1). If x , g E V (1) for some w E Wn, then, since (V(1),E1) is connected, there exists a path r = z o , z l , . . . ,zk = r in (V(1),E1) connecting r and r We have zi-1, zi E v(~ ,wl for some wi E W1, for each 1 < i < k. Then if z[ = r ! zi_ 1, z iI E Fw~.~ and so {z~_l,zl } E En+l. Thus x , y are connected by a path in (V (=+~),E.+I). For general x, y E V (n+l), as (V('~),En) is connected there exists a path Yo,...,Y,~ in ( v ( n ) , E n ) such that { Y i - l , y i } E En and x,yo, and Y, Ym, lie in the same n + 1-cell. Then, by the above, the points x, Y0, Y l , . . . , Ym, Y can be connected by chains of edges in En+l. To show that F is path-connected we actually construct a continuous path 7 : [0, 1] --~ F such that F = {7(t),t C [0, 1]}. Let x 0 , . . . ,XN be a path in (V(I),E1) which is "space-filling", that is such that V (1) C { x 0 , . . . , XN}. Define 7 ( i / N ) = x~, A1 = { i / N , 0 <_ i < N } . Now x o , x l E V~(~ for some w E W1. Let x0 = Yo,Yl,... ,Y,~ = xl be in a space-filling path in (V~(1),E2). Define 7 ( k / N m ) = Yk, 0 _< k < m. Continuing in this way we fill each of the sets V(1), w E W1, and so can define A2 C [0,1] such that A~ C A2, and 3'(t), t E A2 is a space filling path in the graph (V (2), E2). Repeating this construction we obtain an increasing sequence (An) of finite sets such that 3'(t), t E AT, is a space filling path in (V ('~), E,~), and UnA,~ is dense in [0, 1]. I f t C An, and t' < t < t" are such that (t',t") M An = {t}, then 3'(s) is in the same n-complex as 3'(t) for s e (t', t"). So, if t e [0,1] - A, and sn,tn E An are chosen so that sn < t < tn, (Sn, tn) M An = 0, then the points 3'(u), u E A M (s, t) all lie in the same n-complex. So defining 3'(t) = limn 3'(tn), we have that the limit exists, and 3' is continuous. The construction of 3' also gives that 3' is space filling; if w E W then for any n > 1 a section of the path, 3'(s), an < s <_ bn, s E A,~, fills V(~). It follows immediately from the existence of 3' that F is pathwise connected. [] Remark. This proof returns to the roots of the subject - the original papers of Sierpinski [Siel, Sie2] regarded the Sierpinski gasket and Sierpinski carpet as "curves". C o r o l l a r y 5.36. A n y A N F is pathwise connected. R e m a r k 5.37. If F is a p.c.f.s.s, set, and the graph (V(1),E1) is not connected, then it is easy to see that F is not connected. For the case of ANFs, we wish to examine the structure of the graphs (V (=), En) a little more closely. Let ( F , r ,r be an ANF. Then let

a:min{I x-y[:

x, y 6 V (~ x # y } ,

76 and set

E6 = {{x,y} E V(~ Ix-yl = a } ,

E'~ =

I{x,y}

E E,~:x--r162

W E W~, { x , y }tE

for some E0} ' , n _> 1.

P r o p o s i t i o n 5.38. L e t F be an A N F . (a) L e t x , y , z E V (~ be distinct points. T h e n there exists a p a t h in (V(~ connecting x and y and not containing z. (b) L e t x , y E V (~ T h e r e exists a p a t h in (V(1),E~) connecting x , y which does not contain a n y point in V (~ - {x, y}. (c) L e t x, y, x', y' E V (~ with Ix - Yl -- Ix' - Y'I. T h e n there exists g E GR such that =

g(y')

= y.

Proof. If # ( Y (~ = 2 then E0 = E~, so (a) is vacuous and (b) is i m m e d i a t e from Corollary 5.36. So suppose # ( V (~ _~ 3. (a) Since (see R e m a r k 5.25(3)) V (~ is either a d-dimensional tetrahedron, or a ddimensional simplex, or a regular polygon, this is evident. (For a proof which does not use this fact, see [L1, p. 34-35]). (b) This now follows from (a) by the same kind of argument as t h a t given in Proposition 5.35. (c) Write g[x, y] for the reflection in the hyperplane bisecting the line segment Ix, y]. Let gl = g[Y,Y'], and z = g l ( x ' ) . T h e n if z = x we are done. Otherwise note t h a t I x - Yl -- !x ' - Y'[ = [ z - y[, so if g2 = g[x~z] t h e n g2(Y) = Y. Hence gl og~ works. [] Metrics on N e s t e d FractaJs. Nested fractals, a n d A N F s , are subsets of I~d, and so of course are metric spaces with respect to the Euclidean metric. Also, p.c.f.s.s, sets have been assumed to be m e t r i c spaces. However, these metrics do not necessarily have all properties we would wish for, such as the mid-point p r o p e r t y t h a t was used in Section 3. We saw in Section 2 t h a t the geodesic metric on the Sierpinski gasket was equivalent to the E u c l i d e a n metric, b u t for a general nested fractal there may be no p a t h of finite length between distinct points. (It is easy to construct examples). It is however, still possible to construct a geodesic metric on a A N F . For simplicity, we will just t r e a t the case of nested fractals. Let (F, ( r M be a n e s t e d fractal, w i t h length scale factor L. Write d,~(x,y) for the n a t u r a l graph distance in the graph (V(~),E~). F i x x0, Y0 E V (~ such t h a t {x0,Y0} E E~, and let an = dn(xo, Yo), and b0 be the m a x i m u m distance between points in (V (~ E~). Lemma

5.39. I f x, y E V (~ then an < d,~(x,y) < bonn.

Proof. Since x , y are connected by a p a t h of length at most b0 in (V(~ the u p p e r b o u n d is evident. F i x x, y, and let k = d,~(x, y). If {x, y} E E~ then d~(x, y) = d~(xo,Yo) = as, so suppose {x,y} • E~. Choose y' E Y (~ such t h a t { x , y ' } E E~o, let H be the h y p e r p l a n e bisecting [y, y~] and let g be reflection in H. Write A, A ~ for the c o m p o n e n t s of I~d - H containing y, y' respectively. As Ix - y'[ < Ix - Yl we have x E A ~. Let x = Z o , Z l , . . . , z k = y be the shortest p a t h in (V(n),E,~) connecting x a n d y. Let j = min{i : z, E A}, a n d write z[ = z, if i < j , z[ = g(zi)

77 i f i _>j. T h e n z~, 0 < i < k is a p a t h in ( V ( n ) , E n ) connecting x and y', and so [] dn(x, y) = k > d , ( x , y') = an. Lemma

5.40. Let x, y E V (n) 9 Then for m > 0

(5.24)

a m d , ( x , y) < d , + m ( x , y) <_ boamd,(x, y).

In particular (5.25)

anam <_ a,~+m <_ boanam,

n > O, m > O.

Proof. Let k = dn(x, y), a n d let x = z0, z a , . . . , zk = y be a shortest p a t h connecting x a n d y in ( V ( " ) , E n ) . T h e n since by L e m m a 5.39 d , ~ ( z i - l , z i ) < boam, the u p p e r b o u n d in (5.24) is clear. For the lower b o u n d , let r = d~+,~(x, y), and let (zi)i~0 be a shortest p a t h in (V ("+m), En+,~) connecting x, y. Let 0 = i0, i l , . . . , is = r be successive disjoint hits by this p a t h on V (n). (Recall the definition from Section 2: of course it makes sense for a d e t e r m i n i s t i c p a t h as well as a process). We have s = d~(x, y) >_ an. T h e n since zi~_l,zij lie in the same n-cell, ij - ij-1 = dm(zi~_l,zij) >_ am, by L e m m a 5.39. So r = E ; : I ( i j - i j - 1 ) -> a , am. [] C o r o l l a r y 5.41. There exists 7 E [L, boa1] such that (5.26)

bo17 n _< an _< 7".

Proof. Note t h a t log(b0an) is a subadditive sequence, and t h a t log an is superadditive. So by the general t h e o r y of these sequences there exist 00, 01 such t h a t 00 = l i m n -1 log(b0an) = inf n -1 log(b0a~), 01 = l i m n -1 l o g ( a , ) = sup n -1 l o g ( a , ) . n---* o o

n~0

So 00 = 0x, a n d setting 7 -- ee~ (5.26) follows. To o b t a i n b o u n d s on ~/ note first t h a t as a , <_ boalan-1 we have -y < boa1. Also, ]xo - Yol <- anL-'~lxo - Yo], s o ' y _> L.

[]

D e f i n i t i o n 5.42. We call d~ = l o g T / l o g L the chemical exponent of the fractal F , a n d 7 t h e shortest path scaling factor. T h e o r e m 5.43. There exists a metric d F on F with the following properties. (a) There exists Cl < oo such that for each n >_ O, w E Wn, (5.27)

dF(x, y) (_ el"/-n for x, y E Fw,

and (5.28)

dF(X, y) > c2~/-n for x E V (n), y E N n ( x ) c.

(b) d e induces the same topology on F as the Euclidean metric.

78 (c) dF has the midpoint property, (d) The Hausdorff dimension of F with respect to the metric dF is log M

d r ( F ) - - log 7

(5.29)

Proof. W r i t e V = UnV(n). By L e m m a 5.41 for x, y E V we have (5.30)

bo17md,~(x, y) < dn+m(x, y) < boTmdn(x, y).

So (7-"~dn+m(x,y),m _> 0) is b o u n d e d above and below. By a diagonalization a r g u m e n t we can therefore find a subsequence nk ~ cx) such t h a t

d f ( x , y ) = lim 7 - ' ~ d , ~ ( x , y ) exists for each x,y 9 V. k---~ o o

So, if x, y c V(~ where w 9 W~ then

(5.31)

Co17 - ~ < dF(x, y) < c07 - ~

It is clear t h a t dR is a metric on V. Let n > 0 and y 9 V ('~). For m = n - 1 , n - 2 , . . . , 0 choose inductively Ym 9 V (m) such t h a t Ym is in the same m-cell as Y,~+I,...,Y,. T h e n _ max{dl(X',y') : z', d,,~+l(ym,Ym+l) <

9 V (1) } = c < oc.

y'

So by (5.30) d,~(y~, Ym+l) < b07 n - ( " + l ) c , and therefore oo

d(y~, y) < ~ ~

7 -~-1 = ~'7 -k

i=k

So if x, y E V are in the same k-cell, choosing xk in the same way we have (5.32)

dF(x,y) < dF(x,x~) + dF(xk,yk) + dF(yk,Y) <_c17 k,

since dk(zk, Yk) < bo. Thus dE is uniformly continuous on V • V, and so extends by continuity to a m e t r i c dF on F . (a) is i m m e d i a t e from (5.31). If x,y 9 V (n) and z # y then dF(x,y) >_ bo17 -n, This, together with (5.30), implies (b). If z, y E V ('~) t h e n there exists z 9 V ('~) such t h a t

I ~ ( ~ , z ) _ 1 ~dn(x,y)l <_ 1,

u = x,y.

So the metrics d,~ have an a p p r o x i m a t e m i d p o i n t property: (c) follows by an easy limiting argument. Let /t be the measure on F associated with the vector ~ = ( M - 1 , . . . , M - X ) . Thus #(F~,) --- M -I~~ for each w 9 W,~. Since we have d i a m d F ( F ~ ) • 7 - I ' l l , it follows t h a t , writing d I -- log M/log 7,

~sr d' _< ~(Bd~(~,r)) _< c6r d', and t h e conclusion then follows from Corollary 2.8.

~ 9 F []

79 R e m a r k 5.44. T h e results here on the metric dF are not the best possible. The construction here used a subsequence, and did not give a procedure for finding the scale factor 7. See [BS], [Kum2], [FHK], [Ki6] for more precise results.

6. R e n o r m a l i z a t i o n o n F i n i t e l y R a m i f i e d Fractals. Let ( F , r ,r be a p.c.f.s.s, set. We wish to construct a sequence y ( , 0 of r a n d o m walks on the sets V ('0, nested in a similar fashion to the r a n d o m walks on the Sierpinski gasket considered in Section 2. T h e example of the Vicsek set shows t h a t , in general, some calculation is necessary to find such a sequence of walks. As the r a n d o m walks we t r e a t will be symmetric, we will find it convenient to use the t h e o r y of Dirichlet forms, and ideas from electrical networks, in our proofs. F i x a p.c.f.s.s, set (F, (r M and a Bernouilli measure # -- P0 on F , where each 01 > 0. We also choose a vector r : ( r l , . . . , rM) of positive "weights": loosely speaking ri is the size of the set r : Fi, for 1 < i < M . We call r a resistance

vector. D e f i n i t i o n 6.1. Let D be the set of Dirichlet forms C defined on C(V(~ From Section 4 we have t h a t each element C 9 D is of the form CA, where A is a conductance m a t r i x . Let also I~ 1 be the set of Dirichlet forms on C(V(1)). We consider two operations on D: (1) R e p l i c a t i o n - i.e. extension of s 9 I~ to a Dirichlet form C n 9 lI)l. (2) D e c i m a t i o n / R e s t r i c t i o n / T r a c e . Reduction of a form C 9 D1 to a form ~ 9 D. N o t e . In Section 4, we defined a Dirichlet form (C, :D) with domain 9 C L 2 (F, #). But for a finite set F , as long as p charges every point in the set it plays no role in the definition of s We therefore will find it more convenient to define E on C(F) = { f : f - ~ R}.

D e f i n i t i o n 6.2. Given s 9 D, define for f,g 9 C(V(1)),

M

CR(f,g) = ~ r~-lC(f o r

(6.2)

or

i----1 (Note t h a t as r

: V (~ --* V (1), / o r

9 C(V(~

Define R : ]D --~ D1 by

R(c) = c R

L e m m a 6.3. Let C = CA, and let M

(6.3)

R

axy : ~

--1

l(zEVi(0))l(yEVi(o,)r i ar

).

i=1

Then (6.4)

Cn(f,g) = ~12 a~,n(f(x) - f(y))(g(x) - g(y)).

80 A R = (a~y) R is a c o n d u c t a n c e m a t r i x , and s

is the associated Dirichlet form.

R _> 0 i f x ~ y , a n d a ~R < 0 . are injective, it is clear t h a t a~y R = ay~ R is i m m e d i a t e from the s y m m e t r y of A. Writing xi = r (x) we have Also azy Proof. As the m a p s r

E

ely:

~-~r/-llv(o)(X)~

yEV(1)

1v(o,(y)ar162

i

yEV(1)

i

yEv(O)

)

so A R is a conductance matrix. To verify (6.4), it is sufficient by linearity to consider the case f -- g = 6~, z 9 Y (I). Let B = {i 9 W1 : z 9 Vi(~ I f i r B, then f o r cannot equal z. If i 9 B, then f o r = 6~,(z), where zi : r s

o

r

f

o

r

sincer So,

= C(6,,, ~z,) ----- a z , z,.

Thus s

f ) = -- ~

M r ~ l a z i z , = -- ~ ri-11vi(O)(z)acrl(z),erl(z )

iEB

=

R

--azz

~

i=l

while

X:

Zc~

So (6.4) is verified.

_

S(y))

_fTAR f

R

[]

T h e most intuitive explanation of the replication operation is in terms of electrical networks. T h i n k of V (~ as an electric network. Take M copies of V (~ and rescale the i t h one by multiplying the conductance of each wire by r~-1. (This explains why we called r a resistance vector). Now assemble these to form a network with nodes V (1), using the i t h network to connect the nodes in V~(~ T h e n g R is the Dirichlet form corresponding to the network V (1). As we saw in the previous section, for x , y E VO) there m a y in general be more t h a n one 1-cell which contains b o t h x and y: this is why the sum in (6.3) is necessary. If x a n d y are connected by k wires, with conductivities c l , . 9 9 ck t h e n this is equivalent to connection by one wire of conductance c1 + .. 9 + c~. R e m a r k 6.4. T h e replication of conductivities defined here is not the same as the replication of t r a n s i t i o n probabilities discussed in Section 2. To see the difference, consider again the Sierpinski gasket. Let V (~ -- {Zl,Z2, z3}, and Y3 be the midp o i n t of IZl,Z:], a n d define y~, Y2 similarly. Let A be a conductance m a t r i x on V (~ a n d write aij = a ~ z j . Take r l : r2 = r3 = 1. While the continuous time Markov Chains X (~ X (1) associated with s and CA R will d e p e n d on the choice of a measure on V (~ and V (D, their discrete time skeletons t h a t is, the processes X (i)

81

sampled at their successive j u m p times do not - see Example 4.21. Write these processes. We have ~Ys

(]i1(

1) E {Z2, Yl

})

y(i)

for

a12 + a31 = 2a12 + a31 + a23

On the other hand, if we replicate probabilities as in Section 2,

in general these expressions are different. So, even when we confine ourselves to symmetric Markov Chains, replication of conductivities and transition probabilities give rise to different processes. Since the two replication operations are distinct, it is not surprising that the dynamical systems associated with the two operations should have different behaviours. In fact, the simple symmetric random walk on V (~ is stable fixed point if we replicate conductivities, but an unstable one if we replicate transition probabilities. The second operation on Dirichlet forms, that of restriction or trace, has already been discussed in Section 4. Definition

6.5. For E 9 D1 let

(6.5)

T(s

Define A : D --* D by A(8) = that if O > 0,

=

T(R(8)).

Tr(EIV(~ Note that A is homogeneous in the sense

A(SE) = 8A(s E x a m p l e 6.6. (The Sierpinski gasket). Let A be the conductance matrix corresponding to the simple random walk on V (~ so that a~y=l,

x~y,

a~ =-2.

Then A R is the network obtained by joining together 3 symmetric triangular networks. If A(EA) = CB, then B is the conductance matrix such that the networks (V(1),A R) and (V(~ are electrically equivalent on V (~ The simplest way to calculate B is by the A - Y transform. Replacing each of the triangles by an (upside down) Y, we see from Example 4.24 that the branches in the Y each have conductance 3. Thus (V (1), A R) is equivalent to a network consisting of a central triangle of wires of conductance 3/2, and branches of conductance 3. Applying the transform again, the central triangle is equivalent to a Y with branches of conductance 9/2. Thus the whole network is equivalent to a Y with branches of conductance 9/5, or a triangle with sides of conductance 3/5. Thus we deduce A(EA) = EB,

where

3 B = ~A.

82 T h e example above suggests t h a t to find a decimation invariant r a n d o m walk we need to find a Dirichlet form s E ID such t h a t for some A > 0 (6.6)

h(s

= As

Thus we wish to find an eigenvector for the m a p A on ]I). Since however (as we will see shortly) A is non-linear, this final formulation is not p a r t i c u l a r l y useful. Two questions i m m e d i a t e l y arise: does there always exist a non-zero (E, A) satisfying (6.6) a n d if so, is this solution (up to constant multiples) unique? We will abuse terminology slightly, a n d refer to an E E I]) such t h a t (6.6) holds as a fixed point of A. (In fact it is a fixed point of A defined on a quotient space of :D.) Example

6.7. ("abc gaskets" - see [HHW1]).

Let m l , m2, m3 be integers with mi _> 1. Let Zl, z2, za be the corners of the unit triangle in R 2, H be the closed convex hull of {zl, z2, z3 }. Let M = m l + m2 + ma, and let r 1 < i < M be similitudes such t h a t (writing for convenience eM+j = ej, 1 < j < M ) Hi = r C H , and the M triangles Hi are arranged round the edge of H , such t h a t each triangle Hi touches only H i - a a n d H i + l . (//1 touches HM and H~. only). In addition, let za E / / 1 , z2 E Hm~+l, za E H m , + m l + l . So there are ma + 1 triangles along the edge [zl,z2], and m l + 1, m2 + 1 respectively along [z~,z3], [z3,zl]. We assume t h a t r are rotation-free. Note t h a t the triangles /-/2 and HM do not touch, unless m l = m2 = m3 = 1. Let F be the fractal o b t a i n e d by Theorem 5.4 from ( r 1 6 2 To avoid unnecessarily c o m p l i c a t e d n o t a t i o n we write r for b o t h r and r

Figure 6.1: abc gasket with mx = 4, m2 = 3, m3 = 2. It is easy to check t h a t (F, r , eM) is a p.c.f.s.s, set. Write r = 1, s = m 3 + 1 , t = m3 + m , + 1. We have ~(/~) = ~ ( ( / + 1)§ for 1 < i < m3, ~(ii) = ~ ( ( i + 1)~) form3+l
(~), (i)},

v (~ = { ~ , z z , z 3 } .

and

83 W h i l e it is easier to define F in R 2, r a t h e r t h a n abstractly, doing so has the misleading consequence t h a t it forces the triangles Hi to be of different sizes. However, we will view F as an a b s t r a c t metric space in which all the triangles Hi are of equal size, a n d so we will take ri = 1 for 1 < i < M . We now s t u d y the renormalization m a p A. If C = s E D, then A is specified by the conductivities Oil = a z 2 , z 3 ,

0~2 -~ az3,z~,,

(~3 = a z l ,z2.

Let f : ~3 ~ ]~3 be t h e renormalization m a p acting on ( a l , a2, a3). (So if A = A ( a ) t h e n A(~) = CA(f(,~))). It is easier to c o m p u t e the action of the renormalization m a p on the variables fli given b y t h e / ~ - Y , transform. So let ~o : (0, ~ ) 3 __, (0, c~) 3 be the A - y m a p given in E x a m p l e 4.24. Note t h a t ~o is bijective. Let fl = ~ ( a ) be the Y - c o n d u c t i v i t i e s , a n d write ~ = (ill, f12, f13) for the renormalized Y - c o n d u c t i v i t i e s : then = ~ ( f ( a ) ) . A p p l y i n g the A - y transform on each of the small triangles, we o b t a i n a network with nodes zl, z2, z3, Yl, Y2, Y3, where {zi, yi} has conductivity 8i, a n d if i ~ j {Yi, Yj} has c o n d u c t i v i t y 8i, and i f / ~ j, {Yi,Yi} has conductivity

fi

where k = k(i,j) is such t h a t k e {1,2,3} - {i,j}. A p p l y the /k - y transform again to {Yl,Y2,Y3}, to o b t a i n a Y, with conductivities 61, 52, 53, in the branches where 6i7 i :

S ~---71 "~2 ~- "~2 73 "~- "/371,

1 < i < 3.

Then

(6.7)

~;-i = ~{-1 + 6{-1 = 8;-i + (~+~,),~i s .

Suppose t h a t a E ( 0 , ~ ) 3 is such t h a t ~ ( a ) = Aa for some ), > 0. T h e n since ~(Oa) = 0 ~o(a) for any 0 > 0, we deduce t h a t ~ = ~(f(a)) = Aft. So, from (6.7), ~2~~2 A-I~{-I = Z;-I + ( ~ + ),.,s,

which implies t h a t ) - 1 > 1. Writing T = ] ~ 1 ~ 2 f l 3 / S , and 0 = TA(1 - A ) - ' , we therefore have ml(Z2 + 8 3 ) = 0, a n d (as (6.8)

S,T

are s y m m e t r i c in the ~i) we also o b t a i n two similar equations. Hence f12+83-0/ml,

fl3+fl, =8/m2,

fl1+82 =0/m3,

which has solution (6.9)

281 =

O(m~ 1 + m31 - m~l),

etc.

84 Since, however we need the/~i > 0, we deduce that a solution to the conductivity renormalization problem exists only if m~-1 satisfy the triangle condition, that is that (6.10)

m~- l + m ~ -1 > m l 1,

m31+ml

I >m21,

m~- l + m 2 1

> m ~ -1.

If (6.10) is satisfied, then (6.9) gives fl~ such that the associated c~ = ~ - l ( f l ) does satisfy the eigenvalue problem. In the discussion above we looked for strictly positive ~ such that ~(~) -~ ha. Now suppose that just one of the a~, ~3 say, equals 0. Then while z, and z2 are only connected via z3 in the network V (~ they are connected via an additional path in the network V (1). So, ~(a)3 > 0, and ~ cannot be a fixed point. If now al > 0, and a2 = a3 = 0 then we obtain ~(a)2 -- ~(~)3 : 0. So ~ = (1, 0, 0) satisfies ~(a) = h a for some A > 0. Similarly (0, 1, 0) and (0, 0, 1) are also fixed points. Note that in these cases the network (V (~ A(a)) is not connected. The example of the abc gaskets shows that, even if fixed points exist, they may correspond to a reducible (ie non-irreducible) E E ~). The random walks (and limiting diffusion) corresponding to such a fixed point will be restricted to part of the fractal F. We therefore wish to find a non-degenerate fixed point of (6.6), that is an s E I]) such that the network (V (~ A) is connected. D e f i n i t i o n 6.8, Let ]])~ be the set of E E D0 such that s is irreducible - that is the network (V(~ is connected. Call E E]I) strongly irreducible if C -- s and a ~ > 0 for all x :~ y. Write D ~ for the set of strongly irreducible Dirichlet forms on V(~ . The existence problem therefore takes the form: P r o b l e m 6.9. (Existence). Let ( F , r 1 6 2 Does there exist E EI]) i, A > 0, such that

(6.12

A(E)

be a p.c.f.s.s, set and let ri > 0.

:

Before we pose the uniqueness question, we need to consider the role of symmetry. Let (F, (r be a p.c.f.s.s,set, and let T/be a symmetry group of F.

Definition 6.10. C E ~ is 7-t-invariant if for each h E E(foh, goh)=s

f, g E C(V (~

r is 7-t-invariant if rh(i) = ri for all h E ~ . (Here h is the bijection on I associated with h). L e m m a 6.11. (a) Let C = EA. Then E is ~-l-invariazit if and only if: (6.13)

ah(x) h(y)

:

a~y for all x,y E V (~

h E 7~.

(b) Let s and r be ~=invariant. Then AE is 7~-invariant. Proof. (a) This is evident from the equation s

ly) -- -a~y.

85 (b) Let f E C(V(D). Then if h E 7-/,

s

o h , f o h) = Z

r~l ~(f o h o r

ohor

i

I f g C C ( V (~

--~ ~

?,~-1 ~,(f 0 ~)h(i) O h, f o eh(i) o h, )

_- ~

?.;(1i)E(f o ~bh(i), f o ~b~(i)) = ER(f,f).

then writing ~ = A(s

~(goh, goh) ~_s

if fly(o) = g then as f o hlv(o ) = g o h,

oh, f oh)=ER(f,f),

and taking the infimum over :, we deduce that for any h C ~, ~'(goh,goh) _< ~'(g,g). Replacing g by g o h and h by h -1 we see that equality must hold.

[]

If the fractal F has a non-trivial symmetry group ~ ( F ) then it is natural to restrict our attention to ~(F)-symmetric diffusions. We can now pose the uniqueness problem. P r o b l e m 6.12. (Uniqueness). Let (F, (r be a p.c.f.s.s, set, let 7"/be a symmetry group of F, and let r be 7~-invariant. Is there at most one 7g-invariant C E D i such that A(s = ),s (Unless otherwise indicated, when I refer to fixed points for nested fractals, I will assume they are invariant under the symmetry group GR generated by the reflections in hyperplanes bisecting the lines Ix, y], x, y E V(~ The following example shows that uniqueness does not hold in general. E x a m p l e 6.13. (Vicsek sets - see [Me3].) Let (F, r 1 < i < 5) be the Vicsek set - see Section 2. Write {zl, z2, z3, z4,} for the 4 corners of the unit square in R 2. For a,fl,3, > 0 let A(a,f~,~) be the conductance matrix given by a12 : a23 : a34 ---- a41 = or,

ot13 = ~ ,

a24 = - y ,

where aij : az~ z~. If 7t is the group on F generated by reflections in the lines [Zl, z3] and [z2,z4] then A is clearly 7~-invariant. Define &,/~, ~ by

A(EA)= CA(W,~, 7)" Then several minutes calculation with equivalent networks shows that

(6.14)

~ =

a ( a + 13)(a + 7) 5a 2 + 3a/3 + 3a'y + / ~ 7 '

5(" + ~) -

~,

7= 89 If (1,13,3,) is a fixed point then ( ~ , ~ , ~ ) -- (0,0~,0~,) for some 0 ~ 0, so that S o ~ - I - 5 , and this implies that f17 = 1. We therefore have that (1, fl,f~-l) is a fixed point (with A -- 89 for any f~ E (0, oo) Thus for the group 7/ uniqueness does not hold.

=~,~=~7.

86 However if we replace 7-/by the group ~n = G(F), generated by all the symmetries of the square then for EA to be QR-invariant we have to have/3 = 7. So in this case we obtain (6.15)

~ ( . + f~)2

~(a, fl) -- 5a 2 + 6~fl + ~ 2 ,

~(~,~)

+~)

~.

This has fixed points (0,~),/3 > 0, and (a, a), a > 0. The first are degenerate, the second not, so in this ease, as we already saw in Section 2, uniqueness does hold for Problem 6.12. This example also shows that A is in general non-linear. As these examples suggest, the general problem of existence and uniqueness is quite hard. For all but the simplest fractals, explicit calculation of the renormalization map A is too lengthy to be possible without computer assistance - at least for 20th century mathematicians. LindstrCm ILl] proved the existence of a fixed point E E I) si for nested fractals, but did not treat the question of uniqueness. After the appearance of [L1], the uniqueness of a fixed point for LindstrCm's canonical example, the snowflake (Example 5.26) remained open for a few years, until Green [Gre] and Yokai [Yo] proved uniqueness by computer calculations. The following analytic approach to the uniqueness problem, using the theory of quadratic forms, has been developed by Metz and Sabot - see [Me2-Me5, Sabl, Sab2]. Let l~+ be set of symmetric bilinear forms Q(f, g) on C(V (~ which satisfy

Q(1,1) = 0, Q ( f , / ) _> 0 for all f 9

C(V(~

For Q1, Q2 9 l~+ we write Q1 _> Q2, if Q2 - Q1 9 • + or equivalently if Then I3 C l~+; it turns out that we need to consider the action of A on 1~+, and not just on 13. For Q 9 l ~ + , the replication operation is defined exactly as in (6.2)

Q2(f,f) >_Ql(f,f) for all f 9 C(V(~

M

(6.16)

Qn(f,g)=Er~lQ(for162

f,g 9

i=1

The decimation operation is also easy to extend to IVY+ :

T(Qn)(g,g) = inf{QR(f,f): f 9 C(V(~ we can write

= g};

T(Q R) in matrix terms as in (4.24). We set A(Q) = T(Qn).

L e m m a 6.14. The m a p A on M+ satisfies: (a) A : M + ~ ~+, and is continuous on int(M+). (b) A(Q1 + Q2) _> A(Q1) + A(Q2). (c) A(OQ) = 0A(Q)

Proof. (a) is clear from the formulation of the trace operation in matrix terms.

87

Since the replication operation is linear, we clearly have QR = (0Q)R = ~QR. (c) is therefore evident, while for (b), ifg ~ C(V(~

+ Qf,

T(QR)(g,g) = inf{Q~ ( f , f ) + Q~ ( f , f ) : ftv(o) -:- g} > i n f { Q ~ ( f , f ) : fly(o) = g} +inf{Q2R(f,f): fly(o) = g} = T(Q1R)(g,g) + T(QR)(g,g).

[]

Note that for 8 E D i, we have E(f, f ) = 0 if only if f is constant. D e f i n i t i o n 6.15. For 81,82 C ~)i set

= inf{ E1(f, f) constant}. 82(f, f ) : f non Similarly let M(81/82) = sup ~r ~$1(f, f ) : f non constant}. Note that (6.18)

M(~/82)

=

m(82/$~) -~.

L e m m a 6.16. (a) For s

E ]I)i, 0 < m(81,82) < oo . (b) l-f El, & e D*i then m(81/~2) = M(E1/&) ff and only if82 = 481 for some

A>O.

(c) If •,E2,E3

E D i then

m(E1/e ) >_re(elf82) m(82183), M(E1/E3) _< M(81/E2) M(s

Proof. (a) This follows from the fact that Ei are irreducible, and so vanish only on the subspace of constant functions. (b) is immediate from the definition of m and M. (c) We have m(,-qx/Sa) = ii~f 81 (f, f) 82(f, f) > m(81/$2)rn(82/E3); 82(f,f)

E3(f,f)

--

while the second assertion is immediate from (6.18). D e f i n i t i o n 6.17.

[]

Define M(81s dH(81,82)----log m(Eas

81,82 E

Di"

Let pD i be the projective space D i / ~-, where 81 ~ E2 if 81 = 482. dH is called Hilbert's projective metric - see [Nus], [Me4].

88 = 0 if and only if s = )~s for some )~ > O. (b) dH is a pseudo-metric on D ~, and a metric on p D ~. (c) Ifs163163 E E) ~ then for aO,al > O,

P r o p o s i t i o n 6.18. (a) d H ( ~ 1 , s

d~(s ~0 G0 + ~1s

< ma~(dz(s E0), dz(s

In particular open balls in dH are convex. (d) (pD i, dH) is complete. Proof. (a) is evident from Lemma 6.17(b). To prove (b) note that dH(s163 > O, and that dH(s s = dH(s s from (6.18). The triangle inequality is immediate from L e m m a 6.17(c). So dH is a pseudo metric on D i. To see that dH is a metric on p D i, note that m(Ael/e2) = A-~(s163

A > 0,

from which it follows that dH(As s = dH(s s and thus dH is well defined on p/l) i. The remaining properties are now immediate from those of dH on I])i. (c) Replacing s by ('~(s163163 s we can suppose that m(s163 = m(s163 Write Mi = M ( s 1 6 3

= m.

Then if ~" = so Go + a1s

M(s

= inf aos

I

f) + a l s s

f)

~ o~0m(s163 -~- o ~ 1 m ( s Similarly M ( s

= 0~0 "4- 0~1.

~ aoMo + aiM1. Therefore

exp dH(E,~) < (~o/(~o + ~l))(Mo/m) + (~1/(~o + < B(s

m~x(Mo/~,M~/~).

It is immediate that if s E B ( s then dH(s163 is convex. For (d) see [Nus, Thin. 1.2].

Theorem

~1))(M1/~)

6.19. Let s163

+ (1 -- A)s

< r, so that []

E IDi. Then

(6.19)

m(A(s163

> m(s163

(6.20)

M(A(s163

<_ M(s163

In particular A is non-expansive in dH : dH(A(s163

(6.21)

<_ dH(s163

Proof. Suppose a < m(El,s Then Q = s - as non-constant f E C(V(~ So by Lemma 6.14 A(s

= A(Q + as

E

~ + , and Q ( f , f ) > O, for all

_> A(Q) + ah(s

89 and since A(Q) _> 0, this implies that A(s - h A ( & ) _> 0. So a < m(A(E1),A(s and thus m(s s <_ m(A(s A(s proving (6.19). (6.20) and (6.21) then follow immediately from (6.19), and the definition of dg. [] A strict inequality in (6.21) would imply the uniqueness of fixed points. Thus the example of the Vicsek set above shows that strict inequality cannot hold in general. So this Theorem gives us much less than we might hope. Nevertheless, we can obtain some useful information. C o r o l l a r y 6.20. (See [HHW1, Cor. 3.7]) Suppose s A(s - Ais i = 1, 2. Then A1 = A2.

s are fixed points satisfying

Proof. From (6.19) re(e,/e=) <_ m (A(s so that A1 _> A2. Interchanging s

= (A1 / ~=)m(s163

and s

we obtain )h = A2.

[]

We can also deduce the existence of 7-/-invariant fixed points. P r o p o s i t i o n 6.21. Let ~ be a symmetry group o f F . I r A has a fixed point El in D i then A has an Tl-invariant fixed point in D i .

Proof. Let A = {s E IDi : s is 7-/-invariant.}. (It is clear from Lemma 6.11 that A is non-empty). Then by Lemma 6.11(b) A : A -+ A. Let s E A, and write r = dH(s163 B = BdH(gl,2r). By Theorem 6.20 A : B - - * B. S o A : A N B A A B. Each of A, B is convex (A is convex as the sum of two 7~-invariant forms is 7-(-invariant, B by Proposition 6.18(c)), and so A fq B is convex. Since A is a continuous function on a convex space, by the Brouwer fixed point theorem A has a fixed point s 9 A N B, and s is ~-invariant. [] We will not make use of the following result, but is useful for understanding the general situation.

(with E1 ~ )~s

C o r o l l a r y 6.22. Suppose A has two distinct fixed points s and s for any A). Then A has uncountably many fixed points.

Proof. (Note that the example of the Vicsek set shows that 89163+ s is not necessarily a fixed point). Let F C D i be the set of fixed points. Let go,s 9 F; multiplying s by a scalar we can take m(s163 --- 1. Write R = dH(s163 If s ----As + (1 -- A)g0 then as in Proposition 6.19(c) expdn(Ex,s

) <_ (1 - A) + AM(s163

and so

d7~(s163

<_ log((1 + eR)/2).

Thus there exists 6, depending only on R, such that

A = {s 9 D i : s 9 B(s

- 6)R) N B ( s

- ~)R)}

is non-empty. Since A preserves A, A has a fixed point in A. F thus has the property: if s

s

are distinct elements of ]F then there exists s

such that 0 < d~(s163

< dn(s

9 F

90 As ]F is closed (since A is continuous) we deduce that F is perfect, and therefore uncountable. []

-

This if as far as we will go in general. For nested fractals the added structure s y m m e t r y and the embedding in R d, enables us to obtain stronger results. If (F, (r is a nested fractal, or an ANF, we only consider the set D i N {s : s is GR-invariant}, so that in discussing the existence and uniqueness of fixed points we will be considering only GR-invariant ones. Let (F, (r be a nested fractal, write G = GR and let EA be a (G-invariant) Dirichlet form on C ( V (~ CA is determined by the conductances on the equivalence classes of edges in (V (~ Co) under the action of g. By Proposition 5.38(c) if Ix-y] = Ix' - y'[ then the edges { x , y } and {x',y'} are equivalent, so that A~v = A~,y,. List the equivalence classes in order of increasing Euclidean distance, and write a l , a 2 . . . , a ~ for the common conductances of the edges. Since A = A(A) is also g-invariant, A induces a m a p A' : R~_ --* R~_ such that, using obvious notation, A(A(a)) = A(A'(a)). Set D* = {a : a l > a2 >~ ... > 0% > 0}. Clearly we have I~* C D s~. We have the following existence theorem for nested fractals. T h e o r e m 6.23. (See ILl, p. 48]). Let (F, (r Then A has a tlxed point in D*.

be a nested fractal (or an ANF).

Proof. Let s E D*, and let a l , ...ak be the associated 0, Q~, z E V (~ be the continuous time Markov chain (Y,~,n > 0, Q ~ , z E V (~ be the discrete time skeleton L ~~ ~~ 0 0 ) ' " ' " ' ~~(k) be the equivalence classes of edges 0 a i if {x, y} E E~ i). Then if {x, y} E E~ j),

conductivities. Let (Yt, t > associated with CA, and let of Y. in (V (~ ) Eo), so that A~ v

ai Q'(Y1 = V) - ~ y # ~ A~v" As cl = ~ A~ v does not depend on x (by the s y m m e t r y of V (~ the transition vgz probabilities of Y are proportional to the a i. Now let R ( A ) be the conductivity matrix on V (1) attained by replication of A. Let (Xt, t >_ O, I?~, x e V (1)) and (2~,n > 0, P~,z c VO)) be the associated Markov Chains. Let To,T1, ... be successive disjoint hits (see Definition 2.14) on V (~ by -~,~. Write .4 = A(A), and ~ for the edge conductivities given by A. Using the trace theorem, IP~(2T~ -= y) = ~ j / c l if {x, y} 9 E~ i). Now let x l , y l , y 2 (6.23)

9 V (~ with Ix - Yll < ]x - y2]. We will prove that ]~z'(2T~ = Y2) < IP*'(2T~ = Yl).

Let H be the hyperplane bisecting [Yl,Y2], let g be reflection in H , and x2 = g(xl). Let T = min{n > 0: .~,~ 9 V (~ - {xl}},

91

so that T1 = T F~l-almost surely. Set

/~(z) = E ~ i(r <.)(i~ (Xr) - ly~(2r)). Let p(x,y), x,y E V (~ be the transition probabilities of ,~. Then

(6.24

/~+~(~) = IA(~):o(~) + IAo(~) ~

p(~,y)/,,(y).

Y

Let J12 = {x E V (1) : tx - Yll < tx - Y~I}, and define J21 analogously. We prove by induction that f,~ satisfies

f=(x) > O, f,,(x) + f,~(g(x)) >_ O,

(6.25a) (6.25b)

x E J12, x E J12.

Since f0 = 1~ - ly2, a n d y l E J12, f0 satisfies (6.25). Let x E A c U J 1 2 and suppose fn satisfies (6.25). If p(x,y) > 0, and y C J~2, then x , y are in the same 1-cell so if y' = g(y), y' is also in the same 1-cell as xl and Ix - y'] < Ix - y]. So (since $A 6 •*), p(x,y') > p(x,y) and using (6.25b), as f,~(y') >_ O,

p(~, y)/n(y) + p(~, y')/,~(y')>_ p(~, y)(/,,(y)+ .:,,(g(y))_> o. Then by (6.24), fn+l(x) > 0. A similar argument implies that fn+l satisfies (6.25b). So (f,~) satisfies (6.25) for all n, and hence its limit f ~ does. Thus foc(xl) = I?~(-~T = Yl) -- ~(XT = Y2) > 0, proving (6.23). From (6.23) we deduce that ~l > ~2 > ... > ~k, so that A : D* ~ D*. As A'(Oa) = 0A'(a), we can restrict the action of A' to the set {.

> ... >

>

0, y :

=

This is a closed convex set, so by the Brouwer fixed point theorem, A ~ has a fixed point in ]D*. [] R e m a r k 6.24. The proof here is essentially the same as that in LindstrOm ILl]. The essential idea is a kind of reflection argument, to show that transitions along shorter edges are more probable. This probabilistic argument yields (so far) a stronger existence theorem for nested fractals than the analytic arguments used by Sabot [Sabl] and Metz [Me7]. However, the latter methods are more widely applicable. It does not seem easy to relax any of the conditions on ANFs without losing some link in the proof of Theorem 6.23. This proof used in an essential fashion not only the fact that V (~ has a very large symmetry group, but also the Euclidean embedding of V (~ and V (1). The following uniqueness theorem for nested fractals was proved by Sabot [Sabl]. It is a corollary of a more general theorem which gives, for p.c.f.s.s, sets, sufficient conditions for existence and uniqueness of fixed points. A simpler proof of this result has also recently been obtained by Peirone [Pe].

92 be a nested fractal. Then A has a unique ~R-invariant

T h e o r e m 6.25. Let (F, (r non-degenerate fixed point.

Definition 6.26. Let s be a fixed point of A. The resistance scaling factor of C is the unique p > 0 such that

A(E) = p-1 E. Very often we will also call p the resistance scaling factor of F : in view of Corollary 6.21, p will have the same value for any two non-degenerate fixed points.

Proposition 6.27. Let (F, (r

be a p.c.s set, let (r,) be a resistance vector, and let s be a non-degenerate fixed point of A. Then for each s E {1, ...M} such that r(~) C V (~ (6.27)

r~ p-1 < 1.

Proof. Fix 1 < s < M, l e t x = 7r(~), and let f = 1~ E C(V(~ EA(f,f) =

E A~v = IA~I. yE V (~ y#x

=- p-lEA,

Let g : lx C C(V(1)). As A(s (6.28)

Then

p-l[A~z[ = A ( E A ) ( f , f ) < EAR(g,g) :

since g is not harmonic with respect to CA R, strict inequality holds in (6.28). By Proposition 5.24(c), x is in exactly one 1-complex. So

SAR(g,g) = ~

r:-lSA(g o r

or

= r:llAzzl,

i and combining this with (6.28) gives (6.27).

[]

Since r~ = 1 for nested fractals, we deduce C o r o l l a r y 6.28. Let (F, (r

be a nested fractal. Then p > 1.

For nested fractals, many properties of the process can he summarized in terms of certain scaling factors. D e f i n i t i o n 6.29. Let (F, (r be a nested fractal, and E be the (unique) nondegenerate fixed point. See Definition 5.22 for the length and mass scale factors L and M. The resistance scale factor p of F is the resistance scaling factor of E. Let (6.29

~ = Mp ;

we call T the time scaling factor. (In view of the connection between resistances and crossing times given in Theorem 4.27, it is not surprising that ~- should have a connection with the space-time scaling of processes on F.) It may be helpful at this point to draw a rough distinction between two kinds of structure associated with the nested fractal ( F , r The quantities introduced in Section 5, such as L, M, the geodesic metric dF, the chemical exponent 7 and the dimension dw(F) are all geometric- that is, they can be determined entirely by a geometric inspection of F. On the other hand, the resistance and time scaling

93 factors p and ~- are analytic or physical- they appear in some sense to lie deeper than the geometric quantities, and arise from the solution to some kind of equation on the space. On the Sierpinski gasket, for example, while one obtains L = 7 = 2, and M = 3 almost immediately, a brief calculation (Lemma 2.16) is needed to obtain p. For more complicated sets, such as some of the examples given in Section 5, the calculation of p would be very lengthy. Unfortunately, while the distinction between these two kinds of constant arises clearly in practice, it does not seem easy to make it precise. Indeed, Corollary 6.20 shows that the geometry does in fact determine p: it is not possible to have one nested fractal (a geometric object) with two distinct analytic structures which both satisfy the s y m m e t r y and scale invariance conditions. We have the following general inequalities for the scaling factors.

Proposition 6.30.

Let (F, (r

be a nested fractal with scaling factors L, M, p, T.

Then (6.30)

L>I,

M_>2,

M>L,

T=Mp>L

2.

Proof. L > 1, M > 2 follow from the definition of nested fractals. If0 = diam(V(~ then, as VO) consists of M copies of V (~ each of diameter L -10, by the connectivity axiom we deduce M L - I O > O. Thus M > L. To prove the final inequality in (6.30) we use the same strategy as in Proposition 6.27, but with a better choice of minimizing function. Let 7-/be the set of functions f of the form f ( x ) = O x + a , where x C L~d and O is an orthogonal matrix. Set 7-/,~ = {fly(~), f E 7-/}. Let 0 = sup{C(f, f ) : f E 7-/0}: clearly 8 < ~ . Choose f to attain the supremum, and let g E 7-/ be such that f = gig(o). Then if fx = gig(l) M

p-10 = p-lE(f, f) = h(E)(f, f) < ER(g~,gl) = ~

E(gl o r

or

i=1

However, gl o r is the restriction to V (~ of a function of the form L - l O x + ai, and so E(g o r o r <_ L -20. Hence p-10 < M L - 2 8 , proving (6.30). [] The following comparison theorem provides a technique for bounding p in certain situations.

Proposition

6.31. Let ( F I , { r 1 < i < M1}) be a p.c.s set. Let Fo C 1;1, Mo ~_ M1, and suppose that ( F o , { r ~_ i _~ Mo}) is also a p.c.s set, and that V FI (~ = V Fo (~ " Let (r~k),l < i < Mk) be resistance vectors for k -- 0,1, and suppose that r~~ >_ r~ 1) for 1 < i < Mo. Let A~ be the renormalization map for (Fk, (r Mk /( r i(k)xMk h=l J" I f Ck are non-degenerate Dirichlet forms satisfying Ak(Ck) = p~-lEk, k = 0, 1, then Pl _< P0.

Proof. Since V(~ ) C V(1 ), we have, writing Ri for the replication maps associated with Fi,

Rle(;, f) > Roe(f, f),

f e

c(v(~)).

94

So AI(E) >_ A0(E) for any s 9 D. I f m =

p11s = AI(s

_> A l ( m s

m(s163

_> Ao(ms

then =

mPols

>_poIs , []

which implies that Po _> Pl.

7. D i f f u s i o n s

on p.c.f.s.s, sets.

Let (F, (r be a p.c.f.s.s, set, a n d r, be a resistance vector. We assume that the graph (V (1), E l ) is connected. Suppose that the renormalization map A has a non-degenerate fixed point s (~ = CA, so that A(s (~ : p-1s176 Fixing F, r, and s in this section we will construct a diffusion X on F, as a limit of processes on the graphical approximations V ('0. In Section 2 this was done probabilistically for the Sierpinski gasket, but here we will use Dirichlet form methods, following [Kus2, Ful, Ki2]. D e f i n i t i o n 7.1. For f 9 C(V(n)), set (7.1)

E(n)(f,f):p

n ~ r=lC(~ wEw~

This is the Dirichlet form on V (n) obtained by replication of scaled copies of s (~ where the scaling associated with the map Cw is p'~r~1. These Dirichlet forms have the following nesting property. 7.2. (a) For n > 1, Tr(s ('~-D : s (b) Is 9 C(V(")), and g = flv(--~) then $(")(f, f) _> g("-l)(g,g). (c) s is non-degenerate.

Proposition

Proof. (a) Let f 9 C(V(~)). Then decomposing w 9 W~ into v. i, v 9 W ~ - l , (T.2)

E(n)(f,f)=P n ~

r:l~'~r/1E(~176162176162176162176

vEW~_~ = pn--I

i

r:lE(1)(f~'f=)'

E yEW,,_ ~

where f~ = f o Cv E C(VO)). Now let g E C(V('~-I)). fv[v(o) = g o r = g,. As s (~ is a fixed point of A, (7.3)

inf{s

=g~}

=pinf{Rs176 =

=

Summing over v E W,~-I we deduce therefore

v

If

flv(.-l) = g

then

95 For each v C W n - 1 , let h~ E C(V (U) be chosen to a t t a i n the infimum in (7.3). We wish to define f E C(V('O) such t h a t (7.4)

f or

-- hv,

v C W~-I.

Let v E W n - 1 . We define f(r

= h~(y),

Y E V(1).

We need to check f is well-defined; b u t if v, u are distinct elements of W,~-I and x = r = r t h e n x C V (n-D by L e m m a 5.18, and so y, z E V (~ Therefore f(r

= h~(v) = g~(y) = g ( x ) = f ( r

so t h e definitions of f at x agree. (This is where we use the fact t h a t F is finitely ramified: it allows us to minimize separately over each set of the form V(1)). So E(n)(f, f ) = ~ ( n - 1 ) ( g , g ) , a n d therefore Tr (s = g'('~-U. (b) is evident from (a). (c) We prove this by induction. E (~ is non-degenerate by hypothesis. s is non-degenerate, a n d t h a t E ( " ) ( I , f ) = 0. From (7.2) we have

E(")(f,f) = p ~

r:~E("-~)(f o r

Suppose

f or

vEWx

a n d so f o r is constant for each v E W1. Thus f is constant on each I-complex, a n d as (V (1), E1 ) is connected this implies t h a t f is constant. [] To avoid clumsy n o t a t i o n we will identify functions with their restrictions, so, for example, if f E C(V(~')), a n d m < n, we will write s i n s t e a d of

E(m) (flv(~), flv(~)). D e f i n i t i o n 7.3. Set V (~ = U~=0 V(~). Let U -- { f : V (~) --* R}. Note t h a t the sequence (C(n)(f, f))~=1 is non-decreasing. Define 7)' ----{f e U : supC(n)(f, f ) < (x)}, n

s

= sups

f,g C 7)'.

E ~ is the initial version of the Dirichlet form we are constructing. Lemma

7.4. s is a s y m m e t r i c

Markov

form on 7)~.

Proof. s clearly inherits the properties of symmetry, bilinearity, and positivity from the s If f E 7)', a n d g = (0 V f ) A 1 then s <_s as the E ( ' ) are Markov. So s <s [] W h a t we have done here seems very easy. However, more work is needed to o b t a i n a ' g o o d ' Dirichlet form s which can be associated with a diffusion on F . Note the following scaling result for s

96

Lemma

7.5. For n > 1, f E 7:)',

E'(f,f)= E

(7.5)

flnr~iE'(fOCw,f O•w).

wEW~

Proof. W e

have, for m > n, f E If,

E(m)(f'f)= E

P'~r~'IE(m-n)(f~162176162

wEW~

Letting m --* oo it follows, first that f o r

E Z)~, and then that (7.5) holds.

[]

If H is a set, and f : H ---, ]~, we write (7.6)

O s c ( f , B ) = sup I f ( x ) - f(Y)l,

B C H.

~,y6B

Lemma

7.6. There exists a constant co, depending only on $, such that

f E C(V(~

_< coE(~

O s c ( f , V (~

Proof. Let /~0 = {{x,y} : A,y > 0}. As s is non-degenerate, (V(~ is connected; let N be the maximum distance between points in this graph. Set a = m i n { A , y , { x , y } e E0}. If z, y e Y (~ there exists a chain x = x o , x l , . . . , x n = y connecting x, y with n < N, and therefore,

If(z) - f(y)l 2 _<

[ f ( ~ ) - f(z~_~)l

< n~

lf(xl) - f(x,_1)l2

i=l

< n a - ' ~-~A~,_x,.,[f(zi ) - f ( x i _ l ) l 2 i=1

<_ N a - l ~(~

, f).

[]

Since V 0) consists of M copies of V (~ we deduce a similar result for V (1). C o r o l l a r y 7.7. There exists a constant cl = cl (F, r, A) such that

(7.7)

Osc(f, v (~))<_ cle(1)(f,f),

f 9 ~)'.

Proof. For i 9 W1, f 9 C(V(1)),

Osc(f, v~(~ = Osc(f o r v (~ < c0G(~

r

f or

So, as V (1) is connected,

Os~(f,y(')

< ~

Osc(f, v, (~ i

E c0r i

~ 1 6 2 o~bi) ~ ClC,(1)(f,f),

97

where cl is chosen so t h a t co <_ clpr~ 1 for each i C W1.

[]

C o r o l l a r y 7.8. Let w E W,', and x, y E V(1). Then

Osc(f,v(1)) < el ~,p

E (f,f),

Proof. We have O s c ( f , V O ) ) = O s c ( / o r g (1) < 8 ' , a n d by (7.5)

E'(f o r

f o

r

f 9

_< c1s

o C w , f o Cw). Since

< r~p-,'E'(f, f),

the result is i m m e d i a t e .

[]

D e f i n i t i o n 7.9. We will call the fixed point E (~ a regular fixed point if

ri < p

(7.8)

for

1
M.

P r o p o s i t i o n 6.27 implies t h a t (7.8) holds for any s 9 { 1 , . . . , M } such t h a t lr(k) 9 V (~ In p a r t i c u l a r therefore, for nested fractals, where every point in V (~ is of this form and r is constant, any fixed point is regular. It is not h a r d to p r o d u c e examples of non-regular fixed points. Consider the LindstrOm snowflake, b u t with ri = 1, 1 < i < 6, r7 = r > 1. Writing p(r) for the resistance scale factor, we have (by Proposition 6.31) t h a t p(r) is increasing in r. However, also by P r o p o s i t i o n 6.31, p(r) < Po, where P0 is the resistance scale factor of t h e nested fractal o b t a i n e d just from r 1 < i < 6. So if we choose r7 > P0, t h e n as r7 > P0 ~ p(rT), we have an example of an affine nested fractal with a non-regular fixed point. From now on we take s to be a regular fixed point. (See [Kum3] for the general situation). W r i t e 7 = m a x / r i / p < 1. For x, y 9 F , set w(x,y) to be the longest word w such t h a t x, y 9 F ~ . Proposition

7.10.

(Sobolev inequality). Let f 9 T~'. Then if s

is a regu/ar

fixed point (7.8)

i f ( x ) _ f(y)]2 ~ c2r~(,,y)p-l~(z,y)ls

x,y 9 V (~176

Proof. Let x, y 9 Y(,'), let w = w(z,y) and let Iwl = m. We prove (7.8) by a s t a n d a r d kind of chaining argument, similar to those used in continuity results such as Kolmogorov's lemma. (But this argument is deterministic and easier). We m a y assume n > m. Let u 9 W," be an extension of w, such t h a t x 9 V(~ such a u certainly exists, as x 9 V~( ~ Write uk = ulk for m < k < n. Now choose a sequence zk, m < k < n such t h a t z," = x, and zk 9 V~(~ for k < m < n k 9 { m , . . . , n - 1} we have zk, zk+l 9 V(~ ). So

(7.9)

If(z,,)- f(z.~)l < ~

lf(zk+1) - f(zk)l

n--1

< Z le,=m

(e:~

I/"

1. For each

98

n--1 Tuk ^_k+m~l/2 k=m As s is a regular fixed point, 7 = m a x i ri/p < 1, so the final sum in (7.9) is b o u n d e d CO by ( ~ k = m 7 k - m ) U2 = c3 < co. Thus we have

If(x) - f(zm)l ~ < c l c : ~ p - " E ' ( f , f), a n d as a similar b o u n d holds for If(y) - f(zm)[ 2, this proves (7.8).

[]

We have not so far needed a measure on F . However, to define a Dirichlet form we need some L 2 space in which the domain of ~ is closed. Let # be a p r o b a b i l i t y measure on (F, B(F)) which charges every set of the form F ~ , w E Wn. L a t e r we will take # to be the Bernouilli measure P0 associated with a vector of weights 8 E (0, ~ ) M , b u t for now any measure satisfying the condition above will suffice. As # ( F ) = 1, C(F) C L 2 ( F , # ) . Set

= {f e C ( F ) :

e(f,f) Proposition

=

flv(~) 9 :D'}

e'(flv(~),flv(~)),

f 9 :V.

7.11. (~, :D) is a dosed symmetric form on LZ(F, p).

Proof. Note first t h a t the condition on p implies t h a t if f , g 9 l) then [[f - g [ [ 2 = 0 implies t h a t f = g. We need to prove t h a t V is complete in the norm ][f[[~, -e ( f , f ) + Ilfll]. So suppose (fn) is Cauchy in [[. lie,- Since ( f , ) is Cauchy in H" [Is, passing to a subsequence there exists f 9 L 2 ( F , # ) such t h a t f,~ ~ f #-a.e. F i x x0 9 F such t h a t fn(xo) --~ f(x). T h e n since fn - f m is continuous, (7.8) extends to an e s t i m a t e on the whole of F and so If,,(x) - f,~(x)[ _< l(f,, - f,,,)(x) - (f,, - f,,,)(x0)l + I(f,, - fm)(x0)l I12 ,-,:. <( C 2

C(.]'n -- fro,

fn

-- fro) I/2

"~ If.(x0)

-

fm(x0)[.

So (f~) is Cauchy in the uniform norm, and thus there exists f 9 C(F) such t h a t f~ --* f tmiformly. Let n _> 1. T h e n as s is a finite sum, s

=

lim s m--coo

_< lim s u p e ( f m , f m ) m---*oo

_< sup Ill-lie, < ~ . m

Hence E('~)(f, f ) is bounded, so f 9 l). Finally, by a similar calculation, for any N>I, E(N)(f~ -- f, fn -- f) (_ limoo E(f,~ -- fm, f,~ -- f.',). So E(f,~ - f, fn - f) --~ 0 as n ---* oo, and thus Ill - f.II~,

--+ o.

[]

To show t h a t (C, Z~) is a Dirichlet form, it remains to show t h a t 2) is dense in

L2(F, p). We do this by s t u d y i n g the harmonic extension of a function.

99 D e f i n i t i o n 7.12. Let f e C(V(')). Recall that C('~)(f,f) -~ inf{E("+l)(g,9) : giv(-) -- f}- Let H n + i f E C(V (~+D) be the (unique, as E (n+l)) is non-degenerate) function which attains the infimum. For x G V (~) set

H~f(x) = lim H , ~ H , ~ - I . . . H , ~ + l f ( X ) ; note that (as H ~ + l f -- f on V (~)) this limit is ultimately constant. 7.13. Let g be a regular fixed point. (a) H,~f has a continuous extension to a function H~f C 7) M C(F), which satisfies

Proposition

~(H,J, Hnf) = E(n)(f, f).

(b)

f, a c C(F) E(H~f,g) -- g('~)(f,g).

(7.10)

Proof. From the definition of H,~+I, ~(n+i)(Hn+lf, H ~ + l f ) -- s

Thus

E(m)(H,~f,H~f) = C(~)(f, f ) for any m, so that H ~ f E 7)~ and

E(Hr, f, Hnf) = C(n)(f,f),

f E C(V(")).

If w E Win, and x, y E V (~) M F,o then by Proposition 7.10 (7.11)

] H u f ( x ) - H , I ( v ) ] 2 <_ c2rwp-mE(n)(f, f).

Since r~p -'~ < Tin, (7.11) implies that O s c ( H ~ f , U (~) f) Fw) converges to 0 as Iwl = m -~ co. Thus H,~f has a continuous extension H~f, and H~f E 23 since (b) Note that, by polarization, we have C (n+l) (Hn+a f, Hn+l 9) --- E (n) (f, g). Since g ( n + l ) ( H n + l f , h) = 0 for any h such that hiv(. ) = 0, it follows that E(n+l)(H,~+lf, g ) = E(n)(f,9). Iterating, we obtain (7.10). Theorem

[]

7.14. (C, 7)) is an irreducible, regular, 1ocM Dirichlet form on L 2 (F, #).

Proof. Let f E C(F). Since for any n > 1, w C Wn we have i n f f _< Hnf(x) < sup f, F~ F~

x C F~,

it follows that H~f --* f uniformly. As Hnf E 73, we deduce that 7) is dense in C(F) in the uniform norm. Hence also 7) is dense in L2(F,#). As (4.5) is immediate, we deduce that 7) is a regular Dirichlet form. If s = 0 then E(~)(f,f) = 0 for each n. Since s is irreducible, fly(,) is constant for each n. As f is continuous, f is therefore constant. Thus E is irreducible.

100

To prove that E is local, let f, g be functions in 79 with disjoint closed supports,

S/, Sg say. If $(=)(f,g) # 0 then one of the terms in the sum (7.1) must be non-zero, so there exists wn 9 W,~, and points xn 9 S I N V(~ yn 9 Sg A V(w~ Passing to a subsequence, there exists z such that xn ---+z, Yn ---* z, and as therefore z 9 S I A Sg, this is a contradiction.

[]

By Theorem 4.8 there exists a continuous #-symmetric Hunt process (Xt, t > 0, F =, x 9 F ) associated with (E, 79) and L 2 (F, #). R e m a r k 7.15. Note that we have constructed a process X = X (t') for each Radon measure # on F. So, at first sight, the construction given here has built much more than the probabilistic construction outlined in Section 2. But this added generality is to a large extent an illusion: Theorem 4.17 implies that these processes can all be obtained from each other by time-change. On the other hand the regularity of ($, 79) was established without much pain, and here the advantage of the Dirichlet form approach can be seen: all the probabilistic approaches to the Markov property are quite cumbersome. The general probabilistic construction, such as given in ILl] for example, encounters another obstacle which the Dirichlet form construction avoids. As well as finding a decimation invariant set of transition probabilities, it also appears necessary (see e.g. [L1, Chapter VI ]) to find associated transition times. It is not clear to me why these estimates appear essential in probabilistic approaches, while they do not seem to be needed at all in the construction above. We collect together a number of properties of (E, ~). Proposition

7.16. (a) For each n > 0

(7.12)

E(/'g)=

E P%'~lE(f~162176162 wGW~

(b) Fo~, / 9 79, (7.13) (7.14)

(7.15)

If(x) -/(y)l 2

<

Clr~p-nE(f,f)

f f2d# <

c2$(f,f)+

is x,y 9 Fw, w 9

Wn

(f fd,)

I ( z ) 2 < 2 fj f2d# + 2ClE(f,f),

x C F.

Proof. (a) is immediate from Lemma 7.5, while (b) follows from Proposition 7.10 and the continuity of f. Taking n = 0 in (7.13) we deduce that

(f(x) -- f(y))2 < ev~(f,f),

f 9 79.

101

So as # ( F ) = 1,

/ / ~,E(s,S).(dx).(d~,)=e,~(S,S)

=2//2rig-2 (/)' f d#

,

proving (7.14). Since f ( x ) 2 <_ 2 f ( y ) 2 + 2[f(x) - f(y)]~ we have from (7.13) that

f(x) 2 =

/

< 2f

f(x)U#(dY)

f(y)'~(dy)+ 2~ f ~(f, 1)~(dy), []

which proves (7.15). We need to examine further the resistance metric introduced in Section 4. D e f i n i t i o n 7.17. Let R(x, x) = 0, and for x ~ y set

R(x, y ) - I : inf {g(f, f ) : f(x) = O, f(y) = 1, f 6 7)}. Note t h a t

f(y)l 2

R(x,y) = sup( If(x)yff,?~ (

(7.16)

: fCV,

f non constantS.

P r o p o s i t i o n 7.18. (a) I~ = # y then 0 < R(=, y) <_ cl < o~. (b) I f w e W,~ then

(7.17)

R(x,y) < Clrwp-",

x,y e Fw.

(c) For f ~ 79 (7.18)

If(=) - f(y)l 2 -< R(=, y)E(I, I).

((t) R is a metric on F, and the topology induced by R is equa/ to the originM topology on F. Proof. Let x, y be distinct points in P. As 73 is dense in C ( F ) , there exists f E 7) with f ( x ) >_ 1, f(y) <_ O. Since g is irreducible, s > 0, and so by (7.16) R(x, y) > 0. (7.17) is immediate from Proposition 7.16, proving (b). Taking n = 0, and w to he the e m p t y word in (7.17) we deduce R(x,y) _< cl for any x, y E F, completing the proof of (a). (c) is immediate from (7.16). (d) R is clearly symmetric. The triangle inequality for R is proved exactly as in Proposition 4.25, by considering the trace of s on the set {x, y, z}.

102

It remains to show that the topologies induced by R and d (the original metric on F ) are the same. Let R(xn, x) ---. O. If e > 0, there exists f E 7:) with f ( x ) = 1 and s u p p ( f ) C Bd(X,e). By (7.16) R(x,y) >_ s > 0 for any y e Bd(X,e) c. So xn E Bd(X, ~) for all sufficiently large n, and hence d(x,~, x) --* O. If d(x,~, x) ~ 0 then writing

Nm(x)=U{F~o:wEW,,,

,

xEF~,}

we have by L e m m a 5.12 that xn E N,,(x) for all sufficiently large n. However if 7 = maxir~/p < 1 we have by, (7.17), R(x,y) < c1~/m for y E Win(x). Thus

R ( x , , z) ~ O.

[]

R e m a r k 7.19. The resistance metric R on F is quite well adapted to the study of the diffusion X on F. Note however that R(x, y) is obtained by summing (in a certain sense) the resistance of all paths from z to y. So it is not surprising that R is not a geodesic metric. (Unless F is a tree). Also, R is not a geometrically natural metric on F. For example, on the Sierpinski gasket, since r~ = 1, and p = 5/3, we have that if x, y are neighbours in (V (~), En) then

n(x, y) • (3/5?. However, for general p.c.f.s.s, sets it is not easy to define a metric which is well-adapted to the self-similax structure. (And, if one imposes strict conditions of exact self-similarity, it is not possible in general - see the examples in [Ki6]). So, for these general sets the resistance metric plays an extremely useful role. The next section contains some additional results on R. It is also worth remarking that the balls BR(x, r) = {y : R(x, y) < r} need not in general be connected. For example, consider the wire network corresponding to the graph consisting of two points x, y, connected by n wires each of conductivity 1. Let z be the midpoint of one of the wires. Then R(x, y) = I/n, while the eonductivities in the network { x , y , z } axe given by C(x,z) = C(z,y) = 2, C(x,y) = n - 1. So, after some easy calculations,

R(x,z)-

n+l 4n-1

1 >~"

Soifn=4, R ( x , y ) = x while R(x, z) = ~. Hence i f ~1 < r < l t h e b a l l B R ( x , r ) is not connected. (In fact, y is an isolated point of BR(x, 88 = { x ' : d(x,x') < 88 (Are the balls BR(x, r) in the Sierpinski gasket connected? I do not know). Recall the notation $~(f, g) = s of X. Since by (4.8) we have

g ) + a ( ] , g). Let (U~, a > 0) be the resolvent

E~(vof, g) = (f,g), if U~ has a density u~(x,y) with respect to #, then a formal calculation suggests that E~(u~(~,.),9)

= E~(u~=,9)

= (~=,g) = g(~).

We can use this to obtain the existence and continuity of the resolvent density u~. (See [FOT, p. 73]).

103

T h e o r e m 7.20.

(a) For each x 9 F there exists u s 9

(7.19)

for all

Ea(u~, f ) = f ( x )

suchthat

f 9

(b) Writing u~,(x, y) = u~(y), we have

u~(=,y) = ,~.(y,=)

:or a11 =,y e F.

(c) us(.,-) is continuous on F • F and in particular lu~(=,y) - ~ ( = , y ' ) l 2 < R(y,y')~o(=,=).

(7.20)

(d) ~.(=,y) is the resolvent density ~or X: ~or f e C ( f ) , E~

:

e-~tf(Xt)dt = U~f(z) =

/

u~(z,y)f(y)~(dy).

(e) There exists c2(a) such that (7.21)

~ ( ~ , y ) <_ c2(~),

=,y 9 F.

Proof. (a) The existence of u~ is given by a standard argument with reproducing kernel Hilbert spaces. Let x 9 F , and for f 9 7? let r = f ( x ) . Then by (7.15)

lr

= ]/(x)l2 < 211fll2 +

2ciE(f,f) < c~s

f),

where co = 2 m a x ( c l , a - 1 ) . Thus r is a bounded linear functional on the Hilbert space (7), II lifo), and so there exists a u~ 9 7) such that r

----,~',~(u~,f ) -= f(x),

f 9 7).

(b) This is immediate from (a) and the symmetry of E:

~(=)

E . ( u .Z, ~ . )y

= E.(~,~X)

= ~(y).

(c) As u~ 9 7), u~,(x,x) < oo. Since E(u~,u~) = u~,(x,x) < c~, the estimate (7.20) follows from (7.18). It follows immediately that u is jointly continuous on F • F. (d) This follows from (7.19) and linearity. For a measure u on F set

: 9 C(F) As us is uniformly continuous on F • F, we can choose u~---~# so that V ~ . f --~ V f uniformly, and vn are atomic with a finite number of atoms. Write V,~ = Vv., V = Vv. Since by (7.19) E , ( V J , g) = ~ - ~ ( { z } ) / ( z ) E o ( u : , g ) x

104

we have

Ea(Vnf - Vmf , Vnf - Vmf) =

Thus g~(V,J

- V~LV.I

- Vmf)

--, 0 as m, n ~

~,

and so, as g is closed, we .,J o

So

deduce that

$ a ( Y f, g) : Ea(U~f ,g) for all g, and hence Y f = V ~ f . (e) As R(y, y') < Cl for y, y' C F, we have from (7.20) that

(7.22)

u~(x,y) > ~ ( x , x ) - (c~,~o(x, ~)) 1/~.

Since I u,(:e, y)#(dy) = o~-1, integrating (7.22) we obtain

~o(x,x) < (cl~o(x,~)) 's~ + ~-~, and this implies that u,~(x, x) < c2(a), where c(a) depends only on a and el. Using (7.20) again we obtain (7.21). [] T h e o r e m 7.21. (a) For each x C F, x is regular for {x}. (b ) X has a jointly continuous local time ( L ~ , x e F, t > O) such that for all bounded

measurable f l(x.)es

=

~)Lt.(d~),

a.s.

Pro@ These follow from the estimates on the resolvent density u~. As u~ is bounded and continuous, we have that x is regular for {x}. Thus X has jointly measurable local times (L~,x E F , t > 0). Since X is a symmetric Markov process, by Theorem 8.6 of [MR], L~ is jointly continuous in (x, t) if and only if the Gaussian process Yx, x E F with covariance function given by EYaYb = ul(a,b)~

a,b E F

is continuous. Necessary and sufficient conditions for continuity of Gaussian processes are known (see [Tall), but here a simple sufficient condition in terms of metric entropy is enough. We have

E(Ya - yb)2 = ul(a,a) - 2ul(a,b) + ul(b,b) < clR(a,b) U2. Set r(a, b) = R(a, b) 1/2 : r is a metric on F. Write N~(e) for the smallest number of sets of r-diameter e needed to cover F. By (7.17) we have R(a, b) < c~/'~ if a, b e Fw and w E W~. So N~(ct~/n/2) < #W,~ = M ~, and it follows that N~(c) < e2c - ~ ,

where fl = 2 log M~ log ~-1. So

fo + (lOg~V~(~))~/~a~ < ~,

105

and thus by [Du, Thm. 2.1] Y is continuous.

[]

We can use the continuity of the local time of X to give a simple proof that X is the limit of a natural sequence of approximating continuous time Markov chains. For simplicity we take # to be a Bernouilli measure of the form It -- It0, where 0i > 0. Let It,, be the measure on V ('0 given in (5.21). Set

A'~ = ./~ L~It,~(dx), "r: = inf{s : A~ > t}, x;' = x v.

(a) (X?,t >_ 0 , I ? ' , x 6 V (~)) is the symmetric Markov process associated with $ ('~) and L2(V ("), It,). Theorem

7.22.

(b) X~ --+ Xt a.s. and uniformly on compacts. Proof. (a) By Theorem 7.21(a) points are non-polar for X. So by the trace theorem (Theorem 4.17) X n is the Markov process associated with the trace of $ on L 2 ( V ( ' 0 , I t , ) . But for f 6 :D, by the definition of s Tr

(elv ( - ) ) ( f , :) =

e(n)

(flv(-),fl,:~-)).

(b) As F is compact, for each T > 0, (LT, 0 < t < T, x 6 F ) is uniformly continuous. So, using (5.22), if T2 < T1 < T then A t ~ t uniformly in [0,T1], and so T~ --+ t uniformly on [0, T2]. As X is continuous, X~ ~ X uniformly in [0,T2]. [] R e m a r k 7.23. As in Example 4.21, it is easy to describe the generator Ln of X ~. Let a('O(x, y), x, y G V('*) be the conductivity matrix such that 1

E(")(f,:) = ~ E a(n)(~,V)(f(~)- f(V)) ~ z,y

Then by (7.1) we have (7.23)

a('q(x,y) ---

E

l(,,yev!0))p n r~--1A ( r

--1 ( x ) , r w --1 (y)) ,

w6W~

where A is such that E (~ -- CA, and A(x,y) = A=y. Then for f E L2(V('q,#,~), (7.24)

Lnf(x) -- # n ( { x } ) - '

E

a(n)(x'Y)( f ( y ) - f(x)).

yEV(~)

Of course T h e o r e m 7.22 implies that if (Y'~) is a sequence of continuous time Markov chains, with generators given by (7.24), then Y'~ TM ~X in D([0, oo), F).

106 8. Transition D e n s i t y Estimates. In this section we fix a connected p.c.f.s.s, set (F, ( r a resistance vector ri, and a non-degenerate regular fixed point ~A of the renormalization m a p A. Let P = P0 be a measure on F , and let X -- ( X t , t > O, IP~,z C F ) be the diffusion process c o n s t r u c t e d in Section 7. We investigate the transition densities of the process X : initially in fairly great generality, but as the section proceeds, I will restrict t h e class of fractals. We begin by fixing the vector 0 which assigns mass to the 1-complexes r in a fashion which relates # e ( r with ri. Let/3i = r l p - l : by (7.8) we have

fli < l,

(8.1)

l < i < M.

Let a > 0 be t h e unique positive real such t h a t M

(8.2)

ZZ?

-- 1.

i=I

Set

(8.3)

Oi =

fiT,

1<

i < M,

and let # = P0 be the associated Bernouilli t y p e measure on F . Write fl+ = maxi/3i, /3_ = m i n i f l i : we have 0 < fl_ < fl~
w.W=(w.v,v 9149 Lemma

1
8.1. (a) F o r A > 0 let

Wx = {w 9 W~ :/3~ _< ~,/~,~ > ~}. T h e n the sets {w 9 W , w 9 Wx} are disjoint, a n d

U wEWx

w.W=W.

Iwl}.

107

(5) Fo~ S 9 L ~(F, ~),

iSd#= ~ 8~Sf~d# wEWx

e(s, s) -- Z / 3 2 E ( f w , s~) wEWx

Proof. (a) Suppose w, w ~ E W~ and v E ( w . W) N (w I. W). T h e n there exist u, u t E W such t h a t v = w - u = w ~. u ~. So one of w, w ~ (say w) is an ancestor of the other. But if/~w _< A, fl~w > A t h e n as/3i < 1 we can only h a v e / 3 ~ , > A if w ~ = w. So if w # w ~, w 9W a n d w I 9W are disjoint. Let v E W. Then/3,1~ ~ 0 as n --~ cx~. So there exists m such that vtm e W~, a n d t h e n v e (vtm) 9W, completing the proof of (a). (b) This follows in a straightforward fashion from the decompositions given in (7.12) a n d L e m m a 5.28. [] Note t h a t / 3 _ > 0 and that (8.4)

f l A < / 3 ~ < A,

(/3_)~A ~ _ ~

< As,

w9

D e f i n i t i o n 8.2. The spectral dimension of F is defined by

ds = d~(F, EA) = 2(~/(1 + or). Theorem

(8Z)

8.3. For f 9 79,

llSll~?-~/~.<_c, (E(s,s)+ IlStl~)ItSll~/~:.

Proof. It is sufficient to consider the case f non-negative, so let f E 79 with jr > 0. Let 0 < A < 1: by L e m m a 8.1, (7.14) and (8.4) we have (8.6)

IlSll~= ~ 8,~S Ssdp wEW,~

< ~ 8,o elg(f~,, f~,) +

f~ d#

(i

w

_< eaA~+I E / 3 ~

(fw,f~,)+c2a

< c3a'~+~g($, f ) + c4,~-~

o~, J f,~ d~

fwd#

108 The final line of (8.6) is minimized if we take :~'+~ ~llfll~ then ~ < 1 and so we obtain from (8.6) that

(8.7)

llfl12~. _ cs

--

c~ IIflI~/C(f, f). If E(.f, .f)

>_

f)<~l(2,~+,)(llfll~)(<,+:)I(~<'+~),

which implies t h a t t h a t

(8.8)

11.:112~ +4/'~. < cE(/,f)ll:ll~ ia.

if E(:,:)

if

E(y,f) _> r

_< c511:112~then by (7.14)

Ilfll~ -< .: (c(f, f)

+

ilfll:,) <_ cll:ll~,

and so

2

41d.

IIf1122+*/<'. -< cll:ll211flll

(8.9)

if e(:,:)

_< c511:112~. []

Combining (8.8) and (8.9) we obtain (8.5). F r o m the results in Section 4 we then deduce Theorem

8.4. X has a transition density p(t, x, y) which satisfies

p(t, x, y) < Cl t-d~

(8.10)

(8.11)

0 < t < 1,

I p ( t , x , y ) - - p ( t , x , y / ) l 2 ~ c2t-l-d'/2.R(y,y/),

x, y E F, 0 ~ t ~ 1~

:c,y,y I E -P.

Proof. By P r o p o s i t i o n 4.14 X has a jointly measurable transition density, a n d by Corollary 4.15 we have for x, y E F , 0 < t < 1,

p ( t , x , y ) < ct-d'/2ea < clt -d~ By (4.17) the function qt,= = p(t,x,-) satisfies E(qt,=,qt,=) _< ct - 1 - a ~ qt,= E T~ a n d is continuous. Further, by Proposition 7.18

Ip(t,x,y) - p ( t , x , y ' ) l ~ <_ cR(y,y')t -d'/~-~,

and so

x , y , y ' 9 F.

Thus p(t,-, .) is j o i n t l y H61der continuous in the metric R on F .

[]

R e m a r k s 8.5. 1. As a > 0, we have 0 < d~ = 2a(1 + a) -1 < 2. 2. T h e e s t i m a t e (8.10) is good if t E (0, 1] and x close to y. It is poor if t is small c o m p a r e d with R(x, y), a n d in this case we can o b t a i n a b e t t e r estimate by chaining, as was done for fractional diffusions in Section 3. For this we need some additional p r o p e r t i e s of the resistance metric. Lemma

8.6. If v, w E Wx and v ~ w then F~ M F~ = V~(~ M V (~

Proof. This follows easily from the corresponding p r o p e r t y for V4n. Let v, w E W~, w i t h I v ] = m < _ ]w I = n , v ~ w . LetxEF.MF~. Setw'=wim;thenasFw CFw,, x E F v f q F , , , and so by L e m m a 5.17(a) x E V( ~ 1 7 6 Further, as x E F , there exists v t E W,, such t h a t v'lm = v, and x C F , , . Then x E F . , M F ~ = l/-(~ . , , :q V~(~ .

So ~ c v~ ~ n v~ ~

[]

109

D e f i n i t i o n 8.7. Set

Let G~ = (V~(~

LI

~

v o)

be the graph with vertex set V(0), and edge set EA such that

{x,y} is an edge if and only if x, y 9 V(~ for some w 9 W~. For A C F set N A ( A ) -- U { F ~ : w 9 W~, F~ M A ~ O}, /V~(x) = Y~ (Y~({x})). As we will see, N~(x) is a neighbourhood of x with a structure which is well adapted to the geometry of F in the metric R. We write N~(y) = N~ ({y}). Lemma

8.8. (a) I f x, y 9 V (~ and x ~ y then R ( x , y ) >_ c1)~.

(b) g { = , v } 9 E~ then n(=,V) < c2~. Proof. (b) is immediate from the definition of WA and Proposition 7.18(b). For (a), note first that if x 9 F then by Proposition 5.21 x can belong to at most M1 = M # ( P ) n-complexes, for any n. So there are at most M1 distinct elements w 9 WA such that x 9 F~. As V (~ is a finite set, and E(A~ is non-degenerate, there exists c3, c4 > 0 such that, (8.12)

c4 >_ R ( x , V (~ - {x}) > e3,

x 9 V (~

(Recall that this resistance is, by the construction of s the same in (F, E) as in (V(~176 Now fix x E V(~ I f w 9 W~, and x 9 V(~ let x' = r and g~ be the function on F such that g~(x') = 1, g~(y) = O, g 9 Y (~ - {x'}, and :

R

v(~

-

_>c ,

Define g " on F~, by g " -- g,, o r 1, and extend g" to F by setting g " = 0 on F - F ~ , . Now l e t g ~ = 0 i f x ~ V (~149 g =

E 'gv.

vEWx

Then g(x) = 1, g(y) = 0 if y 9 V(~

y ~ x, and

wEW~

--- E / S w

--1

l(~cF~)E(gw,gw) <_ C5A-1/1 .

110

Hence if y # x, y E V(~

we have

n(x,y) -1 <_ E(g,a) <_ A-Imlc[ 1, so that R(x,y) > c6A.

[]

R e m a r k . For x E V(~ the function g constructed above is zero outside Nx({x}). So we also have (8.13)

R(x,y) >_c6A,

x E V(~

y E Nn({x}) ~.

P r o p o s i t i o n 8.9. There exist constants c~ such that/'or x E F, A > 0, (8.14)

(8.15)

(8.16) (8.17)

BR(X,clA) C Ar~(x) C BR(X, c2A), c3~ ~ < ~ ( B R ( x , A ) ) < c,A ~

chA < n(~, ~(x)~) < ~ , c~A < n(~,B.(~,A) ~) < csA.

Proof. Let x C F. I f y C Nn({x}) then by (7.17), R(x,y) <_ cA. So i f z C Nn(x), since there exists y E Nn({x}) with z E Nn({y}), R(x,z) < c'A, proving the right hand inclusion in (8.14). If x E V~(~ then by (8.13), if c9 = c8.7.6, BR(x, egA) C NA(x). Now let x ~ V~(~ so that there exists a unique w C Wa with x E F,o. For each y E V(~ let fv(') be the function constructed in Lemma 8.8, which satisfies fy(y) = 1, fy = 0 outside N),(y), fy(z) -- 0 for each z e V(~ - {y}, and E(]y, fy) < cl0 A-1. Let f = ~ y fy: then ] ( y ) = 1 for each y e V(~ g=lF~

So if

+lEaf,

E(g,g) < ~ ( L f ) < #(Y~(~ -1 _< ~ 1 ~ - 1 . As g(=) = 1, and g(~) = 0 for ~ ~t N~(x), we have for z r N~(x) that R(x,z) -1 <_$(g,g) <_cnA -1. So BR(x, cllA) C N~,(x). This proves (8.14), and also that R(x,~rx(x) ~) > c51A. The remaining assertions now follow fairly easily. For w E W~ we have cl2A ~ < #(F~,) < c13A~. As N~(x) contains at least one A-complex, and at most M 2 # ( P ) 2 A-complexes, we have ,(N~(~)) • ~, and using (8.14) this implies (8.15). If A C B then it is clear that R(x,A) > R(x,B). So (provided)~ is small enough) if x ~ F we can find a chain x, yl, y2, y3 where y~ ~ V(~ {y~,y~+l} is an edge in E~, Y3 ~ N~(x), and x and y are in the same A-complex. Then R(x, y3) <_cA by (7.17), and so, using Lemma 8.8(b) we have R(x,y3) <_c'A. Thus R(x, Ar~,(x)~) < R(x, y3) <_c'A proving the right hand side of (8.16): the left hand side was proved above. (8.17) follows easily from (8.14) and (8.16). []

111 C o r o l l a r y 8.10. In the metric R, the Hausdorff dimension o f F is a, and further

0 < n~(F) <

o0.

Proof. This is immediate from Corollary 2.8 and (8.15).

[]

P r o p o s i t i o n 8.11. For x E F , r > 0 set ~-(x, r) = TBR(x:)o. Then

(8.18)

e : ~+x < Ex~(z,r) _< c : ~+1,

z 9 F,

r > 0.

Proof. Let B = Ba(x, r). T h e n by Theorem 4.25 and the estimates (8.15) and (8.17) Z~T(x,r) <_ # ( B ) R ( x , B c) <_ car ~+1, which proves the upper bound in (8.18). Let ( X f f , t _> 0) be the process X killed at T = TBo, and let g(x,y) be the Greens' function for X B. In view of Theorem 7.19, we can write

g(x,y) = E~L~,

x,y 9 F.

Then if f(y) = g(x, y)/g(x, x), f 9 7) and by the reproducing kernel property of g we have E(f, f) = g(x, x)-2E (g(x, .), g(x, .)) = g(x, x) -1 , and as in Theorem 4.25 g(x, x) = R(x, B ~) >_ car. By (7.18) If(x) - f(y)l 2 < R(z,y)C(f,f) <_ R(z,y)(c,r)-~ < if R(x, y) <_ 88

1

Thus f(y) >_ 89on BR(x, 88 r), and hence E ~ = f o g(x, y),(dv)

>

(B.

88

>

proving (8.18).

[]

We have a spectral decomposition of p(t, x, y). Write (f, g) = fF fgd#. T h e o r e m 8.12. There exist [unctions qo~ 9 7), Ai >_ O, i >_ O, such that (qoi, ~i) = 1,

O-- )~o < A1 <_...,and $(Wi, f) = Ai(~oi, f),

f 9 7).

The transition density p(t, x, y) of X satisfies (8.19)

p(t, x, y) = ~

e-X't~,(x)~,(y),

i=0

where the sum in (8.19) converges uniformly and absolutely. So p is jointly continuous in (t, x, y).

Proof. This follows from Mercer's Theorem, as in [DaS]. Note that ~o0 = 1 as s is irreducible and # ( F ) = 1. [] The following is an immediate consequence of (8.19)

112

C o r o l l a r y 8.13.

(a) F o r x , y E F ,

t >0,

p(t, ~, y)2 < p(t, x, ~)p(t, y, y). (b) For each x, y E F lim p(t, x, y) -- 1. t -"'* O 0

Lemma (8.20)

8.14.

p ( t , x , y ) > co t-d~

0 < t < 1,

R ( x , y ) < clt UO+~).

Proof. We begin with the case x --- y. From Proposition 8.11 and L e m m a 3.16 we deduce that there exists c2 > 0 such that ~ = ( T ( x , r ) _< t) < (1 -- 2C2) + c3tr - ~ Choose c4 > 0 such that c3tro a-1 -- c 2 if ro = c4tU(l+~). Then ~=(Xt e B n ( x , ro)) _> ~ ( T ( x , r0) _< t) _> C~. SO using Cauchy-Schwarz and the symmetry of p, and writing B = BR(X, ro), 0 < c~<_ ( / B P ( t ' x ' Y ) # ( d Y ) 2 )

<<

f.( .... ).(dy) .(B)) p(2t, ,x)

< cst~/(l+~)p(2t, x, x). Replacing t by t/2 we have

p ( t , x , x ) >_ Cot-d'/2. Fix t,x, and write q(y) -= p(t, x,y). By (4.16) and (8.5) $(q,q) < c~t -1-~'/2 for t <_ 1, so using (7.18), if R ( x , y ) <_ cTt 1/(1+~) then, as 1 + ds/2 = (1 + 2 a ) / ( 1 + a),

q(y) > q(~) - I q ( ~ ) - q(y)] ~_ COt-a/(l+cO (R(x, y)E(q, q))1/2 - -

= t-~l(l+~)(c

Choosing c7 suitably gives (8.20).

o _ (c7c~)~12).

[]

We can at this point employ the chaining arguments used in Theorem 3.11 to extend these bounds to give upper and lower bounds on p(t,x,y). However, as R is not in general a geodesic metric, the bounds will not be of the form given in T h e o r e m 3.11. T h e general case is given in a paper of Hambly and Kumagai [HK2], but since the proof of Theorem 3.11 does not use the geodesic property for the upper bound we do obtain:

113

Theorem

8.15.

The transition density p(t, x, y) satisfies

p(t,x,y) <_ c l t - a / ( l + " ) e x p ( - c 2 ( R ( x , y ) l + ( ~ / t ) l / ~ ) .

(8.21)

N o t e . The power 1/a in the exponent is not in general best possible. T h e o r e m 8.16. Suppose that there exists a metric p on F with the midpoint property such that for some 8 > 0

clp(x,y) e <_ R(x,y) < c2p(x,y) e x,y E F.

(8.22)

Then it" d~ -- 8(1 + a), df = aS, (F, p, It) is a fractional metric space of dimension dl, and X is a fractional diffusion with indices di, dw. Proof. Since Bp(x, (r/c~) 6) C Ba(x, r) C B;(x, (r/cl)e), it is immediate from (8.15) that (F, p) is a F M S ( d l ) . Write ~-p(x, r) = inf {t: Xt ~ Bp(x, r)}. Then from (8.18) and (8.22) cr~ <_E~'rp(x, r) < c2r~ So, by (8.10) and (8.20), X satisfies the hypotheses of Theorem 3.11, and so X is a

FD(df,dw). Remark.

[] Note that in this case the estimate (7.20) on the Hhlder continuity of

ux(x, y) implies that 1

(8.23)

<

while by T h e o r e m 3.40 we have (8.24)

[u~(x,y) - u~(x',y)[ < cp(x,x') e.

The difference is that (8.23) used only the fact that ux(., y) 9 :D, while the proof of (8.24) used the fact that it is the A-potential density.

Diffusions on nested f_ractals. We conclude by treating briefly the case of nested fractals. Most of the necessary work has already been done. Let (F, (r he a nested fractal, with length, mass, resistance and shortest path scaling factors L, M, p, 7. Recall that in this context we take r~ = 1, 04 = l / M , 1 < i < M, and # -- #e for the measure associated with 8. Write d = dR for the geodesic metric on F defined in Section 5. Lemma (8.24)

8.17. Set 8 -- l o g p / l o g % Then

c,d(x,y) ~ <_ R(x,y) < c2d(x,y) ~

x,y E F.

Proof. Let A E (0, 1). Since all the r, are equal, /Y~(x) is a union of n-complexes, where p - n < A ~ p - ~ + l . So by Theorem 5.43 and Proposition 8.8, since ~/-'~ = (8.26)

y E /Y~(x) implies that R(x,y) <_ clA, and d(x,y) < e2A e,

114

(8.27)

y ([ Nx(x) implies that R ( x , y ) > c3A, and d(x,y) >_ c4A~

The result is immediate from (8.26) and (8.27).

[]

Applying Lemma 8.17 and Theorem 8.15 we deduce: T h e o r e m 8.18. Let F be a nested fractal, with scaling factors L, M, p, ~. Set df -- l o g M / l o g v ,

d~ = l o g M p / l o g v .

Then ( F, dF, #) is a fractional metric space of dimension d f, and X is a F D( dl, d~). In particular, the transition density p(t, x, y) of X is jointly continuous in (t, x, y) and satisfies

(8.28)

e,t-"""

(d(x, y)'-/t) "("o-'))

~_ p ( t , x , y ) (_ c3t-d'/d~ exp (--c4(d(x,y)d=lt)U(d~-l)) .

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(1987). [Tel] A. Telcs: Random walks on graphs, electric networks and fractals. Prob. Th. ReI. Fields 82, 435-451 (1989). [Tet] P. Tetali: Random walks and the effective resistance of networks. J. Theor. Prob. 4, 101-109 (1991). [Tri] C. Tricot: Two definitions of fractional dimension. Math. Proc. Camb. Phil. Soc. 91, 54-74, (1982).

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[Watl] H. Watanabe: Spectral dimension of a wire network. J. Phys. A. 18, 28072823 (1985). [Y1] J.-A. Yan: A formula for densities of transition functions. Sem. Prob. XXII, 92-100. Lect Notes Math. 1321.

ANALYSIS ON WIENER AND ANTICIPATING

SPACE

STOCHASTIC

CALCULUS

David NUALART

Contents 1 Derivative a n d d i v e r g e n c e o p e r a t o r s on a G a u s s i a n space 1.1 Derivative operator 1.2 Divergence operator 1.3 Local properties 1.4 Wiener chaos expansions 1.5 The white noise case 1.6 Stochastic integral representation of random variables 2 0 r n s t e i n - U h l e n b e c k s e m i g r o u p a n d equivalence of n o r m s 2.1 Mehler's formula 2.2 Hypercontractivity 2.3 Generator of the Ornstein-Uhlenbeck semigroup 2.4 Meyer's inequalities 3 A p p l i c a t i o n of Malliavin calculus to s t u d y p r o b a b i l i t y laws 3.1 Computation of probability densities 3.2 Regularity of densities and composition of tempered distributions with elements of ID-~r 3.3 The case of diffusion processes 3.4 Lp estimates of the density and applications 4 Support theorems 4.1 Properties of the support 4.2 Strict positivity of the density and skeleton 4.3 Skeleton and support for diffusion processes 4.4 Varadhan estimates 5 A n t i c i p a t i n g s t o c h a s t i c calculus 5.1 Skorohod integral processes 5.2 Extended Stratonovich integral 5.3 Substitution formulas 6 Anticipating stochastic 6.1 Stochastic differential 6.2 Stochastic differential 6.3 Stochastic differential

differential e q u a t i o n s equations in the Stratonovieh sense equations with boundary conditions equations in the Skorohod sense

Introduction The first two chapters of these notes are devoted to present the basic elements of the stochastic calculus of variations or Malliavin calculus on a Gaussian space. That is, we develop an infinite dimensional differential calculus for functionals of an arbitrary Gaussian process. The stochastic calculus of variations on the Wiener space was introduced by Malliavin in [62] in order to provide a probabilistic proof of HSrmander's hypoellipticity theorem. One of the main results of this subject is the equivalence between the L p norms of the iterations of the derivative and the Ornstein-Uhlenbeck operators, that was established by Meyer in [66]. Chapter 3 deals with the application of the techniques of the stochastic calculus of variations to study regularity properties of probability laws. First we discuss the existence and smoothness of a density for an m-dimensional random vector assuming that its Malliavin matrix is nondegenerate. The basic tool to obtain these results is an integration by parts formula. We deduce also L p estimates of the density and apply them to the problem of the existence, uniqueness and convergence of approximation schemes for ordinary and partial stochastic differential equations. In Chapter 4 we discuss the properties of the support of the probability distribution of a random element in the underlying Gaussian space. For nondegenerate smooth random vectors, the points where the density is strictly positive are characterized by means of the notion of skeleton. We show this type of characterization and we apply it to deduce Varadhan-type estimates when the variance of the white noise tends to zero. In chapters 5 and 6 we present some results on the anticipating stochastic calculus. The divergence operator coincides with an extension of the It& stochastic integral, due to Skorohod, for processes which are not adapted to the Brownian motion. The Skorohod integral turns out to have properties which are similar to the classical It& integral. For instance, one can deduce a change-of-variable formula which generalizes the It& formula. On the other hand, one can also define an extended Stratonovich integral by means of suitable approximation by Riemann sums. We discuss the relationship between the Skorohod integral and the extended Stratonovich integral and we establish a change-of-variable formula for the Stratonovich integral which is similar to that in classical analysis. The anticipating stochastic integrals allow us to formulate and solve stochastic differential equations whose coefficients are random, or such that some boundary conditions are in force. We deduce some existence and uniqueness results for these type of equations in Chapter 6.

David Nualart Barcelona, July, 1995

Chapter 1 Derivative and divergence o p e r a t o r s on a G a u s s i a n s p a c e In this chapter we study the differential calculus on a Gaussian space. That is, we introduce the derivative operator and its adjoint the divergence operator. We show that these operators are local, and we study their behavior when they are applied to a Wiener chaos expansion.

1.1

Derivative operator

We work in the following general context. Suppose that H is a real separable Hilbert space whose norm and inner product are denoted by I1" IIH and (., ")H, respectively. We associate with H a Gaussian and centered family of random variables 7-ll = {W(h), h e H} such that

E(W(h)W(g)) = (h, g)H, for all h, g C H. The family T/1 is defined in some complete probability space (~, ~ , P). This means that ~1 is a closed Gaussian subspace of L2(~) isometric to H. We will assume that the a-field ~- is generated by 7-/1.

Examples: 1. Let {Wt' , t > 0, i = 1 , . . . , d} be a d-dimensional Brownian motion. In this case the Gaussian space is obtained by taking H ---- L2(]R+;]Rd), and for any h C H the variable W(h) is the Wiener stochastic integral ~i=ld f ~ h~dW~. 2. Suppose that (T, B, #) is a measure space such that B is countably generated, and {W(A), A 9 B, #(A) < oc} is a family of random variables such that each W(A) has the distribution g ( 0 , #(A)), W(A U B) = W(A) + W(B) if A and B are disjoint, and { W ( A 1 ) , . . . , W(An)} are independent whenever the sets { A 1 , . . . , An} are pairwisedisjoint. In this case the Gaussian space is obtained by taking H = L2(T, B, p), and W(h) is given by the stochastic integral

W(h) = IT h(t)W(dt). We will say that W is a white noise on (T, B, #). Note that Example 1 is a particular case of example 2 by setting T = IR+ • { 1 , . . . , d} and tt equals to the product of the Lebesgue measure times the uniform measure on { 1 , . . . , d}.

127

3. Let v be a zero mean Gaussian measure on a real separable Banach space 113 with full support. Consider the inclusion j : II3" --* L2(v) which is linear, continuous and one-to-one. Set 7-ll = j(IB*). Then 7-ll is a Gaussian subspace of L~(v).

Remarks: 1. Consider the case of a d-dimensional Brownian motion W defined on the canonical probability space f~ = C0(N+, IRa). The subspace H 1 of f~ formed by the functions of the form ~o(t) = fg r ~b E H = L2(IR+; IRa) is called the Cameron-Martin space. Equipped with the scalar product <~01,(~2>H 1 = <@1,@2>"

H 1 is isometric to H, and the inclusion i : H 1 -+ f~ is continuous. 2. In the context of Example 3, one can show (eft Gross [34], Kuo [55]) that the dual map j* of j is a compact operator from 7-/1 into IB with a dense image. Then the image H 1 := j*(~x) is a subspace of 113, and the triple (IB, H 1, u) is called an abstract Wiener space. When t13 = Co(N+, IRd), and u is the Wiener measure, then H 1 is the Cameron-Martin space. In fact, given a point measure m = Chto E 113", c E IRd to >_ 0, clearly j ( m ) = c . W(to) and j*j(m)(t) = c(to A t) because

j*j(m)~(t)

= (j*j(m),e~ht)~,~. = (j(m),j(e~t))L2(,) = E ( ( c . W(to)(e,. W(t)) = c,(to A t),

where e, is the ith vector of the canonical basis of IRa. Notice that the function ~(t) = c(to A t) is absolutely continuous and its derivative has an L2-norm equals to v~llcll which is the L2-norm of j ( m ) in L2(v). As a consequence, j*(7-(1) consists of all absolutely continuous functions ~ : IR+ ~ IRd which vanish at zero at have a square integrable density, that is, j*(7-/1) is the Cameron-Martin space. Let us first introduce the derivative operator D. We will follow an approach analogous to the definition of the Sobolev spaces in finite dimensions. We denote by C ~ ( I R n) (resp. C~(IRn)) the set of all infinitely differentiable functions f : IR'~ IR such that f and all of its partial derivatives have polynomial growth (resp. are bounded). Let S (resp. Sb) denote the class of random variables of the form

F = f ( W ( h l ) , . . . , W(hn)),

(1.1)

where f belongs to Cp(IR n) (resp. C~(IR'~)), h i , . . . , h,~ are in H, and n > 1. These random variables are called smooth. We will denote by P the subset of S formed by random variables of the form (1.1) where f is a polynomial. If F has the form (1.1) we define its derivative D F as the H-valued random variable given by

O F = f i ~fx ( W ( h l ) , . . . , W ( h n ) ) h , .

(1.2)

4=1

Notice that for any element h E H the scalar product (DF, h}n coincides with the directional derivative ~a ~,~hl1~=0,where F~h is the shifted random variable

F `h = f ( W ( h l ) + c(h, h , ) H , . . . , W(h~) + c(h, hn)n).

128

In order to show that the operator D is closable we need the following integrationby-parts formula:

1.1.1 Suppose that F is a smooth random variable and h E H. Then

Lemma

E((DF, h)g) = E(FW(h)).

(1.3)

Proof: We can assume t h a t the norm of h is one. There exist orthonormal elements of H, e l , . . . , en, such t h a t h = el and F is a smooth random variable of the form F = f ( W ( e l ) , . . . , W(e,)). Let r

denote the density of the s t a n d a r d normal distribution on IRn. Then we have

E(
=

f(x) (x)xldx = E(FW(el)),

which completes the proof of the lemma.

[]

Applying the previous result to a product FG, we obtain the following consequence. Lemma

1.1.2 Suppose that F and G are smooth functionals, and let h E H. Then

we have E(G(DF, h)H) = E ( - F ( D G , h>H + FGW(h)).

(1.4)

As a consequence of the above lemma, D is closable as an operator from DO(f/) to DO(f~; H) for any p _> 1. In fact, let {FN, N >_ 1} be a sequence of smooth random variables such t h a t FN converges to zero in D'(ft) and the sequence of derivatives DFN converges to r / i n LP(f/; H). Then, from Lemma 1.1.2 it follows t h a t r / i s equal to zero. Indeed, for any h E H and for any smooth random variable F C Sb such t h a t FW(h) is bounded, we have

E((~, h)HF) =- limE((DFN, h)HF) N

= limE(-FN(DF, h)H + FNFW(h)) = O, N

because FN converges to zero in D ~ as N tends to infinity, and the random variables (DF, h)H and FW(h) are bounded. This implies r / = 0. We will denote the domain of D in LP(t2) by ID I'p, meaning t h a t 1DI'p is the closure of the class of smooth r a n d o m variables S with respect to the norm IIFIh,, = [E(IFI" ) + E(IIDF[[;)][. For p = 2, the space ]131'2 is a Hilbert space with the scalar product (F, G)l,2

:

E(FG) + E((DF, DG)H).

We can define the iteration of the operator D in such a way that for a smooth random variable F, the derivative DkF is a random variable with values on H |

129

Then for every p > 1 and any natural number k we introduce a seminorm on S defined by k

IIFIl[,p : E(IFI p) + ~ E(IIDJFII~|

(1.5)

j=l

As in the case k = 1 one can show t h a t the operator D k is closable from S C LP(ft) into LP(f~;H| p >_ 1. For any r e a l p _> 1 and any natural number k _> 0, we will denote by 119k'p the completion of the family of smooth random variables S with respect to the norm I1 9 IIk,~. Note t h a t IDj'p C I19k'q if j _> k and p >_ q. Let V be a real separable Hilbert space. We can also introduce the corresponding Sobolev spaces of V-valued r a n d o m variables. More precisely, S v will denote the family of V-valued smooth r a n d o m variables of the form

F = f i F~b,

b ~ V,

Fj ~ S.

j=l

We define D k F = ~j~=l DkFj | b , k _> 1. Then D k is a closable operator from SV C LP(~; V) into LP(f~; H ~k | V) for any p >_ 1. For any integer k > 1 and any real number p _> 1 we can define the seminorm on S v by k P IIFIIk,p,v = E(IIFII~/) + ~ E(IIDJFII~|174

9

j=l

We denote by IDk'P(V) the completion of Sv with respect to the norm k = 0 we put IlFIIo,~,v

=

p

1

[E(llfllv)]~, and ID~

II 9 Ilk,p,V. For

= LP(a; V).

Consider the intersection

ID~176 = n,>_~ n,<>, IDk"(V). Then IDa(V) is a complete, countably normed, metric space. We will write ID~176 = ID~. For every integer k _> 1 and any real number p _> 1 the operator D is continuous from IDk'P(V) into IDk-I'P(H | V). Consequently, D is a continuous linear operator from ID~176 into I D ~ ( H | V). Moreover, if F and G are r a n d o m variables in ID~ , then the scalar product (DF, DG)H is also in I19~176 The following result is the chain rule, which can be easily proved by approximating the random variable F by smooth r a n d o m variables, and the function ~ by (~ * ~ ) , where ~ is an approximation of the identity. 1.1.1 Let ~ : IRm ~ IR be a continuously differentiable function with bounded partial derivatives, and fix p >_ 1. Suppose that F = ( F I , . .. , F m) is a random vector whose components belong to the space ID I'p. Then ~(F) E I19<~, and

Proposition

=

i:l

As a consequence of the chain rule, the space ID~176 is an algebra. We remark t h a t in the infinite dimensional case, and assuming t h a t f~ is a Banach space, the elements of ID~ may not be differentiable or even continuous. For instance, the random variable F = f~ WtldWt2 is not continuous on C([0, 1]; IR 2) and it belongs to ID~176 We will write DhF = (DF, h}H.

130

1.2

Divergence

operator

We will denote by 5 the adjoint of the operator D as an unbounded operator from L2(f~) into L2(f~;H). That is, the domain of 5, denoted by DomS, is the set of H-valued square integrable random variables u such that

IE((DF, U}H)I < cllFII2,

(1.6)

for all F E ID 1'2, where c is some constant depending on u. If u belongs to DomS, then 5(u) is the element of L2(f~) characterized by

E(FS(u)) = E((DF, U)H)

(1.7)

for any F E ]]31'2. The operator 5 will be called the divergence operator and is closed as the adjoint of an unbounded and densely defined operator. We denote by SH the class of smooth elementary elements of the form

= ~ F~hj,

(1.8)

j=l

where the Fj are smooth random variables, and the hj are elements of H. From the integration-by-parts formula established in Lemma 1.1.2 we deduce that an element u of this type belongs to the domain of 5 and moreover that n

5(u) : ~ FjW(h,) - ~-](DFj, hj)H. j=l

(1.9)

j=l

Notice that if u C SH then Du is an H | H-valued random variable. For smooth elements u, v and F we have the three following basic relations between the operators D and 5:

Dh(5(u)) :


(1.10)

E(5(u)5(v)) = E((u,v)H) + E(Tr(Duo Dv)) 5(F~) = F S ( u ) - (DF, u)H. Proof of (1.10):

(1.11) (1.12)

Suppose that u has the form (1.8). From (1.9) we deduce n

n

Dh(5(u)) = y~ Fj(h, hj}H + ~ (DhFjW(hj) - (D(DhFj), hi)H), j=l j=l which implies (1.10).

Proof of (1.11): Let {ei, i > 1} be a complete orthonormal system on H. Using the duality relationship (1.7) and property (1.10) we obtain

E(5(u)5(v))

=

E((v,D(5(u)))H) = E

=

E

:

E(<~,v>.) + E

(v, ed.D~(5(u))

v, ed~l ((ei, ~ ) . + 5(De,~


,,)
131

Proof of (1.12):

For any smooth random variable G we have

E(GS(Fu))

=

E ( ( D G , FU)H) = E ((u, ( D ( F G ) - GDF)}H)

=

E (G(FS(u) - (u, D E ) H ) ,

and the result follows. Property (1.11) implies the following estimate

E[5(u) 2] - E(IMI~) + E(IIDulI~/|

2 = Ilulll,~,H.

As a consequence, the space of weakly differentiable H-valued variables IDla(H) is included into Dom6, and (1.11) holds for any u C IDI'2(H). By an approximation argument property (1.10) holds for an element u C ID2a(H) (actually, (1.10) holds if u E IDI'2(H) and DhU belongs to the domain of 5, as it can be proved using Lemma 1.4.1). By the definition of the operator 5, property (1.12) holds whenever F C ]13la, u belongs to DomS, F u C L2(Ft;H), and the right-hand side of (1.12) belongs to L2(a).

1.3

Local

properties

In this section we will show that the operators D and 5 are local. The next result shows that the operator D is local in the space ID1'1. P r o p o s i t i o n 1.3.1 Let F be a random variable in the space ]])1,1 such that F = 0 a.s. on some set A c ~ . Then D F = O a.e. on A.

Proof: We can assume that F C ID 1'1 N L~(f~), replacing F by arctan(F). We want to show that I{F=oIDF = 0 a.s. Suppose that r : IR -+ IR is an infinitely differentiable function such that r >_ 0, r = 1 and its support is included in the interval [-1, 1]. Define the function r = r for all e > 0. Set r

=

f

r oo

By the chain rule r belongs to ]1)1'1 and Dr H-valued random variable of the form

= r

Let u be a smooth

n

u = E F,~j, j=l

where Fj C Sb and hj C H. Observe that the duality relation (1.7) holds for F in ID1'1 n L ~ ( ~ ) and for an element u of this type. Note that the class of such processes u is total in L l ( a ; H) in the sense that if v C L l ( a ; H) satisfies E({v, U}H) = 0 for all u in the class, then v - 0. Then we have

IE (r

= =

IE ((D (r IE (r

_< ellr

).

Letting e l 0, we obtain

E ( I { F : o } { D F , U}H) = 0 , which implies the desired result. Let us now state and prove the local property of the divergence.

[]

132

1.3.2 Let u E IDI'2(H) and A C ~ , such that u(w) = O, P a.e. on A. Then 5(u) = 0 a.e. on A.

Proposition

Proof."

Let F be a smooth random variable of the form

F = f(W(h,),...,

W(h~)),

with f E C~~ n) ( f is an infinitely differentiable function with compact support). We want to show t h a t 5(U)I{IMIH=0} = 0, a.s. Consider a function r : IR ~ IR such as that in the proof of Proposition 1.3.1. It is easy to show t h a t the product FCdll~llS) belongs to ~ 1 , ~ Then by the duality relation (1.7) we obtain

Z (5(u)r

= E ({u,D[Fr :

E

D F > . ) + 2E

We claim t h a t the above expression converges to zero as e tends to zero. In fact, first observe t h a t the r a n d o m variables

v~ = Cdllull~)(~, DF)H

+

2Fr

t

2

D~u)u

converge a.s. to zero as e ~ 0, since [[U[[H = 0 implies ~ = 0. Second, we can apply the Lebesgue d o m i n a t e d convergence theorem because we have

Ir162

DF}H]

IIOlI~IlulIHIIDFIIH, <_ IIr162174

_< suPlzr174 x

[]

The proof-is now complete.

Note t h a t D is also local when acting on Hilbert-valued random variables. We can 1,p localize the domains of the operators D and 5 as follows. We will denote by IDIo~(V), p _> 1, the set of V-valued random variables F such t h a t there exists a sequence {(f~n, F~), n _> 1} C ~- x IDI'p(V) with the following properties: (i) f~n T ft , a.s. (ii) F = F~ a.s. on f~n. We then say t h a t ( f ~ , F~) localizes F in IDI'P(V), and D F is defined without ambiguity by D F = DF~ on f ~ , n > 1. The spaces ID~o~(V) can be introduced analogously. Then, if u C ]D1,2(H~ lor j, the divergence ~(u) is defined as a random variable determined by the conditions 5(u)ln ~ = 6(un)la ~ where (fin, un) is a localizing sequence for u.

for

all

n _ 1,

133

1.4

W i e n e r chaos expansions

We will denote by {H~, n E IN} the sequence of the Hermite polynomials defined from the series expansion t2 exp(tx - ~ ) = ~ tnHn(x). (1.13) n=0

Then { v ~ . H ~ , n E IN} is a complete orthonormal system in L2(IR,#) where # is the normal distribution N(0, 1). We will denote by A the set of all sequences a = (al, a2,...), a~ e IN, such that lal = el + a2 + ' " < oc. Let {ei, i > 1} be a complete orthonormal system in H. For any a E A we set oo

(~a : ~

H Hai(W(ei)) ' i=1

where a! = I]~1 a~!. Then the family {Be, a C A} constitutes an orthonormal basis of L2(~,5 ~, P) (we recall that we assume that the a-field jr is generated by W). For any n > 0 we will denote by ~,, the closed subspace of L2(f~) spanned by {(I)~, a C A, la[ = n}. Then ~ n is called the Wiener chaos of order n and we have the orthogonal decomposition =

~)n=0

n-

We will denote by Jn the orthogonal projection on the nth Wiener chaos 7-/n. The Wiener chaos expansion can be extended to the space L2(~; V) of Hilbert valued random variables: L2(~; V) = Q~=0Hn(V), where Tln(V) = 7-[~ | V. The following result characterizes the domain of the operator D in L 2 in terms of the Wiener chaos expansion. P r o p o s i t i o n 1.4.1 We have E(IIDFII2H) : ~ nlldn(F)ll[ ,

(1.14)

n=l

in the sense that this sum is finite if and only if F E ]D 1'2. Moreover, for all n > 1 we have D ( J n ( F ) ) = J n - I ( D F ) . Proof: It suffices to compute the derivative of a random variable of the form (I)~, using the relationship H'~ = H~-l, which follows immediately from (1.13): oo

D(~a)----~a~.E

fi

Ha~(W(ei))Haj-l(W(ej))eJ"

j=l i=l,i#j

Then D((I)~) 9 7"/n_l(H) if [a I = n, and E(HD(Oa)H2H) =

j=l

H~ ai!(aj - 1)! = lal' i=ljr

Now the proposition follows easily. By iteration we obtain E(IIDkF[I2H|

= ~ n(n - 1 ) . . . (n - k + 1)llJn(F)ll~, n=k

[]

(1.15)

134

and F E IDk'2 if and only if ~ - - 1 nkllJ~(F)ll~ < oo. Moreover, for all n > k we have

Dk(Jn(F)) = J~_k(DkF). The next lemma is a consequence of the expression of the operator D in terms of the Wiener chaos. L e m m a 1.4.1 Let G E L2(f~) and q E L2(f~; H) be such that

E(ae(v)) = E((~, ~).), for all v E IDI'2(H). Then G E ID 1'2 and D G = ~. Proof."

We have E((r/, v>t~) = E(Gh(v)) =

E(J~(G)5(v)) = ~ E((D(J~(G)), v)t4), n=l

n=l

hence, Jn-lrl = D(Jn(G)) for each n > 1 and this implies the result.

[]

Remarks: If F is a r a n d o m variable in the space ID 1'1 such t h a t D F = 0, then F = E ( F ) . This is obvious when F E ID 1'2 if we use the Wiener chaos expansion of F and Proposition 1.4.1. In the general case this property can be proved by a duality argument. Indeed, let CN be a function in C~(IR) such that Cy(X) = 9 if Ixl < N, and I~N(x)I _< N + 1. Then one shows t h a t

E(r

- E(F))5(u)) = 0

(1.16)

for any bounded H-valued r a n d o m variable u in the domain of 5. A p p r o x i m a t i n g an a r b i t r a r y random variable u E IDI'2(H) by the sequence of bounded r a n d o m variables {uCk(llull~),k > 1}, where Ck(x) = 1 if Ixl < k, Ck(x) = 0 if Ixl > 2k, and Ir < 2/k, we obtain t h a t (1.16) holds for any u E D I ' 2 ( H ) . Finally, by L e m m a 1.4.1 this implies t h a t C N ( F - E ( F ) ) = E ( r - E ( F ) ) ) , and letting g tend to infinity we deduce the result. As an application of the chain rule and the above p r o p e r t y we can show the following result (see Sekiguchi and Shiota [96]).

L e m m a 1.4.2 Let A E ~ . Then the indicator function of A belongs to ]131'1 if and only if P ( A ) is equal to zero or one. Proof:

Applying the chain rule to a function ~ C C ~ ( ] R ) which is equal to x 2 on [0, 1] yields D I A = D(1A) 2 = 21AD1a.

Consequently, D I A = 0 because from the above equality we get t h a t this derivative is zero on A c and equal to twice its value on A. So, by the previous remark we obtain

1A = P(A).

[]

135

1.5

The white

noise case

Consider the particular case of a white noise {W(A), A 6 I3, #(A) < oo} defined on a measure space (T, B, #). Suppose in addition that the measure # is a-finite and without atoms. In this case (cf. It5 [49]) the multiple stochastic integral of order n provides an isometry between the space L2,T st n , B n , u.n~) of square integrable and symmetric functions of n variables and the nth Wiener chaos ~n. Let us briefly describe how the multiple stochastic integral is constructed. Consider the set g~ of elementary functions of the form k

f(h,...,tn)=

~

air..i lA~lx...xA~(t],...,tn) ,

(1.17)

il,...,in=l

where Ax, A2,..., Ak are pairwise-disjoint sets of finite measure, and the coefficients ai~...i~ are zero if any two of the indices i l , . . . ,in are equal. Then we set k

In(f) =

~

aw..~W(Ail)..'W(A,~).

il,...,in=l

The set gn is dense in L2(Tn), and I~ is a linear map from gn into L2(~) which verifies I~(f~) = I ~ ( s and 0

n](f,O)L2(T~ )

E(I~(f)Iq(g))=

if if

n=/q, n=q,

where f denotes the symmetrization of f. As a consequence, I~ can be extended as a linear and continuous operator from L2(T n) into L2(f~). The image of L2(T n) by In is the nth Wiener chaos 7-/n. This is a consequence of the fact that multiple stochastic integrals of different order are orthogonal, and that In(f) is a polynomial of degree n in W ( A J , . . . , W(Ak) if f has the form (1.17), and those polynomials are included in the sum of the first n chaos. As a consequence, any square integrable random variable F admits an orthogonM expansion in the form F

= E(F) + fi/~(f~),

(1.18)

n=l

where the functions fn E L2(T n) are symmetric and uniquely determined by F. The operators D and 6 can be represented in terms of the Wiener chaos expansion. P r o p o s i t i o n 1.5.1 Let F C ID1'2 be a square integrable random variable with an

expansion of the form (1.18). Then we have O~

DtF : ~ nln-l(fn(',t)).

(1.19)

n=l

Proof: Suppose first t h a t F = L,(f,,), where f,, is a symmetric and elementary function of the form (1.17). Then Dtr= f

~

j = l il,...,in=l

ah...,.W(A~l)...1A,~(t)...W(Ai~)=nln-l(f=(.,t)).

136

Then the results follows easily.

[]

Note that the L2(t2; H)-norm of the right-hand side of (1.19) coincides with expression (1.14). We remark that the derivative DtF is a random field parametrized by T. Suppose that F is a random variable in the space IDN'2, with a Wiener chaos expansion of the form F = End__0In(fn). Then, applying Proposition 1.5.1 N times we obtain that DNF is a random field parametrized by T x given by

D~,..,tNF = ~ n ( n - 1 ) . . - ( n -

N + 1)In-N(f~(',tl,...,tN)).

n=N Note that the L2-norm of this expression is given by (1.15). As a consequence, if F belongs to ID~ = ANID N'2 then fn = ~ E ( D ~F) for every n k 0 (cf. Strooek [100]). We will now describe the divergence operator in terms of the Wiener chaos expansion. Any element u E L2(f~; H) ~ L2(T x f~) (which can be regarded as a square integrable process parametrized by T) has an orthogonal expansion of the form

oo u(t) = ~ In(In(., t)), n=0

(1.20)

where for each n >_ 1, fn E L2(T n+l) is a symmetric function in the first n variables. P r o p o s i t i o n 1.5.2 Let u E L2(T x f~) with the expansion (1.20). Then u belongs to Dom ~ if and only if the series 6(u) = ~ In+10~)

(1.21)

n=0

converges in L2(f~). Equation (1.21) can also be written without symmetrization, because for each n, In+~ (fn) = I~+1 (fn). However, the symmetrization is needed in order to compute the L 2 norm of the stochastic integrals.

Proof: Suppose that G = In(g) is a multiple stochastic integral of order n > 1 where g is symmetric. Then we have the following equalities: E ((u, DG)H) = fT E (In-1 (fn-1 (', t))nIn-1 (g(', t))) #(dr) = n(n - 1)! fT(f~_l(., t), g(., t))L2(T--~)p(dt) = n!(fn-a,g)L2(TN = n!ifn-l,g)L2(TN =

S (In(]n_i)In(g))

= E (In(fn_l)a)

.

Suppose first that u E Dom 5. Then from the above computations and from formula (1.7) we deduce that

E(6(u)O) = E([n(L-1)G) for every multiple stochastic integral G = In(g). This implies that In(fn-a) coincides with the projection of 6(u) on the nth Wiener chaos. Consequently, the series in

137

(1.21) converges in L2(f~) and its sum is equal to 5(u). The converse can be proved by a similar argument. [] For any set A C T we will denote by ~A the a-field generated by the random variables {W(B), B C A, B E Bo}, where B0 = {B e B, #(B) < co}. L e m m a 1.5.1 Let A E B and let F be a random variable in L2(f~, ~-Ac, P) N ]D 1'2. Then D t F = 0 for all (t,w) E A • f~ a.e. with respect to the measure # | P. Proof: Suppose that F = f ( W ( B 1 ) , . . . , W(Bn)), where f E C~(IRn), and Bi C A c, B~ E B0 for all i = 1 , . . . , n. Then we have DtF = 0~(W(B,),...,

W(B~))IB,(t),

which implies that D t F = 0 for all (t, co) E A • fk Finally in the general case it suffices to consider a sequence Fn as above converging to F in L ~. [] The next lemma allow us to interpret the operator 5 as a stochastic integral with respect to the white noise W. L e m m a 1.5.2 Let A E Bo and let F be a random variable in L2(f~,.TA~,P). the process u = F I A belongs to Dom 5 and

Then,

5(F1A) = F W ( A ) . Proof:

Suppose first that F E ID 1'2. Then, using (1.9) and Lemma 1.5.2 we obtain 5(FIA) = F W ( A ) - fT DtF1A(t)#(dt) = F W ( A ) .

The general case follows by a limit argument.

[]

The following lemma provides additional examples of processes in the domain of which are not smooth. L e m m a 1.5.3 Let A E 13o. Let {u(z), x E IRm} be an ~Ac-measurable random field with continuously differentiable paths such that E \lxl
and

E (lu(0)[ 2) < oc,

for every K > O. Let F be a bounded random variable in the space ]D1'4. Then u ( F ) I A belongs to the domain of ~ and rn

5(u(F)IA

= u ( F ) W ( m ) - ~-~(Oiu)(F) fA DtFi#(dt). i=1

Idea of the proof." Approximate u(F) by the convolution of u(x) with an approximation of the identity. []

138

Consider the following space of elementary processes: 7r =

F~IA~, Ai 9 Bo, Fi E L2(~, ~AT, P)

9

By Lemma 1.5.2 we have that 7s C Dora5 and

i=1

i=1

As a consequence we deduce the following result: P r o p o s i t i o n 1.5.3 Let 80 C TOo be a subspace such that for all u 6 ,So the isometry property E[5(u) 2] = E(lluJl~) holds. Then S0011112C Domh, and the isometry property

still holds in the closure of ,So. Notice that the isometry property holds for a process u of the form u = F1A E T4.o, but, in general, it is not true for a general element u in T~0. Particular examples of subspaces of ~0 where the isometry holds are the following: E x a m p l e 1: Suppose that W = {W(t), t E [0, 1]} is a d-dimensional Wiener process. Set n--1

,So = { u : u(t) = ~ Fil(t~,t~+,l(t),Fi E L2(gt, $r[o,td, P; lRd), i=0

0 = to < " " < tn = 1}. Then S011112is the space L] of d-dimensional square integrable and adapted processes. Therefore, L ] c Dora5 and 5(u) coincides with the It6 stochastic integral on this space. If we take n-1

81 = { u : u ( t ) = ~ F~l(t.t,+ll(t), Fi E

L2(~,.T'[t~+I,1],P; ]Rd),

i=0

0 = to < " " < t~ = 1}, then ~11.112 is the space L~ of d-dimensional backward adapted processes and 5(u) coincides with the backward It6 integral on this space. E x a m p l e 2: We will denote by W a white noise on [0, 1]2 with intensity equal to the Lebesgue measure. Then {W(s, t) = W([0, s] x [0, t]), s, t, E [0, 1]} is a two-parameter Wiener process, that is, a zero mean Gaussian process with covariance given by

E[W(Sh tl)W(s2, t2)] = (sl A s2)(tl A t2). Set n-1

s2 = N:

= i,j=0

F~,j E L2(f~,~[0,sd•

P), 0 = to < .-- < t~ = 1,0 = So < .." < sn = 1}.

Then ~11.112 is the space L 21,a of square integrable and 1-adapted processes. Therefore, L 21,~ c Dom 5 and 5(u) coincides with the It6 stochastic integral on this space.

139

1.6

Stochastic integral representation of random variables

Suppose that W = {W(t), t E [0, 1]} is a one-dimensional Brownian motion. We know that any square integrable random variable F, measurable with respect to W, can be written as

F = E(F) +

/01 r

where the process r belongs to L~. When the variable F belongs to the space ]D 1'2, it turns out that the process r can be identified as the optional projection of the derivative of F. This is called the Clark-Ocone representation formula (see [24], [83]):

Proposition 1.6.1 Let F C ID 1'2, and suppose that W is a one-dimensional Brownian motion. Then F : E(F) + E(DtFl~t)dWt. (1.22) In order to proof this stochastic integral representation, we will make use of the Wiener chaos expansions, and we need the following technicM result:

Lemma 1.6.1 Let W be a white noise on ( T , B , # ) . Suppose that F is a square integrable random variable with the representation F = ~n~ o In(f~). Let A E B. Then E(FIYA)

=~

I , ( f , lA~").

(1.23)

rt=0

Proof: It suffices to assume that F = I~(f,), where f , is a square integrable and symmetric kernel. Also, by linearity and density we can assume that the kernel f , is of the form 1Blx...• where B 1 , . . . , B , are mutuMly disjoint sets of finite measure. In this case we have E(FIJZA)

=

E(W(B1)... W(B,)I~A)

=

E ( 1 - [ ( W ( B ~ A A) + W ( B , • A~)) I ~ d )

n

i=1

=

In(l(slnA)•

.• []

Proof of Proposition 1.6.1: Suppose that F : ~~176 0 In(fn). Using (1.19) and Lemma 1.6.1 we deduce

E(m, E I f , ) :

nE(~.-~(A(., t))lf,) n=l

Set r : E(D,FIT,). We can compute 6(r

using the above expression for r and

(1.21), and we obtain 5(r

= ~

I.(A) = F - E(F),

140

which shows the desired result because 5(r r

is equal to the It5 stochastic integrM of []

B i b l i o g r a p h i c a l n o t e s : The integral representation formula (1.22) can been extended to random variables in ]D1'1 (cf. [50]) and to elements of ID - ~ (cf. [104]). The iterated divergence operator 5k is an extension of the multiple stochastic integrals introduced by Hajek and Wong [40] (see [80]).

Chapter 2 Ornstein-Uhlenbeck semigroup and equivalence of norms In this chapter we introduce the operator L, which is the infinitesimal generator of the Ornstein-Uhlenbeck semigroup, and we show the equivalence between the I1" IIk,p norms defined using the derivative operator and the norms 111FIIIk,p= tl ( I - L ) ( F ) k / 2 l l ,. Let W = {W(h), h E H} be a centered Gaussian family associated with a real and separable Hilbert space H, defined on a probability space (ft, Y, P). Suppose that the a-field ~- is generated by W. Consider the one-parameter semigroup {Tt, t >_ 0} of contraction operators on L2(f~) defined by TtF = f i e-ntJnF,

(2.1)

r~=0

where Jn denotes the projection on the nth Wiener chaos. This semigroup is called the Ornstein-Uhlenbeck semigroup.

2.1

Mehler's formula

The following result is known as Mehler's formula.

Proposition 2.1.1 Let W ' = {W'(h), h E H } be an independent copy of W . for any t >_ 0 and F C L2(f~) we have TtF = E ' ( F ( e - t W + v/1 - e-2tW')),

Then

(2.2)

where E' denotes the mathematical expectation with respect to W ' . Note that the right-hand side of (2.2) is well defined because { e - t W ( h ) + x/1 - e-27W'(h), h e H } is a centered Gaussian family with the same covariance as W. Proof: Both Tt and the right-hand side of (2.2) give rise to linear contraction 1 2 operators on L2(ft). Thus, it suffices to show (2.2) when F = e x p ( ~ W ( h ) - ~A ),

142

where h E H is an element of norm one and A E IR, We have

E'(exp(e-tkW(h)+x/X-e-2tkW'(h)-~k~)) oo

/

= exp(e-tAW(h)

= ~ e-ntAnH,~(W(h)) !e-2,;) 2 } n=0

-

\

= TtF,

because J ~ F = A~H~(W(h)).

[]

Mehler's formula implies t h a t the operator Tt is nonnegative and t h a t it is a contraction o n / 2 ( F t ) for any p > 1. Moreover the semigroup {Tt, t >_ O} is continuous in/_2(~) for any p _> 1.

2.2

Hypercontractivity

The operators Tt verify a property (cf. Nelson [73], Neveu [74]) called hypercontractivity, which is stronger than the contractivity in L p, and which says t h a t

[ITtFllq(t) <_ IIFlIp, if q(t) = As a 1< p < fact, let

(2.3)

e2t(p - 1) + 1 > p, t > 0 and F E / y ( ~ , )c, p ) . consequence of the hypereontractivity property it can be shown t h a t for any q < oc the norms It' II~ and II" IIq are equivalent on any Wiener chaos 7~n. In t > 0 such t h a t q = 1 + e~t(p - 1). Then for every F E 7fin we have e-"]lFllq = IlT, Yllq <_ IlFllp.

(2.4)

In addition, for each n _> 1 the operator J~ is bounded in LY for any 1 < p < oo, and

1)~llFllp II&FIIp<-- f~ (( pp -- 1)-~llFllp

if p > 2

if p < 2 .

In fact, suppose first t h a t p > 2, and let t > 0 be such t h a t p - 1 = e st. Using the hypercontractivity p r o p e r t y with the exponents p and 2, we obtain

IIJ, Vll~ = ~'lIv, J~Fllp < e~*lIJ.Fll~ _< e~qIFll2 _< e~'lIFll,.

(2.5)

If p < 2, we use a duality argument. Consider a sequence of real numbers {r linear operator Tr : 5~ -+ P defined by

n k 0}. This sequence determines a

oo

TcF = ~ r

V e P.

n=0

We remark t h a t the operators Tt are of this type, the corresponding sequence being e -'~t. It would be useful to know whether such a multiplication operator is bounded in all Lp, p > 1. The following examples provide sufficient conditions for this property to hold.

143 Examples:

1.

Suppose that r oo -k for n > N and for some ak E ]R such that = ~k=oakn --k < or T h e n the operator Tr is bounded i n / f for any 1 < p < cr

oo ~k=o lakl N

Proof: By duality, and taking into account that Tr is selfadjoint, we can assume oo 7-~ that p _> 2. Moreover it suffices to show that Tr is bounded i n / f on ~n=g ~. Fix M F C ~ = g TI~ with M > N. We have

IIT~FIIp--

akn -k J . F

<

k=0

p

[akl ~ k=0

n-k JnF

n=N

. p

Now, using the equality

n-k = ( fo~ e-~tdt) k = f[o,oo)k e-~(t~++tk) dtl . . . dtk, we obtain

oo

IlTcFIIp <~ ~ lakl Z k=0

10,oo)k

IIT~,+.--+,kPll/t,"'" dtk.

(2.6)

Then it suffices to have an estimate of t h e / f - n o r m of the Ornstein-Uhlenbeck semigroup of the form IITtFIIp _< KN,pe-NtlIYll ,. (2.7) Indeed, from (2.6) and (2.7) we deduce oo

IIT~FIIp <~K~,p ~ laklN-kIIFlI., k=0

which allows us to complete the proof. The estimate (2.7) follows from the hypercontractivity property (2.3). In fact, suppose p > 2 (the case p = 2 is immediate) and choose to such that p = e2t0 + 1. For all t > to we have

lIT,Flip~ = IIT,oTt-toFIl~ <~IITt-~oFIl@= ~ e-2n
< e-=N(~-~o)IlFll ~ _< e-2N(t-~~ which implies (2.7) with KN,p = e Nt~

KN, p =

If t < to Eq.

(2.7) is obvious again with

[]

e NtO.

2. For any a > 0 let us consider the sequence r = (1 + n) -~, n _> 0. Then the operator Tr is a contraction in L p for any p > 1. This follows immediately from the equation (1 + n) -c~ = F(ol) -1 e-(n+Utta-ldt,

/7

which implies Tr = P(O~)-1

e-ttc~-lTtdt. ~0 ~ 1 7 6

With the notations of the next section we have Tr = (I - L)-%

(2.s)

144

Given a sequence of real numbers r = {r n >_ 0}, define T~+ = X:~n=0r 1)jn. The following commutativity relationship holds for a multiplication operator T~: D(TcF) = T4)+(DF ),

(2.9)

for any F E 7". In fact, if F belongs to the nth Wiener chaos, we have

D(TcF) = D ( r

= r

= Tr

In particular we have D(TtF) = e tTt(DF), and by iteration we obtain

nk(Tt F) = e-ktTt(nkF)

(2.10)

for any F E 7", and k >_ 1. The following two properties hold: (A) Let F C ]Dk'p. Then TtF also belongs to IDk'p, and lim tl0

IlTtf

FIIk,p = 0.

-

(2.11)

Indeed, from (2.10) it follows that Tt is a contraction operator with respect to any seminorm II-Ifk,p. This implies that TtF E IDk'p. The continuity at the origin with respect to the norm of IDk'p is immeditate. (B) Let F E LP(fl). Then TtF E fTk>~ IDk'p, and

IIDk(TtF)llp < Ck,pt-kllFIIp,

(2.12)

for any k > 1, t > 0, and p > 1. In particular TtF E ID~ for any random variable F which has moments of all orders. Proof: It suffices to show (2.12) for any polynomial random variable. Suppose first that k = 1. We have from Mehler's formula

D(TtF)

= -

D ( E ' ( F ( e - t W + x/'l - e-2tW'))) e-t ,~_...:~_2tE'(D'(F(e-tW + v/1 _ e-2tW'))). X/1 -- e -'2t

We recall that for any random variable G E ID1'2 we have

IIE(DC)II,

= IIJz6'll2 = c,~llJ, Cllp _< 411all,,.

Hence,

/

E (IID(T,F)II~,) <_ c~ ~ ~

e -t

"~P

)

E(IFIP),

and, by iteration, we deduce (2.12).

[]

Prom properties (A) and (B) it follows that the family 7' of polynomial random variables is dense in IDk'p for any k _> 1 and p > 1. Indeed, we approximate a given random variable F in IDk'p by a polynomial G E 7' in L p, and, then, we use the inequalities k

IIF - r~allk,p _< lie - T, Fllk,p +

~_.cj,~t-Jllf j=O

-

Cll~.

145

2.3

Generator of the Ornstein-Uhlenbeck semigroup

The infinitesimal generator of the semigroup Tt in L2(~) is given by (2.13)

LF = f i - n J n F , n=l

and its domain is c~ D o m L = { F E L2(f~) :

~n2ilJnFIl~ < ~ } . n=l

From Proposition 1.4.1 it follows that Dom L C ID 1'2. Actually, Dom L = ID2'2. The next proposition explains the relationship between the operators D, 5, and L. P r o p o s i t i o n 2.3.1 Let F C L2(f~). Then F E D o t a L if and only if F C ID1'2 and DF E DomS, and in this case, we have 5(DF) = - L F .

Proof: For any polynomial random variable G we have, using the duality relationship (1.7) and Proposition 1.4.1,

E(GS(DF))

= E((DG, DF}H) = f i nE(JnGJnF) n=O

and the result follows easily.

[]

We are going to show that the operator L behaves as a second-order differential operator. P r o p o s i t i o n 2.3.2 Suppose that F = ( F 1 , . . . , F "~) is a random vector whose components belong to ID2'4. Let ~ be a function in CS(]Rm) with bounded first and second partial derivatives. Then ~( F) C Dom L, and

L(~(F)) = f i

0 2 ~ ( F ) < D F i , DFQH + f i ~ ( F ) L F ~. OxiOxj i=I

i,j=l

Proof:

Suppose first that F is a smooth random variable of the form

F = f(W(hl),...,W(hn)), with f E Cp(IRn). In this case using Proposition 2.3.1 we obtain

LF

= -6(DF) = -5

W(hl),...,W(hn))h~

= f i O~f ,.j=l ox,ox-----~j(W(hO,..., W(hn))(h,, hi), -

~(() W

=

hi

,...,W

(h))W() ~

h~ .

=

(2.14)

146

In the general case we approximate F by smooth random variables in the norm ]1" ][2,4 and use the continuity of the operator L in the norm II 9 112,2. [] For any natural k > 0 we can define the norm

IIIflll~,~ =

E(I(/-

L)k/2(F)I2) = f i ( n + 1)kE(l&f[2). n=0

Notice that this seminorm is equivalent to the norm IIF[lk,~, taking into account (1.15). In the next section we generalize this equivalence of norms to the case of p :~ 2. Set C = -x/-L--L, then ID 1'2 coincides with the domain of C.

2.4

Meyer's inequalities

We are going to establish Meyer inequalities following Pisier's approach. In order to do this we need some preliminaries. Consider the function ~ : [ - ~, 0) U (0, -~]2~ 1R+ defined by 1 ~(0) = sign O. (2.15)

V/2~l log cos 2 0l Notice that when 0 is close to zero this function tends to infinity as ~ 1.

Moreover, x2

we have, for all n > 0 (making the changes of variable cos0 = y and y = e - - r ) f0 ~

sin0c~ dO v/lr[ log cos 2 01

1 ~/~

(2.16) + 1)

Suppose that {W'(h), h C H} is an independent copy of the Gaussian family {W(h), h E H}. For any 0 C IR, F C L~ 5c, P) we set

RoF = F ( W cos 0 + W ' sin 0). With these notations we can write the following expression for the operator D ( - C ) -1. L e m m a 2.4.1 For every F C P such that E(F) = 0 we have

:/_} .'(J(RoF)){(0)d0

(2.17)

Proof: Suppose that F = p(W(hl) .... , W(hn)), where h i , . . . , hn C H and p is a polynomial in n variables. For any 0 E ( - 2 , -~) 2 we ]clave ReF = p ( W ( h 0 cos 0 + W'(hl) sin 0 , . . . , W(hn) cos 0 + W'(h,~) sin 0), and therefore

D'(R0P)

-

(W(hl)cos0 + W'(hl) Sin 0, i:l

. . . , W(h,~) cos 0 + W'(h~) sin 0) sin Oh~ = sin ORo(DF).

147

Consequently, using (2.2) we obtain

E'(D'(RoF)) = sin OE'(Ro(DF)) = sin OTt(DF), where t > 0 is such that cos 0 = e -t. This implies

E'(D'(RoF)) = ~ sinO(cosO)~J~(DF). rtmO

Note that since F is a polynomial random variable the above series is actually the sum of a finite number of terms. By (2.16) the right-hand side of (2.17) can be written as

(/-} sinO(cosO)ncP(O)dO)Jn(DF):~ n=0

n~+lJn(DF).

n=O

Finally, applying the commutativity relationship (2.9) to the multiplication operator defined by the sequence r = ~t, n _> 1, r = 0, we get

Tc+DF = DTcF = D(-C)-~F, and the proof of the lemma is complete.

[]

Now with the help of the preceding equation we can show that the operator D C -1 is bounded from LP(ft) into /2(fi; H) for any p > 1. This property will be proved using the boundedness in L p of the Hilbert transform. We recall that the Hilbert transform of a function f E C~(IR) is defined by

H f(x) = f a f ( x + t) --t f ( z - t) dt" The transformation H is bounded i n / 2 ( I R ) for any p > 1 (see Dunford and Schwarz [27], Theorem XI.7.8). Henceforth % and Cp denote generic constants depending only on p, which can be different from one formula to another. P r o p o s i t i o n 2.4.1 Let p > 1. There exists a finite constant cp > 0 such that .for any

F E P with E(F) = 0 we have ]]DC-IFH, < %]]Fl) p. Proof." Using (2.17) we can write E (IIDC-1FII~H) = E = a;~EE '

E'(D'(RoF))~(

( (/: W'

} E'(D'(RoF))~(O)dO

)')

,

where ap = E(I[I p) with [ an N(O, 1) random variable. We recall that for any G E L2(f~',.T',P ') which belongs to the domain of D' the Gaussian random variable W'(E'(D'G)) is equal to the projection J~G of G on the first Wiener chaos. Therefore, we obtain that

E (IIDC-1FllPH) = ap'EE'

J'lRoFqo(O)dO .

148

If g : [ - 2 , ~] --~ 113 is a Lipschitz function with values in some separable Banach space IB the product (z(O)g(O) does not belong to LI([t--~, 2 2];]13) unless g(O) = O. Nevertheless we can define the integral of the product ~g in the following way

?0

~(O)g(O)dO =

/o

~ ~(O)[g(O) - g(-O)]dO.

Notice t h a t d~RoF vanishes at 0 = 0 but this is no longer true for account this remark we can write

RoF. Taking into

E(HDC-11IJHP): o~plFjl~t(J; (/? RoF~(O)dO)p) <_ c~EE'

(;o

RoF~(O)dO ,

for some constant cp > 0. For any ~ E IR we define the process

w d h ) = (W(h) cos ~ + W'(h) sin ~ , - W ( h ) sin ~ + W'(h)cos ~) The law of this process is the same as t h a t of {(W(h), W ' ( h ) ) , h E H}. On the other hand, R~RoF = R~+oF, where wc set R~G((W(hl), W'(hl)),...,

(W(hn), W'(hn))) = G(W~(hl), .. ,W~(hn)).

Therefore, we get

p~ where II' lip denotes the D ~ norm with respect to P • P'. Integration with respect to yields

E (IIDC-XFIIPH) <_apEE'

(/00/00

, )

R(+oFF(O)dOd~ .

(2.18)

Furthermore, there exists a bounded continuous function ~5 and a constant c > 0 such that C ~(0) = ~(0) + ~, on [ - 2 ' ~]. Consequently, using the DO boundedness of the Hilbert transform, we see t h a t the right-hand side of (2.18) is dominated up to a constant by

In fact, the term ~5(0) is easy to treat. On the other hand, to handle the term ~ we write

149

< qo (ff_~ f R~+~

~

%

-

F

p

\7r/ J-~ Ja R~+~

where R o F = @(O)RoF, and r is a smooth function with support included in [_3~, 3~] t 2 2 such that 0 _< @ _< i and @(0) = 1 if 0 c [-TL 7@ Hence,

..,(r /:/v.,o',O,...(.., [] P r o p o s i t i o n 2.4.2 Let p > 1. Then there exist positive and finite constants Cp and Cp such that for any F C 79 we have

CpllOFIIL,(n;io <_ IICFIIp _< CplIDFIIL,(n;m.

(2.19)

Proof: Set G --- C F . Applying Proposition 2.4.1 to the random variable G = C F we have

]IDFllL,(a;H) = HDC tGHL,(a;H) < CpllGlJp = cpllCFllp, which shows the left inequality. The right inequality is proved by means of a duality argument. Let F , G E 79. Set d = C-I(I - J 0 ) ( a ) . Let q be such t h a t ~1 + q1 = 1. Then we have

IE(GCF)I

=

[ E ( ( I - Jo)(G)CF) I = IE(CFCG)I = IE((DF, DG)H)I

<_ IIDFIIL,(a;H)llDOllLqta;m <_ cqIIDF[ILp(a;H)IlCGIIq I = cqlIDFIIL,(a;H) II(I-- Jo)(a)llq -< %llOfllL,(~;mllallq. Taking the supremum with respect to G C 79 with Ilallq < 1, we obtain !

IlCFllp <_ CqIIDFIIGp(a;H).

[] Now we can state Meyer's inequalities in the general case. T h e o r e m 2.4.1 For any p > 1 and any integer k > 1 there exist positive and finite constants Cp,k and Cp,k such that for any F E P,

c,,k

_< (IC Fl')

[E (11 Fll ,o ) + E(IFf')]

(2.20)

Proof: The proof can be done by induction on k. The case k = 1 corresponds to Proposition 2.4.2. In order to illustrate the m e t h o d let us describe the proof of the left-hand side of (2.20) for k = 2.

150 Notice that if {[~, n _> 1} is a family of independent random variables defined on the probability space ([0, 1], B([0, 1]), A) (A is the Lebesgue measure), with distribution N(0,1), and {an, 1 < n < N} is a sequence of real numbers, we have, for each p > 1,

riO,l] N~=an~n(t) Pdt=Ap(n=~la2) ,

(2.21)

x2

where Ap Suppose that F = p(W(hl),..., W(hn)), where the hi's are orthonormal elements of H. We fix a complete orthonormal system {e~, i _> 1} in H which contains the h{s. With these notations, using Eq. (2.21) and Proposition 2.4.2, we can write =

E(IID2FII~|

= E


/o /o E

< Cp <_ c;

fo E

.zi (

<_ %

E

4'E

(D2F, e~ | ej)H~j(t)~(S) \li,j=l

)

D

(DF, ej>H(i(t) ,ei

;'~

)')

H[j(t )



dt

= 4'E(IICDFH~).

Consider the operator R = ~n~--1 V/1 - _11nn. This operator verifies CDF and it is bounded in L p for all p > 1 (see example 1). Hence, we obtain

E(]]D2FII~|

dt

H

i=1

C

dtds

= DCRF

< cpE(IIDCRF]]PH) < e'pE(IC2RF[p) ~ c'~E(]C2FIP). []

Khintchine's inequalities also allows us to establish the inequalities (2.20) for polynomial random variable taking values in a separable Hilbert space V. Let us now introduce a continuous family of Sobolev spaces. For any p > 1 and s E IR we will denote by III ' IIIs,pthe seminorm IIIF[I]s,p = ]](I - f)fFIIp,

where F E 7) is a polynomial random variable. Note that ( I - L)~ = ~n~__0(1+n)fJn. These seminorms verify the following properties:

151

(i) IiiFiIkp is increasing in b o t h coordinates s and p. The monotonicity in p is clear and in s follows from the fact t h a t the operators ( I - L) ~ are contractions in /2, for all a < 0, p > 1 (see (2.8)). (ii) The seminorms I]1" []]s,v are compatible, in the sense t h a t for any sequence F~ in P converging to zero in the norm IBm.]Jkp, and being a Cauchy sequence in any other norm aim-]Ni~,,p,they also converge to zero in the norm I1[" ]]l~',p'. We define IDs'p as the completion of P with respect to this norm. Remarks: 1. [HF[[[o,v= ]]F[[0,v = ]]FI[;, and ]13~ = /2(ft). For k = 1 , 2 , . . . the seminorms [[[' Hik,vand [[. [[k,p are equivalent due to Meyer's inequalities. In fact, we have k

k

[iiPlilk,, = ]i(I - L)~FII, _< iE(F)I + ]]R(-L)~F]I;, k

where R = Zo=I

Notice that R - -

r

{ 0i 88

with

n>ln=0

where h(z) = (1 + x)~ in analytic in a neighbourhood of 0. Therefore, the operator R is bounded i n / 2 for all p > 1. Hence,

IllFlllk,p < co(llFllv + [I(-L)~FIIp) < c'p(llFllp + IIDkFIIL,(n;H~)) I! _< c~llFIIk,v. In a similar way one can show the converse inequality. Thus, the Sobolev spaces IDk'v coincide with those defined by means of the derivative operator. 2.

From p r o p e r t y (ii) we have IDs'~ C ]13r

if p' < p and s ~ <_ s.

3. For s > 0 the operator ( I - L ) - ~ is an isometric isomorphism (in the norm II1"tits,p) b e t w e e n / 2 and IDs'p and between 113-s'p a n d / 2 for all p > 1. As a consequence, the dual of ID~'p is ID -~'q where -1 p +-1q = 1. I f s < 0, the elements o f l D ~'p may not be ordinary random variables and they are interpreted as distributions on the Gaussian space. Set ID - ~ = [Js,v ID~'p. The space 113- ~ is the dual of the space 113~ which is a countably normed space. 4. Suppose t h a t V is a real separable Hilbert space. We can define the Sobolev spaces ID~'v(V) of V-valued functionals as the completion of the class P v of V-valued polynomial r a n d o m variable with respect to the seminorm n[F[[[s,v,v defined in the same way as before. T h e above properties are still true for V-valued functionals. The main application of Meyer's inequalities is the following continuity theorem. P r o p o s i t i o n 2.4.3 Let V be a real separable Hilbert space. For every p > 1 and

s C IR, the operator D is continuous from ID~'P(V) to IDs-~'v(V| and the operator 5 (defined as the adjoint of D) is continuous from IDS'V(V | H) to ID~-LP(V).

152

Proof: We will only proof the continuity of D for V = ]R. The continuity of the adjoint operator follows by duality. For any F E T~ we have (I-

L)~ D F = D R ( I -

L)~ F,

s

where R = ~ = 1 ( h-~ ~ )~ J~. Notice that R = Tr with

{

' o:0o1

where h(x) = ( ~ 1) ~ k in analytic near O. Therefore, the operator R is bounded in L p for all p > 1, and we obtain

I1(I- L)fDFI[p

=

[IDR( I - L)fFIIp <- cpll(/- L ) } R ( I - L)~FII;

=

%IIR(I -

L ) ~ F I I p < c;ll(I - L ) ~ F I I p = c;lllFIIl,+,,p,

which implies the desired continuity.

[]

The following inequality is also an immediate consequence of Meyer's inequalities. P r o p o s i t i o n 2.4.4 Let u be an element ofIDl'V(H), p > 1. Then we have II~(u)ll~ < c,

(llJoullH + IIDulIL~(~;H|

.

(2.22)

Proof." From Proposition 2.4.3 we know that the operator ~ is continuous from ID~'P(H) into LP(f~). This implies that II~(u)llp _< cp

(IlulIL~r

+ IIDulIL,(a;H|

.

On the other hand, we have

IlulIL~ <_ IIJ0ulIH

+ Ilu --

JoUlIL~
and

]]u- JoU[[L~(a;H) =

]](I-- L)- 89RCU]]L~(a;H) A %]]Cu[[L,(a;H)

< c~llD~llL~(a.| where R--

(1 +

Jo

[]

The next lemmas are often useful. L e m m a 2.4.2 Let [Fn, n > 1} be a sequence of random variables converging to F in L p, for some p > 1. Suppose that SUPn HFnHs,p < oo for some s > O. Then F C ]Ds'p.

Proof:

We know that

sup I1(I

-

L)~F~IIp <

oo.

n

Therefore, there exists a subsequence F~(~) such that ( I - L) f F~(~) converges weakly in a ( L p, L q) (where q is the conjugate of p) to some element G. Then, for any polynomial random variable R we have

E[F(I-

L)~R]

=

l i m E [ F n ( , ) ( I - L)~R]

=

limE[(/-

L)~F~(i)R] = E[GR].

Thus, F = (I - L ) - ~ G , and this implies that F E IDs'p.

[]

153 L e m m a 2.4.3 Let ~ C ID- ~ be such that (~?,F) >_ 0 for any nonnegative smooth random variable F E $. Fix a complete orthonormal system {ei, i >_ 1} in H, and let g be the a-field generated by the random variables {W(e~),i > 1}. Then there exists

a finite measure #7 on the a-field G such that /a Fd#~ = (~7,F},

(2.23)

for any random variable F of the f o , ~ F = f ( W ( e l ) , . . . , W ( e , ) ) , n >_ 1, f C~(IRn). In particular #,(f~) = <~, 1>. Proof:

The proof will be done in two steps:

Step 1: Consider the increasing sequence of a-fields .7-n = a { W ( e l ) , . . . , W(en)}. We claim t h a t there exists a measure u. on the a-field 5c~ such t h a t (2.23) holds for any random variable F of the form F = f ( W ( e l ) , . . . , W(e,~)), where f E C~(IR'~). Indeed, consider the function ~ : IRn ~ C defined by

()t

=

77,exp z

P

ej

.

j=l

This mapping is nonnegative definite and continuous, because for all t l , . . . , tN E IR~, Cl,... ,CN G C we have

.oxp(

l,k=l

k=l

j=l

))

Therefore, there exists a finite measure ~,~ on IRa which has moments of all orders and such t h a t ]~. f(x)-P~(dx) = <77,f ( W ( e l ) , . . . , W(e~))), for any function f in C~(IRn). This implies the existence of a finite measure #7 on hen such that (2.23) holds for any F of the form f ( W ( e l ) , . . . , W(en)) with f C C ~ ( I R ' ) .

Step 2: It suffices to show t h a t #7 can be extended to a measure on G. Consider the random variable G = ~n~=l n-2W(en) 2. This variable belongs to ID~176Let r be a smooth function such t h a t r = 0 is x < 0, r = 1 is x _ 1, and 0 < r < 1. For any a > 1, set ~ ( x ) = r - (a - 1))+). For any random variable F of the form f ( W ( e ~ ) , . . . , W(en)) with f E C ~ ( I R n) and such t h a t 0 _< F _< 1, we can write L Fdpn =

<~7,F> = <~, F ( 1 - (~a(G))> -[- (?'],F~ga(G))

_< (r/, 1>liP(1 - ~,,(C))ll~ + <,~,~o(c)>. Hence, the inequality

L Fd#'7<- (rl' I>IIF(1

- V'a(G))l]~ + (7, ~oa(G)>

(2.24)

holds for any F in the class/2 := { f ( W ( e , ) , . . . , W(en)), n > 1, 0 < f < 1, f Borel}. In order to complete the proof of the lemma and taking into account Daniell's theorem, it suffices to show t h a t given a sequence of random variables {Fro n >

154

1} C s

uniformly bounded by one, such that 0 ___ Fn(w) I 0 for all ~ C ~, then

f~ F,~d#n I 0. We can write /~ Fnd#, 7 < (rl, 1)llFn(1 - ~a(G))ll~r + (7, 99a(G))

(2.25)

Suppose that ~ E ]D -k'p. The second summand in the right-hand side of (2.25) can be estimated as follows: (V, ~a(G)) = ((I - L)-+r/, (I - L)~9~a(G)> <

Ill~lll-~,~ll(I

- L)+~a(G)Hq,

where ~ + ] = 1, and this tends to zero as a tends to infinity. The first summand converges to zero as n tends to infinity due to Dini's theorem and the fact that the [] set {{Xn, n > 1} : ~o~ 2-,n=1n -2 xn2 < a} is compact in t 2. B i b l i o g r a p h i c a l n o t e s : The fact that the positive distributions on the Wiener space are measures has been proved in the context of an abstract Wiener space by Sugita in [103]. Other contributions to this problem are [79], [110], [92] and [82]. Spaces of Banach-valued Wiener functionals have been studied in [29], [64] and [42].

Chapter 3 Application of Malliavin calculus to study probability laws In this chapter we will discuss the application of the stochastic calculus of variations on a Gaussian space to the study of different properties of probability distributions of functionals of the underlying Gaussian process. The type of properties that we can analyze with the help of the Malliavin calculus include the absolute continuity, smoothness of the density, estimates (uniform and i n / 2 ) of the density, and application of these estimates to deduce Krylov-type inequalities.

3.1

C o m p u t a t i o n of probability densities

Suppose that W = {W(h), h E H} is a centered Gaussian family associated with a real and separable Hilbert space, defined on a probability space (~, 5~, P). We assume that that the a-field 5~ is generated by W. Suppose that F -- ( F 1 , . . . , F m) is a random vector whose components belong to 1,1 the space lDloc. We associate with F the following random symmetric nonnegative definite matrix: ~ F = ( ( D F i, D F J ) H ) I < _ i , j < m 9 This matrix is called the MaUiavin matrix of the random vector F. The following theorem was established by Bouleau and Hirsch [16] and provides a criterion for the absolute continuity of the law of F. T h e o r e m 3.1.1 Let F = ( F 1 , . . . , F m) be a random vector whose components belong 1,p to the space lD~oc, p > 1, and suppose that the random matrix vF = ( ( D F ~, DFJ) )l<_i,j<_m is invertible a.s. Then the law of F is absolutely continuous with respect to the Lebesgue measure on IRm.

In dimension one the nondegeneracy condition required in the previous theorem reduces to IIDFll g > 0 a.s. The proof of Theorem 3.1.1 is based on the coarea formula. A simple proof in dimension one can be given as follows (see [81]). Proof of Theorem 3.1.1 for m = 1: We can assume that F belongs to the space ]DI'p and that IFI < k for some constant k. We have to show that for any measurable function g : ( - k , k) --* [0, 1] such that fk k g(y)dy = 0 we have E ( g ( F ) ) = O. We can find a sequence of continuously differentiable functions with bounded derivatives

156

g'~: ( - k , k) --* [0, 1] such that as n tends to infinity g'(y) converges to g(y) for almost all y with respect to the measure P o F -1 + A, where A is the Lebesgue measure. Set

r

=

and

~(Y) =

F g"(x)d~,

y c ( - k , k)

k

f

f

k .q(z)dz,

y c (-k, k).

By the chain rule, Cn(F) belongs to the space ]D I'p and we have D[r = gn(F)DF. We have that Cn(F) converges to r a.s. as n tends to infinity, because g" converges to g a.e. with respect to the Lebesgue measure. This convergence also holds in LP(ft) by dominated convergence. On the other hand, D[r converges a.s. to g ( F ) D F because gn converges to g a.e. with respect to the law of F. Again by dominated convergence, this convergence holds in LP(ft; H). Observe that r = 0 a.s. Now we use the property that the operator D is closed to deduce that g ( F ) D F = 0 a.s. Consequently, g(F) = 0 a.s., which completes the proof of the theorem. [] We remark that the above proof also works with p = 1. The following proposition provides an explicit expression for the density of a one-dimensional random variable. P r o p o s i t i o n 3.1.1 Let F be a random variable in the space ]D 1'2. Suppose that ~DF

belongs to the domain of the operator ~ in L 2. Then the law of F has a continuous and bounded density given by DF

Proof: Let r be a nonnegative smooth function with compact support, and set p(y) = ff_~ ~(z)dz, y E IR. We know that ~ ( F ) belongs to ID1'2, and making the scalar product of its derivative with D F obtains (D(~(F)), DF}H =

r

Using the duality relationship between the operators D and 5 (see (1.7)), we obtain

E[r

= E [(D(~(F)), ,,DFN2H

= E [qo(F)5(~)].DF

By an approximation argument Eq. (3.2) holds for r we apply Fubini's theorem to get

(3.2)

= l[a,b](Y)- As a consequence,

DF b

which implies the desired result.

DF

[]

t 57 Proposition 3.1.1 also holds under the hypotheses F C ]D 1'1, and ~ E D F ]DI.P(H) for some p,p' > 1. From expression (3.1) we can deduce estimates for the density. Fix p and q such t h a t ~1 + ~1 = 1. By H61der's inequality we obtain

p(x) < (p(F > x))l/q,,6 (

D~)

,,p.

In the same way and taking into account the relation E[6(DF/llDFll2H)] = 0 we can deduce the inequality

p(x) < ( P ( r <

x))'/qll~ ~

II~-

As a consequence, we get

p(x) < (P(IFI > Ixl))l/qll5

~

IIp,

(3.3)

for all x G JR. Now using the LP-estimate of the operator 6 established in Proposition 2.4.4 we obtain

We have

(

) -IIDFII,~

D ~

'

IIDFII~

and, hence,

DF D ( ~ )

H|

< R_

,IDF,[2H

.

(3.5)

Finally, from the inequalities (3.3), (3.4) and (3.5) we deduce the following result.

t + 1 = 1. Let L e m m a 3.1.1 Let q, a,/3 be three positive real numbers such that ~1 + -& F be a random variable in the space ID2'~, such that E(IIDFIIH2~) < oo. Then the density p(x) can be estimated as follows:

p(x) < %,~,z(P([F I > Ix[)) 1/q

(E(IIDFII~ 1) + IID2FIILo(a;H|

(3.6)

Let us apply the preceding lemma to a Brownian martingale. Example: We will assume t h a t W = {Wt, t C [0, T]} is a Brownian motion, and H = {Hi, t E [t3,T]} is an a d a p t e d process verifying the following hypotheses: (i) E foT H~ds < oo, Ht belongs to the space ID2'2 for each t C [0, T], and A ==

sup s ,rE [0,T]

for some p > 3.

E ( I D ~ H t l p) +

sup r,s e [0,T]

E

((/0~ [D~,~Htl2dt) w2)

< c~,

158

(ii) IH, I > p > 0 for some constant p. Consider the martingale Mt = f~ HsdWs, and denote by pt(x) the probability density of Mr. Then the following estimate holds

P(IMt[>

pt(x) <_

Ixl)Z,

(3.7)

where q > p-~-a" The constant c depends on A, p, and p.

Proof of (3.7): We will apply Lemma 3.1.1 to the random variable Mr. We claim that Mt C lD 2'2 for each t E [0, T]. In fact, note first that Mt E ]131'2 because the process H belongs to ]D2'2(H), due to condition (i), and the operator ~ is continuous from ]D2'2(H) into ]131'2. Furthermore, using (1.10) we get DsMt = H,l{s
DsMt

=

Hs +

D~Mt

= O,

//

DsHrdW~,

s <_ t,

(3.8)

s>t.

Notice that D~IH. E IDI'2(H) for all sl a.e., the process Ds2DsIH. belongs to the domain of ~ for all sl, s2 a.e. (because it is adapted and square integrable), and

E

//

16(D~2D~H.)12ds2 = E

////

ID~D~Htl2ds2dt < oo.

As a consequence, from property (1.10) deduce that for all sl a.e. 6(D~H.) belongs to ID1'2 and

D,2(6(Ds, H.)) = D , , H , 2 + 6(Ds2D~,H.). Hence, Mt C ]D='2 and from (3.8) we deduce

D~,~2Mt = Ds~H82 + D~2H~ ~ +

~v~2Ds~'~H~dW~'

sl, s2 <_ t.

We will take a = p in Lemma 3.1.1. Using H61der's and Burkholder's inequalities we obtain E(IID2MtI[P~H)

=

s2)pj2)

E

s2) )

<_ q, § !

<_ Cp A tp.

t

2

2

159

Set

o(t) := IIOMdl~ =

H, +

//

DsHTdW~)2ds.

We have the following lower estimates for a for any h < 1:

a(t) >_ ftlX_h )(Hs + ~tD, H~dW~)2ds > --

H2ds l-h)

D~H~dW.

ds >

(l-h)

--

2

- Ih(t),

where

Ih(t) = f~il-h)

D~H~d

ds.

Choose h of the form h = ~tp:y, and notice that h _< 1 if y _> a := t-~2. We have

P (a(t) <_ ~) < P (Ih(t) > ~) <_yp/2E(,lh(t),P/2).

(3.9)

Using Burkholder's inequality for square integrable martingales we get the following estimate [/tt

t

2

\p/2\

< Cp sup E(]D,Hr[P)(th) p. I

(3.10)

s,r~[0.t]

Consequently, for 0 < ~, < ~ we obtain, using (3.9) and (3.10),

E[a(t) -7] =

/7 ~,yT-1P(a(t)-I > y)dy

<_ a~ + 7

y~-~P(o(t) <

)ay

<- ( t ~ ) ~ + 7 f,~ E(lIh(t)F/2) YT-I+~dY

< c t -~ +

y~-l-~dy

< c' t -'Y+t ~-~ .

(3.11)

Substituting (3.11) in Eq. (3.6) with a = p, /3 < e2, and with 3' = ~1 and "Y = /3, we get the desired estimation. [] The inequality (3.7) implies

pt(x) <_c

E

(/:

~

IH~12ds

.

If we assume, in addition that the process H is bounded by a constant M, that is, instead of (ii) we impose: (ii)'

M >_ IHt] >_ p > 0 for some constants p and M,

then using the martingale exponential inequality we obtain 1 exp(_ q IzJ ~ 22t ) "

(3.12)

160

3.2

R e g u l a r i t y of d e n s i t i e s a n d c o m p o s i t i o n o f ternp e r e d d i s t r i b u t i o n s w i t h e l e m e n t s of ID- ~

The results obtained in the last section for onedimensional random variables can be extended to the multidimensional case. Furthermore, under additional smoothness and integrability conditions one can show that the probability density of a random vector F is infinitely differentiable. The basic assumptions are introduced in the following definition of nondegenerate random vector. D e f i n i t i o n 3.2.1 We will say that the random vector F = ( F 1 , . . . , F TM) E (lDo*)m is nondegenerate if the matrix 7F is invertible a.s. and

(detTF) - l e

A LP(~) 9

(3.13)

p>l

For a nondegenerate random vector the following integration by parts formula plays a basic role. P r o p o s i t i o n 3.2.1 Let F = (F 1. . . . , F "~) 6 (lD~) 'n be a nondegenerate random vector in the sense of Definition 3.2.1, let G E ]D~ and let g E C~(1Rm). Then (detTF) -1 6 I]9~176 and for any multi-index a 6 { 1 , . . . , m } k, k > 1, there exists an element Ha(F, G) E lDo* such that:

E[(O,g)(F)G] = E[g(F)H~(F, G)].

(3.14)

Moreover the elements Ha(F, G) are recurs@ely given by: m

H(,)(F,G)

E6

:

(G(TF1)i3DFO ,

j=l

Ha(F,G)

H~,(F,H(~ 1...... k_I)(F,G)).

=

Proof: Let us first show that (det 7F) -1 C lDo*. For any N > 1 we have (detTF + I ) -1 C lDo*, because the random variable (det7F + 1 ) - 1 can be written as the composition of det7F with a function in C~r It is not difficult to show using (3.13) that {(det 7F + ~1) -1 , N > 1} is a Cauchy sequence in the norms H' ]]k,p for all k,p, and we obtain the desired result. Also 7/1 E IDO*(IRm• By the chain rule we have rrt

D[g(F)] = ~-~(Oig)(F)DF i, i=1

hence, m

(D[g(F)], DFJ}H = ~"(Oig)(F)7 ij, i=1

and

as a consequence, m

(Oig)(F) = ~-~.(D[g(F)], DFJ)H(TF1) ii. j=l

Finally, taking expectations in the above equality and introducing the adjoint of the operator D we get E[G(Oig) (F)] = E[g(F)g(i)(F, G)],

161

where H(i)(F, G) : ~jm=l ~ (G(~F1)iJDFJ). We complete the proof by a recurrence argument. [] P r o p o s i t i o n 3.2.2 For any p > 1, and for any multi-index c~ there exist a constant

C(p, c~), natural numbers nl, n2, and indices k, d, d', b, b', depending also on p and ct such that ,.y l n]

IIH~(F,G)II~ _ c(p,~) (11 r I1~ IIFII2~IIGII~,,o,) 9 Proof. This estimate is an immediate consequence of the continuity of the operator 5 from IDk+l'p into IDk'p, HSlder's inequality for the II" Ilk,p-norms, and the equality:

D[(~;1) ~j] = _ ~ (~)~k(~l)J~Db~']. k,l=l

[] From Proposition 3.2.1 it follows that the density of a smooth and nondegenerated random vector is infinitely differentiable. We recall that S(IR TM) is the space of infinitely differentiable functions f : IRTM ~ IR such that for any k >_ 1, and for all multi-index/3 E {1,... ,m} j one has sup~ea~ IxlklO~f(x)l < oo. C o r o l l a r y 3.2.1 Let f = ( E l , . . . , F TM) C (ID~176 m be a nondegenerate random vector

in the sense of Definition 3.2.1. Then the density of F belongs to the Schwartz space S(]Rm), and p(x) = E[I{F>x}H(1,2 ....... )(F, 1)], (3.15) where H0,2 ...... )(F, 1) = 5(('TF1DF)mS((',/F1DF) m-1... ~ ( ( ' , / f f l D F ) I ) . - - ) .

Proof." Consider the multi-index a = (1, 2 , . . . , m). From (3.14) we obtain, for any function ~ C C~(IRm), E[(O~O)(F)] = E[{J(F)H~,(F, 1)]. By Pubini's theorem we can write

E[(O~r

=/~m(cq~J)(x)E[l{x
We can take a~r = 1B, where B is a bounded Borel subset of IRm, and in this way we deduce the absolute continuity of the law of F and the expression (3.15) for its density. Moreover, for any multi-index/3 we obtain

O~p(x) = E[O~(I{~
Finally in order to show that the

z~klE[I(~
1))]1 < oo,

162

for a l l j = l , . . . , m .

Ifxj >0wehave

x~kIE[I{=
p(x) = E [ H I{~FJ}H~(F, 1)], []

and we deduce a similar estimate.

Let F be an m-dimensional random vector. The probability density of F at a point x can be formally defined as the generalized expectation E[fi=(F)], where 6= denotes the Dirac function at x. The expression E[~=(F)] can be defined as the coupling ( 6 , ( F ) , 1), provided we show t h a t ~ ( F ) is an element of 119- ~ . The Dirac function ~ is a measure, and more generally we can define the composition T(F) of a Schwartz distribution T E S'(IR m) with a random variable in ID~ the result being a distribution in ID-~ This approach was introduced by W a t a n a b e in [107]. Furthermore, the differentiability of the mapping x --* 6 , ( F ) from IRm into some Sobolev space ID -k'p will provide an alternative proof of the smoothness of the density. Let us describe the main steps of this approach. sequence of seminorms on the Schwartz space S(1Rm): llCll2k = II(1 + Ixl 2 - A)kClloo,

We introduce the following

r ~ S(~m),

for k C Z. Let S2k, k e Z be the completion of S(IR m) by the seminorm II. 112k. Then we have $2k+2 C S2k C ... $2 C So C S-2 C ... C S-2k E S-2k-2, and So = C(IR m) is the space of continuous functions on IRm which vanish at infinity. Moreover, Ak>1 S2k = S(]Rm), and Uk>1 S-2k = S'(IRm). Proposition 3.2.3 Let F =

(FI,...,F m) E (lD~176 m be a nondegenerate random

vector in the sense of Definition 3.2.1. For any k > 1, p > 1, there exists a constant C(p, k, F) such that for any r E S(]R m) we have

IIr Proof:

_< C(p, k, F)llCll-2k-

By a duality principle we have = sup{E(r

IIr

IlCll2~,q < 1},

where q is the conjugate of p. Now using the expression (3.14) we can write

E(r

=

E[((1 + Ixl 2 - A)k(1 + Ixl 2 - A ) - k r

=

E[(1 + Ixl ~ - Z~)-%(F)n~k],

where ~2k ~ ID~176 verifies

C(p, k, F) := sup{E([~2k[), Ilall2~,q _< 1} < ~ ,

163

and this completes the proof.

[]

As a consequence, the mapping r ~ r can be extended uniquely to a continuous linear mapping T ---* T o F from $-2k to ID-2k'p for every p > 1 and k = 0, 1, 2 , . . . . In particular for every Schwartz distribution T E $'(]Rm), we can define a Wiener distribution T o F C ]D - ~ . Actually

T o F E U N ID -2k,p. k=lp>l

We have t h a t for any fixed point x E IRm the Dirac function belongs to S-2k if k > ~ , and the mapping x --~ 5x is 2j times continuously differentiable from IRm to S - 2 k - ~ . This implies t h a t we can define the "composition" ~x(F) as an element of ID -2k'p for any p > 1 and any integer k such that k > ~ . The density of the random vector F is then given by (6~(F), 1}. Moreover, for any r a n d o m variable G E ID~ and for any x such that p(x) 5r 0, we have

( ~ ( F ) , G) = p(x)E[G[F = x].

3.3

The case of diffusion processes

We can apply the previous results to derive the smoothness of the density for solutions to stochastic differential equations. This provides probabilistic arguments to study heat kernels. Suppose t h a t {W(t), t > 0} is a d-dimensional Brownian motion defined on the canonical probability space f~ = C0(N+; IRa). Let Aj : IR m --* IR,n, j = 0 , . . . , d a system of C ~ functions with bounded derivatives of all orders, and consider the following stochastic Stratonovich differential equation on IR'~: d X t = Y~j=I d A j ( X t ) o dW3t + Ao(Xt)dt, N o = Xo

(3.16)

One can show t h a t Xt E (IDa) m for all t _> 0. The following nondegeneracy condition assures t h a t for each t > 0 the random vector X~ is nondegenerate: (H) There exists an integer k0 _> 0 such that the vector space spanned by the vector fields [AJk, [AJk 1 , [ - " [A31,Ajo]]'.-], 0 < k < k0, where j0 E { 1 , 2 , . . . , d } , j~ E { 0 , 1 , 2 , . . . , d } dimension m.

if 1 < i < k, at point x0 has

In the above hypothesis [A, B] denotes the Lie bracket of the differentiable functions A, B : lR m ~ lRm, defined as

A i OB _ Bi OA ~ . i=i

164

T h e o r e m 3.3.1 Suppose that the above condition (H) is satisfied. Then there exists a positive integer u depending only on ko, and for each p > 1 a positive constant c(p, Xo) such that

H(det ~x,)-lllp _< ct-~,

for all t > O. This theorem is a consequence of the precise estimates obtained by Kusuoka and Stroock in [57].

3.4

Lp e s t i m a t e s o f t h e d e n s i t y a n d a p p l i c a t i o n s

The stochastic calculus of variations can be used to establish Krylov-type estimates which are a useful tool in deriving existence and uniqueness of a solution to stochastic differential equations and to partial stochastic differential equations whose diffusion coefficient is nondegenerate and the drift is not smooth. Let us first establish a preliminary result similar to Lemma 3.1.1. L e m m a 3.4.1 Let a, fl be two positive real numbers such that !~ + ~1 < 1. Let F be an m-dimensional random variable (rn > 1) whose components belong to the space 1D2'~ and E(l(7;l)ii[ ~) < oc, i = 1 . . . . , rn. Then for any nonnegative and measurable function f : IRTM ~ IR we have: E[f(E')] ~

Cc~,B,mHfHmSUiP(E(~r

_~=]]D2F]]L.(f~;H|174

.

Proof: We can assume that the function f is continuous and with compact support. Let r be an approximation of the identity in IRm. We can write E[f(F)]

ej.OJIR"~E [ % ( x -

=

limf

<

[[fll~limsup

<

Ilfll,~limsupI-[

(/.

el0

e,LO

F)]f(x)dx

i=1

[E[r

m

F)]l~-~-~dx

IE[O,r

F)lIdz

)'

Here we have used the Gagliardo-Nirenberg inequality which says that for any function f in the space C ~ ( I R m) one has

IIIIIL~--~_ -<

IlO, fllL~. i=1

Applying Proposition 3.2.1 we can write

E[Oi%(x - F)] = E[r

- F)Hi(F, 1)],

where

Hi(F, 1 ) = 6 ( j = ~ D F J ( ~ / F 1 ) i J ) .

(3.17)

165

Now, using (3.17) and (2.22) we obtain, for any p > 1 m

Elf(F)]

< IIfIIml](E(IH~(F, 1)I)) ~ i=1

<_ Cpllfllm I I

E

DFY(~x)'J)IIH +

4=1

m

+llD(~=DFJ(~/~)~J)IiLp(a;.|

))~1 9

1 We conclude the proof choosing p in such a way that ~1 = E1 + ?, applying H61der's inequality, and using the relations m

II ~ DFJ(yF~)~3]12H = (~'F~) ~,

j=l

and

m

IID(~ DFJ( ~ F 1 )

ij)

~-- II ~ j~l

II"|

j-1

D2FJ(77Ft)0 IIH|

+ll ~ DFJ | (~lykD((DFk,DFS).)(~I)~J)IL.| j,k,s=l

[] Lemma 3.4.1 allows us to deduce Krylov-type estimates for solutions to stochastic differential equations. Let {Wt,t E [0, 1]} be an m-dimensional Brownian motion. Consider the m-dimensional stochastic differential equation

dXt=f(X,t)dt+

m gi(X,t)dW~,~ Ei=l

t E [0,1],

(3.18)

X o = XO,

where f, gi: C([0, 1]; IRm) x [0, 1] ~ IRTM, i = 1 , . . . , m, are progressively measurable functions verifying the following conditions (a) If(x,t)L <_ K for some constant K. (b) g is twice Pr~chet differentiable in the first variable, with uniformly bounded first and second order derivatives, and g(0, t) is bounded. (c) There exist positive constants cl and c2 such that

c11012 <_ Ig(x,t)O] 2 <_ c210[2, for all 0 E IRm, x ~ C([0, 1]; lRm), and t E [0,1]. We will denote this equation by Eq(f, g). By a solution to this equation we will mean a continuous and adapted process that satisfies the corresponding stochastic integral equation.

166

P r o p o s i t i o n 3.4.1 Let g, : C([0, 1];lR m) • [0, 1] ~ ]Rm, i = 1,... ,m, be measurable functions verifying conditions (b) and (c), and let { Ft, t E [0, 1]} be an m-dimensional progressively measurable process bounded by K . We denote by {Xt, t C [0, 1]} the solution of the following equation (denoted in the sequel by Eq(F, g)): Xt = Xo +

So'

F, ds +

so'

g,(X, s)dWT,

t C [0, 1].

(3.19)

Then for any measurable function h : lR m • [0, 1] --+ JR, and for all p > m, 7 > 1, we have 1

where the constant C depends on p, 7, K , and the coefficient g. Proof:

Define a new probability/5 by dP = Z Z :

exp (--

where

LI[PtT(g-1)T(x,t)dWt- 2L 11(g-1)(X,t)Ftl2dt).

By Girsanov's theorem = Wt + L t ( g - 1 ) ( X , s ) F f l s

is a Brownian motion under /5. Thus, under /5 the process X has the same law as the process ]z solution of t C [0, 1],

dYt = g(Y,t)dWt,

(3.21)

Y0 = X0~ By H61der's inequality

1

E(L'ih(Xt, t)idt)

1 for every a, fl > 1 such that ~ + ~1 = 1. The random vector Yt belongs to the space ID2'~ for all p > 1 (see Lemma 3.4.2 below). When m = 1 one can use the estimate on the density of Yt obtained in (3.12). Suppose m > 1. Applying Lemma 3.4.1 to the random variable Yt we obtain if ~1 § y1 = i

E (/o' ih( 't)l dt) • (E ( ~ )

(L _<

1 '

+ IID~ll<~,ll(%')"ll~,)dt).

The inequality (3.25) allows to control the term IID2~II<~, by a constant times t. On the other hand, using hypothesis (c) one can show (a sketch of the proof is given below) that E([(7~l)i'] p) < a3t -p (3.22)

167

for all p _> 1. Hence,

1

~

1

1

- cm,~,,Z,

m

]h(y,t)[mZdy

dt.

This allows to complete the proof of the Proposition.

Sketch of the proof of (3.22): First we write E([(3%1)ii] v) _< cm.v [E(HDY, H~(m-1))] 89[E((det 7v~)-2v)] 899 From (3.24) we get that the first factor in the right-hand side of the above equality is bounded by a constant times t p(m-1). In order to estimate the second factor we write detTv~ > inf (vTTv~V) rn= inf [[(v, DYt)[[2Hm. -M= 1 M=I From (3.28) we get

(v,D~)= (v,g~(Y,s))+(v,I t(V(Z,r),(D~))aWr), j = 1,...,~.

(3.23)

Consequently, we deduce, with the convention of summation over repeated indices, for a n y h < 1,

R

~

(1-h) k=l t

t

.

2

k=l

> __Clth_ f t --

2

Jr(l-h)

where

[J(s)[2ds,

-ji'((9'),,. (Y, ,-),

JJ(s) =

D~Y)dWr

Now, using the same technique as in the proof of (3.7) we can show that

E([~ldlII(v, DYt)l]~4~)

<_ E

-

(l-h)

Id(s)[2ds

<_ a4t-2pm,

and this allows us to conclude the proof. Remark:

[]

In the preceding proposition we can take ~, = 1 when m = 2.

L e m m a 3.4.2 Let {Y,, t E [0, 1]} be the solution to equation Eq(0, g) where the function g satisfies hypothesis (b). Then Yt E Mp>l ID2'p, and

E(IIDF, II~) < altO, E(IID2~II~| < a2t p, for some constants al and a2.

(3.24)

(3.25)

168

Proof." We will denote by g' and by g" the Pr6chet derivatives of the function g. We introduce the sequence of Picard approximations defined recursively by y0 = x0, and Ytn+l = xo +

g(Y ,s)dWs,

n >_ O.

Then for each n the random variable Yt~+1 belongs to Np>I IDI'P, and

DJVn+ s*t 1

gj(Y'~,s)+ f t (g'(}~',r),(D~Yn))dWr,

j= l,...,m,

where DY denotes the derivative with respect to the j t h component of the Brownian motion. That is, D~W~ = l{,<_t}Sik. We have linm E[II r ~ - r l l ~ ] = 0,

for all p > 1, and using Grownall's lemma we deduce sup sup n

E[IIDW"+lff~]<

oo,

(3.26)

sE[0,1l

and sup sup n

s,rE[0,1]

2 n+l [[oo3 p < E[IID,,rY

c~,

(3.27)

for all p > 1. This implies that Yt E flv>l IDZP. Finally we use the chain rule (properly extended to C([0, 1])-valued functions) in order to compute the derivative of g(Y, t), and we obtain:

D~Yt = gj(Y, s) +

2

(g'(V, r), (DgY)>dWr,

j = 1,..., m.

(3.28)

Using Burkholder's inequality we can show (3.24) from the expression (3.28). Similar computations can be done for the second derivative. [] Notice that the inequality (3.20) also holds for a process X such for each t E [0, 1],

Xt is the almost sure limit of a sequence of random variables X~ such that X n solves an equation of the form (3.19) with coefficients g and Fn, where Fn is bounded by K. We can now to derive a convergence criterion for solutions to equation (3.19): C o r o l l a r y 3.4.1 Let hn : IRm x [0, 1] --~ IR be a sequence of measurable functions

uniformly bounded by a constant K1 and converging a.e. to h. Consider a sequence of processes X n solutions of equations Eq(Fn, g), where g satisfies conditions (b) and (c) and Fn is a progressively measurable process bounded by K. Then linmE(fo*lh,~(X:,t)-h(Xt, t)]dt ) = 0 . Proof:

Set

Fix r / > 0. Let r be a nonnegative smooth function with support included in [-1, 1], r = 1 and bounded by 1. Choose R > 0 such that E (/01 ( 1 - r ( ~ ) )

dt) < ~ 9

169

Consider a continuous function 9R,v on IR x [0, 1] bounded by 2KI such that g(x, t) = 0 for all Ix I > R and for a fixed p > m

o

,1]

Ih(x't)-gR'~(x't)lPdx

dt <-Tl~'

where p > m and 3' > 1 are the exponents appearing in Proposition 3.4.1. Consider the following decomposition

J~=J~+J~+J3+J4n, where

J1n = E ( f o l l h n ( X : , t ) - h ( X ~ , t ) l d t )

J• = E(follh(X,,t)-gR,v(Xt,t)ldt)

J: = Z(follg~,~(x:,t)-g~,.(x.t)ldt) We write

-- E

(/o'r ( ~ :Ih~(Xt, ) ) t ) - h(Xg, t)ldt

+ E(fo 1 (1-~(X--~R))lhn(X2,t)-h(Xg,

t)ldt ) ,

and we make similar decompositions for the terms 3~ and Jn3. In this way we obtain, using Proposition 3.4.1, lira sup J~ n

~("m:~P(/o~(/ "~)'h~(x ~)--~ ~)'~)~0 ~ 1

+limsupE(fol[gR,~(Xt,t)-gR,~(Xt, t)ldt)). Hence, lim sup Jn < Cr/. n

[] C o r o l l a r y 3.4.2 Let gi : C([O, 1];JRm) x [0, 1] --~ ]R, 1 < i < m, be measurable functions satisfying conditions (b) and (c) of Proposition 3.4.1. Consider sequences of

170 progressively measurable processes F, Fn : [0, 1] x ~ ~ ]Rm, and measurable functions f, fn : [0, 1] x ]R --* ]Rm such that TM

limFn(t) n

limf~(t,x)

=

F(t),

dt|

= f(t,x),

a.e.,

dt|

Suppose that for each n > 1 equation Eq(Fm g) admits a solution X ~, and for each t E [0, 1], X~ converges a.s. to a process Xt. Then X solves equation Eq(F,g).

Proof:

It suffices to pass to the limit each term in the equation

x

=x0+ f 0 t F,(s)ds+

rn

[

t

i = l J0

[] Applying the preceding results we can derive existence and uniqueness results for equations of the form Eq(f, g) where f is a measurable and bounded function of Xt and g is a smooth (up to the second order) nondegenerate function. To do this one usually makes use of comparison theorems for stochastic differential equations, and for this reason, one has to consider particular type of equations. In order to illustrate this method we will describe the onedimensional case. P r o p o s i t i o n 3.4.2 Suppose that f : IR x [0, 1] ~ ]R is a measurable function bounded by K and g : IR x [0, 1] ---* IR is twice continuously differentiabIe, with bounded deriva-

tives and such that 0 < cl <_ [9(x,t)l <_ c2 < oo. Then equation (3.18) has a unique solution. Proof of the existence: Let p be a smooth nonnegative function with compact support in ]R such that f~t p(x)dx = 1. Define (x, t) = j / ~ f ( z , t)p(j(x - z))dz. Moreover, let k

~

A,k-- AfJ,

n<_k

j=n

and j~n

Cleraly, f,,a(x, t) is Lipschitz in x uniformly with respect to t, f,,k(x, t) I fn(x, t) as k 1" oc, and f , ( x , t) I f ( x , t) as n T oc, dx-a.e, for each t. For each n > k equation Eq(fn,k, g) has a unique solution Xn,k. From the comparison theorem for stochastic differential equations the sequence {X,,k, k = n, n + 1,...} is decreasing. Hence it has a limit X~ = lim Xn,k. k~cx~

Since X~,k is bounded above by the solution of Eq(K, g) and below by the solution of E q ( - K , g), we can apply Corollary 3.4.1 and deduce that

lim _~A,k,X,~,k,t,t,dt k Jo

=

/o'A(X~(t),t)dt,

a.s..

171

On the other hand, by the continuity of the function g we get limf0tg(X,,k(t),t)dWt = fotg(Xn(t),t)dWt,

a.s..

Hence the process Xn is a solution to Eq(f~, g).

Proof of the uniqueness: The proof of the uniqueness has been inspired by the work [36]. Let X denote the solution constructed as the limit of the sequence X~, and let Z be another solution. We can write with the above notations dZt =

Fj(t)dt + fj(Zt, t)dt + g(Zt, t)dWt,

where

['j(t) = f(Zt, t) - ]j(Zt,t),

,j > 1.

Let us denote by Zmk ,_ and by Zn,k,+ the solutions of the following equations

dZ~,k,_(t) =

(t)

A Odt + f~,k(Zn,k_(t),t)dt + g(Z~,k_(t),t)dWt,

dZ.,k,+(t) =

(t)

v 0 dt + f.,k(Z.,~,+(t), t)dt + g(Z.,~,+(t), t)dW,.

and

By the comparison theorem we have Zn,k,- < Xn,k < Zn,k,+, and Zn,k_ < Z < Z~,k,+, Moreover, the sequences {Z~,k-, k _> n} and {Z~,k,+, k > n} are decreasing in k. Hence, they converge to some limits {Z~_, k > n} and {Z~,+, k _> n}, respectively, which are increasing in n. Let us denote by {Z , k _> n} and {Z+, k _> n} the limit of these sequences. We have Z_ _< X, Z _< Z+. (3.29) The coefficients Aj=, k /g'j(t) and fn,k converge almost everywhere to 0 and f, respectively, as k and n tend to infinity. Consequently, applying Corollary 3.4.2 we get that the processes Z_ and Z+ solve (3.18). But from Girsanov's theorem, the law of (3.18) is unique. Hence, from (3.29) we deduce that Z_ = X = Z = Z+ which completes the proof of the uniqueness. [] This approach has been used to derive existence and uniqueness of the solution and approximation of the implicit approximation scheme for the unidimensional heat equation perturbed by a space-time white noise. Let us describe the results obtained for these equations.

172

Let { W ( s , t) = W([0, s] x [0, t]), s, t, E [0, 1]} be a two-parameter Wiener process. Let us consider the stochastic partial differential equation

Ou 02u Ot - Ox 2 + f(t, x, u(t, x)) + g(t, x, u(t, x)) OtOxO2W

(3.30)

with Dirichlet boundary conditions u(t,0)=u(t,

1)=0,

tC[0,1];

(3.31)

and with initial condition

u (O,x) = Uo(X),

x e [0, 1],

(3.32)

where u0 is a continuous function on [0, 1] vanishing at 0 and 1. The coefficients f (t, x, r) and g(t, x, r) are locally bounded Borel functions mapping [0, 1]2 • ]R into IR. The existence and uniqueness of the strong solution is well known when f and g satisfy the linear growth condition and are Lipschitz in r. Consider the following hypotheses on the coefficients of Eq. (3.30): ( H 1 ) g has a Lipschitz continuous derivative, it has linear growth, and it satisfies the nondegeneracy condition g2 > r > 0. f is satisfies the one-sided linear growth condition r f ( t , x, r) < C(1 + r2). Under hypothesis (H) the existence and uniqueness of the solution for equation (3.30) has been established in [5]. The main tool used in this paper is a n / 2 estimate, obtained using Malliavin calculus, similar to Eq. (3.20). This existence and uniqueness result is improved in [36], where the smoothness condition on g is replaced by the local Lipschitzness of g in r. Consider the implicit approximation scheme for the equation (3.30). For every integer n > 1 we construct a random field u s in the following way:

x)

=

o(x)

un(tin+l, X) = + +

Z --

A

un(t n, X)

fir+, G n ( x , y ) f ( s , x ,

unct, , ,

Jr?

ftr+l f l a ~ ( x , y ) g ( s , y , Jt?- JO

un(t'~,y))W(ds, dy),

i >_ O,

where t r := 4, 0 < i < n, G~ is the kernel of the operator ( I - -~A ) -1, and A := ~o.2 When t E (t~, t~+l) we define un(t, x) by the polygonal approximation

un(t,x) = U ( t ~i , x ) + n ( t - - t

n) (uWt t t ~i+l, x) - un(t, ni , x ) ) 9

The sequence un is called an implicit approximation scheme for the equation (3.30) because we can write:

u"(tT+l)

=

1

,n,

un(tT) + - A u ~ t ~ + ~ j n

+ ftT+ l f(s,u~(tT))ds Jt7

-I- f t:+l L 1 g(8, un(tn))W(ds, dy). Jr?

173

Under hypothesis (H) one can show (see [37]) that l i m P ( sup n \*,tc[0J}

I~~

~(~,x)l > ~]] = 0,

for all e > 0. The main ingredient of the proof is the following / 2 estimate for the density p~,xof the law of un(t, x). We obtain it using L e m m a 3.1.h sup ~ 1 ~ 0 1 ~

n

n ~ ~o( t , x ) pt,z(r)

drdxdt~()o

for any function qo E C~176(0, 1) 2) with compact support, and for all ~ > 1. B i b l i o g r a p h i c a l n o t e s : For an m-dimensional nondegenerate random vector F, Watanabe has precised the order (s,p) of the negative Sobolev space IDs'p which contains the distribution ~=(F) (cf. [109]). The asymptotic expansion of Xt(eco) using Malliavin calculus has been studied in [108].

Chapter 4 Support theorems In this chapter we apply the stochastic calculus of variations to study the properties of the support of the law of a random vector. We also discuss the characterization of the support of the law using the so-called skeleton approximation, and the application of the Malliavin calculus to the proof of Varadhan-type estimates.

4.1

Properties of the support

Given a random variable F : ~ ~ S with values on a Polish space S, the topological support of the probability distribution of F is defined as the set of points x E S such that P(d(F, x) < ~) > 0 for all c > 0. The connected property of the support of the law of a finite-dimensional random variable vector was established by Fang [28]: P r o p o s i t i o n 4.1.1 Let F E (]DI'P)m for some p > 1. Then the topological support of

the law of F is a closed connected subset of IRm. Proof: Suppose that supp P o F -1 is not connected. There exists two nonempty disjoint closed sets A and B such that supp P o F -1 = A U B. For each integer M >_ 2 let CM : IRm --* IR be an infinitely differentiable function such that 0 < ~2M _~ 1, ~)M(X) = 0 if IXl > M, ~)M(X) 1 if Ixl < M - 1, and SUPx,M IVCM(X)I < ce. Set AM = AN{Ix I ~_ M} and BM = B(3{Ix [ ~_ M}. For M large enough we have AM ~ 0 and BM ~ O, and we can find an infinitely differentiable function fM such that 0 < fM ~_ I, fM = 1 in a neighborhood of AM, and fM = 0 in a neighborhood of =

BM. The sequence (fM~M)(F) converges a.s. and in L p to I{F~A} as M tends to infinity. On the other hand, we have m

D[(fMOM)(F)]

m

= ~(r

i + ~(fMOiOM)(F)DF ~

i=1 m

i=1

= ~-~(IMOiOM)(F)DF i i=1

Hence,

supIID[(fMOM)(F)]IIH M

~

~8up IIO~MII~IIDF~IIH ~ L '~. i=1

M

175

By Lemma 2.4.2 we obtain t h a t l{FcA} belongs to ID I'p and, due to Lemma 1.4.2, this is contradictory because 0 < P(F E A) < 1. [] As a consequence, the support of the law of a random variable F E ]D I'p, p > 1 is a closed interval. The next results provides sufficient conditions for the density of F to be nonzero in the interior of the support. P r o p o s i t i o n 4.1.2 Let F E IDI'p, p > 2, and suppose that F possesses a locally

Lipsehitz density p(x). Let a be a point in the interior of the support of the law of F. Then p(a) > O. Proof:

Suppose p(a) = 0.

Set r = ~p + 2

> 1.

By Lemma 1.4.2 we know t h a t

I{F>~} r ID ~'~ because 0 < P ( F > a) < 1. Fix c > 0 and set

Then qo~(F) converges to l{F>a } in L~(f~), as e $ 0. Moreover, ~ ( F ) C lD 1'~ and 1

D(qo,(F)) = ~-~1[. . . . . +4(F)DF. We have

E(IID(~,(F))II~H) <_ (E(IIDFIIH))p +2

/~_~ p(x)dx

The local Lipschitz p r o p e r t y of p implies t h a t p(x) <_ KIx - al, and we obtain --

P

2

E (llD(~,(F))ll~)< (E(IIDFIIH)) ~ 2 - r g ~+~. By Lemma 2.4.2 this implies l{f>a} E

ID l'r

and we are in contradiction.

[3

The following example shows that, unlike the one-dimensional case, in dimension m > 1 the density of a nondegenerate random vector can vanish in the interior of the support. Example:

Let hi and h2 be two orthonormal elements of H, and define X1

=

arctanW(hl),

X2

=

arctanW(h2).

Then X1, X2 C ID~ and

DXi = (1 + W(h~)2)-lh~, for i = 1, 2, and get 3,x = (1 + W(hl)2)-2(1 + W(h2)2) -2, where 3'x denotes the Malliavin m a t r i x of the nondegenerate random vector X = (X1,X2). Notice t h a t the support of the law of X is the rectangle [ - 2 , ~ ]2, and

176

the density of X is strictly positive in the interior of the support. Now consider the following vector ]/1

( X l ~- ~3r- ) cos(2X2 + it)

:

( X x + ~ -3~r ) sin(2X2 + 7@

Y,~ = We have t h a t Y E (]D~176 2 and

q ~r

d e t 7 u = 4(X, + ~2)2(1 + W(ha)2)-2(1 + W(h2)2) -2. This implies t h a t Y is a nondegenerate vector. Its support is the set {(x, y) : 7r2 _< x 2 + y2 < 47r2}, and the density of Y vanishes on the points (x, y) in the support such that 7r < y < 2% and x = 0. For a smooth and nondegenerate random vector when the density vanishes, then all its partial derivatives also vanish (see Ben Arous and L6andre [9]). P r o p o s i t i o n 4 . 1 . 3 Let F = ( F 1 , . . . ,

Y m) E (]D~176 be a nondegenerate random vector in the sense of Definition 3.2.1, and denote its density by p(x). Then p(x) = 0 implies O~p(x) = 0 for any multiindex a. TM

Proof: Suppose t h a t p(x) = 0. We know t h a t p(x) = (6~(F), 1). We have that (5=(F), R) >_ 0 for any smooth random variable R because

(~(F), R) = ~o E(r

- F)R),

where ~r is an approximation of the identity. So, from Lemma 2.4.3 there exists a measure r/~ on the ~-field g generated by { W ( e i ) , i _> 1} such t h a t (5~(F), R} = fa Rdrl~ for any g-measurable and smooth random variable R. Notice that p(x) = r/~(ft). Therefore, r/= = 0, which implies that 5=(F) = 0 as an element of ID -~176For any multiindex a we have

Oc~p(x) = Oc~(5~(F), 1) = ((O~5x)(F), 1). Hence, it suffices to show that (O~5=)(F) vanishes. Suppose first t h a t a = (i). We can write m

D(5~(F)) = ~-~(O,~5~)(F)DU, i=1

as elements of ID - ~ , which implies m

(O,6=)(F) = ~-~.(D[5=(F)], DFJ)H("/F1) ji = O, j=l

because D [ ~ ( F ) ] = 0. The general case follows by recurrence.

[]

In the case of a diffusion processes, the characterization of the support of the law is obtained by means of the notion of skeleton. A general notion of skeleton is provided by thc following definition. In the sequel we will assume t h a t {gl N, N _> 1} is a sequence of orthogonal projections on H of finite-dimensional range, which converges strongly to the identity. If { e l , . . . , eN} is an orthonormal basis of the image on H N we set H N w = EN=I W(ei)ei.

177

D e f i n i t i o n 4.1.1 Let F : f~ ~ S be a random variable taking values on a Polish space space (S, d). We will say that a measurable function 9 : H ~ S is a skeleton of F if the following two conditions are satisfied." (i) For all e > 0 we have liNmP { d(O~(yINW), F) > ~} = O. (ii) For all h C H, there exists a sequence of measurable transformations T h : ~2 ~ f~ such that P o (Th) -1 is absolutely continuous with respect to P, and for every e>0 l i m s u p P { d ( F o T h , ~ ( h ) ) < c} > 0 . N

P r o p o s i t i o n 4.1.4 Suppose that ~2 is a skeleton o f F in the sense of Definition 4.1.1. Then the support of the law of F is the closure in S of the set {~(h), h ~ H}.

Proof: Step 1: Let us first show that condition (i) implies the inclusion supp (P o F -1) C q~(H). It suffices to show that P ( F E ~5(H)) = 1, and this follows from

P {d(F,O(H))<_ e} >_ P {d(~(gINW),F)<_ ~}--~ 1, as N tends to infinity. Letting c tend to zero yields the result.

Step 2: We have to check that for each h C H and each e > 0, P {d(F, qo(h)) < ~} > O. Since P o (Th) 1 is absolutely continuous with respect to P it suffices to show that P {d(F o

<

> 0

for some N _> 1, and this follows from condition (ii).

[]

R e m a r k s : Both parts of the proof are independent, in the sense that Step 1 uses only assumption (i) and Step 2 uses (ii), and we could have taken different skeletons ~1 and eP2 in both assumptions. Also we can replace H by a dense subspace H0 such that IINW takes values in H0, and IINW by a sequence of random variables ~N taking values on H0.

4.2

Strict positivity of the density and skeleton

Under stronger hypotheses, and assuming that F is a finite-dimensional random variable, one can characterize the set of points where the density is strictly positive in terms of the skeleton. The following propositions are devoted to this problem. More precisely we want to establish the equivalence between the following conditions:

178 (a) The density p of a random vector F satisfies p(y) > 0 at some point y E ]Rm. (b) Fix y E ]Rm. There exists an element h E H such that O(h) = y and det'/~(h) > 0, where ~ : H ---* ]RTM is a differentiable mapping and 7~ = ( D~s D~J)H" P r o p o s i t i o n 4.2.1 Let F = ( F 1 , . . . , F m) be a nondegenerate vector in (]D~176 m. Sup-

pose that rp : H ---* IRm is an infinitely differentiable function such that 9 and all its derivatives have polynomial growth (i.e., IID(k)r~(h)ll <_ C(1 + Ilhll~k)), for all h G H, k > 0). We will also assume that the following condition holds: (H1) limg~oo r

= F, in the norm I1" IIk,p for all k >_ O,p > 1.

Then for each y e IRm (a) implies (b). Proof: Fix y c lRm. We assume that p(y) = E[6~(F)] > 0. For every M > 1 we consider a function o~M ~ C ~ 1 7 6 s u c h that 0 __< O~M __< 1, C~M(X) = 0 if IXl _< ~,1 and OlM(Z) = 1 if Ixl > ~2. The fact that F is nondegenerate implies that lim 0LM(det'yF) = 1,

M~oo

in the norm I1 IIk,~ for all k, p. Consequently, 0 < E[fy(F)] = lim E[fy(F)~M(det'yF)], M~oo

and we can find a positive integer M such that

E[ey(F)~M(det'Yr)] > 0. Our assumptions imply that h ave

(4.1)

~(IINW) E ]D~176 for every N, and for every k > 1 we

Dk(e(II~vW)) = (IIN)| We have

E[~y(F)C~M(det'yf)] = limooE[6y(q~(IINW))~M (det')'r

(4.2)

in all the norms II" Ilk,p- This convergence follows from hypothesis (H1) and the following integration by parts formulas (see (3.14)):

E[~(F)C~M(det ~'f)] = E[l{y
E [6y(+(IINw))~M (dee 7+ 0. Now consider a function r G C+(lR) such that 0 < /3K _< 1, l~g(X) = 0 if IX[ > K, and/3K(x) = 1 if Ix[ < K - 1. Then r converges to 1 in lD ~176 and we can find an integer K > 1 such that E [f~(+(IINw))c~M (det')'+(nNw)) •K(IIIINwII2H)] > O.

179 This implies that

p{I~(HNW))-yI<e,

det~(nsw)>_I,

IIIINWll2H<_K}>O,

(4.3)

for every e > 0. Notice that det 7e(nNw) < det"/r because for every t E ]Rm we have

t' ( D( ~ ( H N W ) ), D( aPJ(IINW ) ) }Htj i,j=l

= ~ t'(IIN[(DO')(H~W)],nN[D('~J)(nNw)])Htj i,j=l m

=

IInN(~t~(Dr

<

i=1

=

II~t~(DO~)(nNW)ll~ i=1

~ ti((D~)(HNW),(DW)(H~W)).P. i,j=l

Hence, (4.3) implies

P

{ Io(nNw)--yl<e, det?~(nNw)>_--~,llnNwll~o.

(4.4)

Finally, from (4.4) we can find a sequence of elements hk E Im(YiN) such that [a2(hk) - Yl < ~1 for every k, Hhk]12H< - K, and det'y~(hk) _> ~1. Bounded and closed sets in the image of 1-IN are compact, so we can select subsequence converging to some element h E H which verifies the desired properties. [] In order to formulate and prove the converse implication (that is, that (b) implies (a)) we need some preliminaries. We will denote by Ba(x) the ball of ]Rm with center x and radius a > 0. The following lemma is a somewhat quantitative version of the classical inverse function theorem. For a function g : BI(0) --* ]Rm which is twice continuous differentiable we will denote by IIglIc~ the norm

Ilgllc2 = zeBx(0) sup {Ig(z)l + Ig'(z)l + Ig"(z)l}. For each/3 > 1 there exist constants cr 6 mapping g : BI(O) ~ ]R verifying g(O) = O,

L e m m a 4.2.1

(0, ~), ~ > 0

(4.5) such that any

TM

Ilgl]c2 <_/3, and

Idetg'(0)l >

1

is diffeomorphic from B~o (0) into a neighborhood of B~ (0). Consider a r a n d o m vector F 6 (IDa) m. Givcn m elements h b . . . , z E IRm we define the shifted Gaussian process

(T~W)(h) = W(h) + ~ zj(h, hi}H, j=l

h r g.

hm 6 H

and

(4.6)

180

By an elementary change of probability argument we know that for any (integrable or nonnegative) r a n d o m variable F we have (4.7)

E(F) = E(F(TzW)Jz), where Jz = exp

(

-- ~ z j W ( h j ) j=l

2 j=l

H)

We set _h = ( h i , . . . , hm), and note that T z W and Jz depend on _h. We also set, given p > m and k > 0

T~h_,k,pF= f{[zl_
d~.

P r o p o s i t i o n 4.2.2 Consider a nondegenerate random vector f E (]D~ "~. Let 9 : H --~ IR TM be a C i mapping. Suppose that the following condition holds." (H2) For all h E H , there exists a sequence of measurable transformations T~ : f~ --* f~ such that P o (Th) - i is absolutely continuous with respect to P, and f o r every e>0,

l i m P {lF O ThN--'~(hD] > e} = 0

l ~ P { l l ( D r ) o T ~ - (Dr

>r

lim supP{(T~Dl,(h),t:,pF) o T h > M }

M~oo N

= 0 =

0

f o r some p > m, and for k = 0, 1, 2, 3. Then (b) implies (a). Proof: We fix a point x E ]Rm and an element h E H such that ~ ( h ) = x and det0'v(h) > 0. Consider the elements of H given by hj = (DrlSJ)(h), j = 1 , . . . ,m. Using these elements we can introduce the random mapping g : IRm + IRTM defined by 9(z) = F ( T z W ) - F, where T z W is defined in (4.6). Notice t h a t the random function g has an infinitely differentiable version and for each multi-index c~ = ( c t i , . . . o~k) E {1, 2 , . . . , m } k its derivatives are given by

(O~g)(z) = ((DkF)(TzW), h~l |

O h~}.~k 9

(4.8)

By Sobolev's inequality, if p > m is the exponent appearing in hypothesis (H2), we have 1 sup Ig(z)l _< ~ Izl_
Izl-
Ig(z)[pdz +

Izl-
[Ojg(z)[Pdz

(4.9)

Consequently, using (4.8) and (4.9) we can estimate the norm (4.5) in the following way

Ilgllc~ _
~

Izl_
la~g(z)Fdz

< G,

181

where G is the random variable defined by 1

a = Cp

El p + ~ _ , ( ~ ]lhyllS) k k=0 j=l

II(DkF)(TzW)llPH|

.

Izl-
For any /3 > 0 we will denote by a z a continuous function such t h a t 0 < c~z < 1, a~(x) = 0 if Ixl _< ~, and aZ(x) = 1 if Ixl > 5" Also kz will be a continuous function such that 0 < k z < 1, k~(x) = 0 if Ixl >/3, and kz(x) = 1 if Ixl < / 3 - 1. Set H a = k~(G)a~( I d e t ( D F ~, DcPJ)HI). Suppose t h a t p : ] R m ~ IR is a strictly positive continuous function such that f~m p(z)dz = 1. Let f : ]Rm ---* ]R be a nonnegative continuous and bounded function. We can write, applying (4.7)

Elf(F)] = ./~ E[f (F )]p(z)dz = ./~ E [f(F(TzW))Jz] p(z)dz . Fix /3 > i and let c0 and 5Z the constants provided by L e m m a 4.2.1. W e can apply this lemma to the function g(z) = F(T.W) - F (notice t h a t this function verifies the hypotheses of the lemma if H~ ~r O, because Ilgllc~ <- G, and detg'(O) = d e t ( D F ~, D~J)H). Hence, making the change of variable y = g(z) we obtain

E [H~ famf(F(TzW))JzP(z)dz ] > E[Hz/IzI E [Hz flul<j,}f(y + F)Jg-~(y)p(g-a(y))ldetOjg~(g-~(y))ldy ] . From the above inequalities we deduce

p(x) > E [gZl{ir_,l<e,}Jg-,(~_r)p(g-l(x -

F ) ) I det

03gi(g-l(x -

F))I]

Notice that if H a r 0 and ]x - F I < 5z then

Hence, in order such a way t h a t

gg-,(,_F)p(g-l(x -- F)) I detOjgi(g-l(x - F)) I > O. to deduce p(x) > 0 it only remains to show that we can

choose/3 in

P { ,F - x[ <_5~, G <_/3, [det(DFi, D~J)H' >_~ } > O. By the absolute continuity of P o (Tub)-1 with respect to P it suffices to show t h a t

P{,FoThN- X, <_6~,GoT~ <_/3,,det((DFi) oT~,D~J)H, >_~} >O

182

for some N > 1. Finally we can write

P {[F o T~ - x[ <_5~,Go T~ <_~,[det((DF~)o T~,D(I~)H, >_~} > I-P{[FoTh-x[>5z}-P{GoTh>/3} - P {Idet<(DF')oT~,D~J>HI >_~}, and letting N tend to infinity and using hypothesis (H2) we complete the proof.

4.3

[]

S k e l e t o n a n d s u p p o r t for diffusion p r o c e s s e s

Let us now describe the application of the above results to the case of a diffusion process. In the sequel we will assume that the underlying Gaussian process is a ddimensional Brownian motion {W(t), t E [0, 1]}, defined in the canonical probability space f~ = C0([0, 1], IRd). Moreover, for each positive integer N, and for any element h 9 H = L2([0, 1]; IRa) we set fiN(h) = ~

2N

i--1)2-N,i2-N]

i=1

h(s)ds

)

X(i2-N,(i+1)2-N).

With this definition ( f i g w ) ( w ) = d) N, where for any continuous function w 9 f~ we denote by w g the element of ~ such that on each interval (i2 - g , (i + 1 ) 2 - g ) , 1 < i < 2 y -- 1, has a constant derivative equal to (w(i2 -N) - w((i - 1))2 g . We will denote by i the isometry between H and the Cameron-Martin space H 1. That is, for each h r H we have i(h)(t) = f~ h(s)ds. Let Aj, B : IRm __, IRm, j = 1 , . . . , d a system of functions such that Aj is of class C 2 (i.e., the partial derivatives of first and second order are continuous and bounded), and B is Lipshcitz. Consider the following stochastic differential equation on IRm: t

d

t

Xt = xo + fo B(Xs)ds + ~ fo Aj(Xs)dW~.

(4.10)

j=l

Let (I)(h) be the solution of

9 (h)t = xo + fot Ao(~(h),)ds + ~ fot Aj(~(h)~)hJds,

(4.11)

j=l where

1m d i k Aio = B~ - ~ ~ ~ akAjA3. k=lj=l

Notice that A0 drift of the diffusion process Art if we write the stochastic differential equation (4.10) in the Stratonovich form. Then one can show that the mapping (I) : H -~ C([0, 1]; IRm) is a skeleton of the process X in the sense of Definition 4.1.1. That is, the following convergences hold ([711): /

liNmP

(sup I (nNw) ktr

\

> d : 0, /

183

and l i m P ( s u p IXt o T h - ~(h)tl > e) = O, N

\re[0,1]

where T h ( w ) = w -- cog -[- i(h). Actually, the above convergences hold if we replace C([0, 1]; IRTM) by the space C~([0, 1]; ]Rm) of a-H61der continuous functions equipped with the a-H61der norm

Ilxll = sup Izd +sup re[0,1] sr for

IXt

Xs I [t--s~' - -

9

If the coefficients Aj, j = 0 , . . . , m are of class C ~~ with bounded partial derivatives of all orders, then, for any fixed t C [0, 1] the convergences of O ( H N W ) t to X t and X t w - w N + i(h) to ~ ( h ) t holds in the topology of ID~ Hence, if the nondegeneracy condition (I-I) holds then we can apply Propositions 4.2.1 and 4.2.2 to the random vector Xt for all t C (0, 1] and we deduce the following characterization of the points where the density of X t is strictly positive (see [11]): P r o p o s i t i o n 4.3.1 The density Pt of the diffusion process Xt satisfies Pt(Y) > 0 at some point y C IR TM if and only if there exists an element h E H such that gP(h)t = y and d e t ( D r Dq~J(h)t)H > O.

E x a m p l e : Consider the following example (cf. [1]). Suppose a ( x ) , /3(t, x), 7(x, y) are smooth functions with bounded derivatives of all orders such that 2 if x e [ - 1 , 1 ] ,

a(x)=x

fl(t,x)#0

iff (t,x) e [ 0 , 1 ) x ( - 1 , 1 ) ,

"),(x,y)r

iff (x, y) • [-1, 0] x [-1,1].

Consider the process { X t , t C [0, 1]} solution to the equation dX:

=

dX2t

=

~(t, X2t ) o d W ) + "),(Xt1, X't2) o dW~J

o dW? +

(X )dt

Xo

=

O.

H6rmander's condition is satisfied in this example, and, applying the techniques of the Malliavin calculus for the case of time dependent coefficients (cf. [31]), one can show that the vector (X~, X~) is nondegenerate. We have {~l(h), h E H} = IR2, and the support of the law of X1 is IR2. However at the point 0 the density vanish because the unique element h e H such that r = 0 is h0 = 0, and det(Dep~(ho), D ~ ( h o ) ) = 0.

4.4

Varadhan

estimates

Fix a small parameter e C [0, 1], and consider the solution X[ to the stochastic differential equation Xt = x +

'/0

B(Xs)ds + e ~ j=l

A j ( X a~) d W jJ .

(4.12)

184

That is, X[ = Xt(~I/V). Suppose that the coefficients Ai and B are infinitely differentiable with bounded derivatives of all orders, and H6rmander's condition (H) holds. We can introduce the following functions on IRm, which depend on the skeleton ~t(h): d2(y)=

Ilhll~,

inf

q~l(h)=y

and

d~(y) =

inf

~1 (h)=y,det 3'~ (h) >0

Ilhll .

Then the following theorem holds (cf. L~andre [58], Ben Arous and L~andre [11] and L~andre and Russo [59]): T h e o r e m 4.4.1 Let us denote by p~(y) the density of X~. Then d2 R(Y),

(4.13)

lim sup 2c2 logp~(y) < -d2(y).

(4.14)

lim inf 2e 2 logp~(y) > _

cl0

and e$0

Moreover, if inf~ 1(h)=y,det~el (h)>0 det 0'el (h) > 0 then lim 2c2 logp~(y) = --d2R(y). eJ,0

(4.15)

We will sketch the proof of this theorem which provides the asymptotic behaviour for the density of a perturbed dynamical system when the noise is small. In order to show the limit results stated in the theorem we will consider a more general case which contains the diffusion case as a particular example. In the sequel we will work in the framework of an arbitrary Gaussian family {W(h), h C H}. Let us first proof the minoration inequality. P r o p o s i t i o n 4.4.1 Consider a family {F~,0 < e < 1} of nondegenerate random

vectors, and a function q~ E C~(H;IR m) such that: lim~10-el (F~(W + _h)e - ~(h)) = Z(h),

in the topology of ID~, for each h C H, where Z(h) is an m-dimensional random vector in the first Wiener chaos with variance ~/r Define d2R(y) =

inf

9 (h)=y,det ~/~(h) >0

Ilhll~,

y e ~m.

Then 2 lim inf 2e 2 logp~(y) _> --dR(y ).

el0

Notice that the hypothesis of Proposition 4.4.1 implies h lim F~(W + - ) = ~(h), e$0

(4.16)

185

in the topology of ]D~, for each h C H. Proof." Fix y C ]Rm, and 7] > 0. If d2R(y) = c~ there is nothing to say. Suppose d~(y) < exp. Let h 9 H be such that (I)(h) = y, detv~(h) > 0, and IIh[[2H< d2R(y)+~?. Let f 9 C~(]l:[m). By Girsanov's theorem

E(f(F~)) = e- 2, E , f ( F ' ( W + ~ ) ) e - - T - ) . Consider a function X 9 C~176

0 < X -< 1, such that x(t) = 0 if t r [-2~, 2~], and

x(t) = 1 if t 9 [-7, r]]. Then, if f > 0, we have E(f(F~)) > e-

~o" E x(eW(h))f(F~(W +

)) .

This implies that

Hence, it suffices to show that

limc21~

)) = 0 .

We have

Then, using hypothesis (ii) the expectation Z (x(~W(h))50( F~(W + ~_)~- + ( h ) ) ) converges, as c tends to zero, to E(5o(Z(h))), and this completes the proof of the proposition. [] The majoration estimate requires large deviation assumptions: P r o p o s i t i o n 4.4.2 Consider a family {F~,0 < ~ < 1} of nondegenerate random

vectors, and a function 9 9 Cp (H; IRTM) such that: (i') sup~c(04 ] ]lF~llk.p < co, for each k > 1 and p > 1. (ii') [I(TF,)-IHk _< e-N(k) for any integer k >_ 1. (iii) The family {F,, 0 < e _< 1} satisfies a large deviation principle with rate function d2(y) = ~(h)=y inf Ilhll~,

y 9 ~.

Then lim sup 2e2 logp~(y) < -d2(y). el0

(4.17)

186

Proof: Fix a point y E ]Itm and consider a function X C C~(Rm), 0 < X -< 1 such that X is equM to one in a neighborhood of y. The density of F~ at point y is given by p,(y) -= E(x(F,)fu(FE)). Using Proposition 3.2.1 we can write

E(x(F,)Sy(F~)) = E (l{F,>~}H(1,2,...,m)(Fc, x(F~))) < E([H(1,2 ...... )(F,, x(F,))[) = E([H(1.2 ...... )(F,, x(F,))[l{f,~suppx})

< (P(F, E suppx))~[Ig(1,%...,m)(F,, x(F,))l]p, where [ + ~ = 1. By Proposition 3.2.2 we know that

IIH(1,=,...,~)(F,, x(F,))II. _< C(,)ll'~:~)llkllF, II.,blix(F,)H.',b', for some constants b, d, b', d'. Thus, hypothesis (ii') implies that lime 2 log IIH(t,2 ...... )(F,, x(F,))IIp = O. el0

Finally, the large deviation principle for F, ensures that for e small enough we have

(P(F, E suppx))~ < e -~-~' (inf~e'uppxd=(y)). [] P r o p o s i t i o n 4.4.3 Consider a family {F~, 0 < e _< 1} of nondegenerate random

vectors, and a function r E C ~ ( H ; ]Rm) such that: (i') sup~e(0.1] IIF~l]k,p < oo, for each k >_ 1 and p > 1. (ii') I]('yF,)-lllk < C-y(k) for any integer k > 1. (iii') The family {(F~, 7F,), 0 < e < 1} satisfies a large deviation principle with rate function A2(y,a) = inf [[h]]~,, (y,a) e ]Rm x ]Rm2. 9 (h)=~,~ (h)=a (iv) limN~o~O(eIIgW) = F,, in the norm I[" IIk.p for all k >_ O, p > 1, where {I[ N, N > 1} is a sequence of orthogonal projections of H of finite-dimensional rangle strongly convergent to the identity. Define d2R(y) = infv(h)=u,det~(h)>o Ilhll2H, y E ]Rm and suppose that ~f :=

inf

~I,(h) = y , d e t ?~ (h) > 0

detT~(h) > 0.

Then lim sup 2~2 logp~(y) < --d2R(y). eJ.0

(4.18)

187

Proof.- The proof is similar to that of Proposition 4.4.2. Consider a function g E 1 Set C~176 0 __ g _< 1, such that g(u) = 1 if lu[ < 88 and g(u) = 0 if [u I > ~7. G~ = g(det 7F~)- With the notations of the proof of Proposition 4.4.2 we have E(x(F~)6y(F~)) = E(G~x(F~)Sy(Fe)) + E((1 - G~)x(F~)Sy(F~)). We have E(G~x(F~)~(F~)) = O, because, otherwise, using condition (i') and applying the method of the proof of Proposition 4.2.1, one can find an element eh E H such that ~(eh) = y and 0 < det ~/r < ~, and this is in contradiction with the definition of 7Proceding as in the Proposition 4.4.2 we obtain E((1 - G~)x(F~)6y(F~)) = E (1F,>yH(,,2 ....... )(F,, (1 - G,)x(F,))) /~(]H(1,2 ...... )(Fr (1 - G,)x(F,))[ )

E([H(1,2,...,m)( F~x( F~)ll F,esuppx,detTF >88 <

(( P

1

F~ E suppx, det'),F~ >

~/

IIH(~,2...... )(F~,x(F~))H p.

Finally, hypotheses (ii') and (iii') imply that for any q > 1 we have limsup2e21~

<

1~( r e x p \( _ 2qe2 ( 1 .

inf ~

[[h]]~)) ) []

When the random variable F~ is X~, conditions (i),(ii),(iii), (iv), (i'), (ii') and (iii') are satisfied, and in this way one can show Proposition 4.4.1. More precisely, condition (ii') is proved by Kusuoka and Stroock in [57], the large deviation principle for both the diffusion X~ and the Malliavin matrix "Yx,~ are known, and the convergences (i), (ii) and (iv) axe easy to check. B i b l i o g r a p h i c a l n o t e s The ideas used in this chapter in order to characterize the support of the law of a diffusion go back to the works of Mackevi[ius [61] and Gy6ngy [35]. The support theorem for the HSlder norms has been studied in [8], [71] and [38]. The results on the skeleton are based on the main references [1] and [11]. The characterization of the support of the law of hyperbolic stochastic partial differential equations has been established by Millet and Sanz-Sol6 in [70], and the ehar~terization of the points of positive density for the solution to such equations has been done by these authors in [72]. The support theorem for parabolic stochastic partial differential equations is proved by Bally, Millet, and Sanz-Sol~ in [6].

Chapter 5 Anticipating stochastic calculus We have seen in Chapter 1 t h a t the divergence operator 6 on the classical Wiener space is an extension of the It6 stochastic integral, in the sense that the class L] of square integrable and a d a p t e d process is included in Dora 5, and the operator 5 restricted to L] coincides with the It6 integral. Actually, the operator 6 coincides with an extension of the It6 integral introduced by Skorohod in [98] using the Wiener chaos expansion. One can develop a stochastic calculus for both the Skorohod integral and the generalized Stratonovich integral. In this chapter we will present the basic facts of the stochastic calculus for anticipating integrals.

5.1

Skorohod

integral

processes

In this section we will assume t h a t W = { W ( t ) , t E [0,1]} is defined on the canonical probability space (12, 5 , P). In order to tation we will assume t h a t W is one-dimensional, but most of the generalized to the multidimensional case. We will say t h a t a square integrable process u = {u(t), 0 < integrable if u belongs to Dom 5, and we will write

(~(U) =

a Brownian motion simplify the presenresults can be easily t < 1} is Skorohod

/0 utdWt.

It may happen t h a t u E Dom 5, but ul[0,tl is not Skorohod integrable for some t E [0, 1]. Example:

Consider the process

ut = l[0,89

} - l(89

t.

The process u is Skorohod integrable and

5(u) = (2W89 - Wail{w,>0}.

(5.1)

Indeed, if we denote by ~k a smooth nonnegative function such that 0 _< 9~k _< 1, ~k(x) = 0 for x _< 0, and ~k(x) = 1 for x _> I , then the sequence of processes

uk(t) = l[o,89

) - l( 89

converges in L2([0, 11 x f~) to u, and

~(~) =

(2% - wa)~(wo

,

189

converges in L2(f~) to the r i g h t - h a n d side of (5.1). Hence, u is Skorohod integrable and (5.1) holds. Nevertheless the process ul[0,89] does not belong to Dom S. In fact, the divergence of this process is the element of ID -~176 given by 5(ul[0,89 = W89

- 250(W1).

Let us denote by IL~ the set of processes u such t h a t ul[0,t I is Skorohod integrabte for any t r [0, 1]. Notice t h a t the space ]DI'2(H) is included into ILL Suppose t h a t u belongs to ]L~ and define

X(t) = ~(ul[0,t]) = /o t usdWs.

(5.2)

The process X = {Xt, t E [0, 1]} is not a d a p t e d , and its increments satisfy the following o r t h o g o n a l i t y property: Lemma

5.1.1 For any process u r ]L~ we have

E( f ~urdW~lfE~,e ) = O,

(5.3)

for all s < t, where., as usual, ~-[s,t]~ denotes the a-field generated by the increments of the Brownian motion in the complement of the interval Is, t]. Proof: To show (5.3) it suffices to take an a r b i t r a r y 5c[~,t]~-measurable r a n d o m variable F belonging to the space lD 1'2, and check t h a t E(F f / u~dW~) = E( fo' urD~Fl[s,t](r)dr) = O, which holds due to the d u a l i t y relation (1.7) and L e m m a 1.5.1.

[]

The Skorohod integral process X is continuous in L 2 for any u C ILL However, there exist processes u in IL~ such t h a t the indefinite integral ft u~dW~ does not have a continuous version. Example:

Consider the process

t

u(t) = sign(W1 -- t) exp(Wt - 7). This process belongs to the space ILs, and

Xt :=

f0'usdW~

= sign(W1 - t) exp(Wt - 7) - signW1.

(5.4)

As a consequence, the process X has a discontinuity on a point t such t h a t W1 = t. We can show (5.4) by means of a d u a l i t y argument. In fact, let G r 8 be a s m o o t h r a n d o m variable. Consider the family of t r a n s f o r m a t i o n s of the W i e n e r space given by (TtW)s = Ws + s A t, t r [0, 1]. Notice t h a t if G = g(W ( hl),..., W ( hn) ), g C Cb(IRn) then

dG(TsW)

~ds g ( W ( h l ) + fo ~ hl(r)dr,...,W(hn) + fo h~(r)dr)

(

= ~-~.(O~g) W(hl) + i=1

= (D~G)(T~W).

/o hl(r)dr,..., W(hn) + /o hn(r)dr ) hi(s)

190

Using Girsanov's theorem we can write

E(L'D~Gusds ) = E(LtDsGsign(W , -s)eW'-:ds) = E

(i?

D~G)(T~W)sign(W:)ds

)

= E ( G ( T t W ) s i g n W : - GsignW:)

---- E ( G (sign(W, - t ) e x p ( W , - ~) and (5.4) follows. A useful tool in proving the existence of a continuous version for stochastic processes is the real analysis lemma of Garsia, Rodemich and Rumsey [32]. Let us recall this lemma: L e m m a 5.1.2 Let p, k~ : IR+ ---, IR+ be continuous and strictly increasing functions vanishing at zero and such that lirnttooqs(t) = :x). Suppose that r : IR m ~ IB is a measurable f u n c t i o n with values in a separable Banach space (IB, I1" II). Let B K = { x E lit m, Ixl <_ K } . Suppose that

9 lie(t) - r

j

dsdt <

0<).

Then, there is a set N C B K of measure zero such that, f o r all s, t E B K -- N , r2It-sl

llr where

)~m is

-

~(s)ii <_ 8Jo

~'-:(:,,m~-2'~)p(du),

a universal constant depending only on m .

Now suppose that X = {X(t), t E B K } is a stochastic process with values on ]13, such that the following estimate holds: E ( i I X ( t ) - X ( s ) l l ~) <_ H i t - sl a

for some constants H > O, 3' > O, a > m, and for all s , t E BK. Then taking d~-2m k~(x) = x ~ and p(x) = x ~ , with 0 < d < a - m, one obtains for all s, t E B K -- N~,

IIx(t) - x(s)ll "~ _< c ~ i t

-

slur,

(5.5)

where N~ is a subset of B K of zero Lebesgue measure depending on w, and E(F) < HC2 for some constants C: and C2 depending only on 7, d, and m. In particular, the above inequality implies that the process X has a continuous version (this is the classical Kolmogorov continuity criterion). For every p > 1 and any positive integer k we will denote by ILk'p the space /2([0, 1]; IDk'P). Notice that ]L:'2 = ]DI'2(H), a n d / 2 ( [ 0 , 1]; IDira) C IDI'P(H) for p > 2. P r o p o s i t i o n 5.1.1 Let u be a process in the class IL :'2. Suppose that E f3 [[Dutll~dt < oo f o r somc p > 2. Then the integral process {f~ usdWs, 0 < t < 1} has a continuous version.

191

Proof: We can assume t h a t E ( u t ) = 0 for each t E [0, 1] because the Gaussian process f~ E ( u s ) d W s has a continuous version. Applying the estimate (2.22) we obtain

Set Ar =

I fo~(Dou,)2dOI~.

Fix an exponent 2 < a < 1 + a2, and assume t h a t p is close to 2. Applying Fubini's theorem we can write -~Z~[~ -- ~ - a 2

asdt)
~ fo I t -

- Ir - s[2

t

)A~drds

2Cp+ 1 -- a ) f01(r~+l_a + [1 -- r[~ + l - a -- 1 ) E ( A ~ ) d r < oo. = ( a - - ~)(~ 2

2

Hence, the r a n d o m variable defined by

is finite almost surely, and by L e m m a 5.1.2 we obtain

IX,-x,I

1

~--2

_< c~,~r; It - sl T ,

for some constant Cp,~.

[] 1,p

Note t h a t the above proposition implies t h a t for a process u in the space ILloc, with p > 2, the Skorohod integral f~ u~dWs has a continuous version. ~klrthermore, if ?~ E]L I'p, p > 2, we have

E( sup I re[0,1]

/:

usdW~l" ) < co.

The next result will show the existence of a nonzero quadratic variation for the indefinite Skorohod integral. Theorem

1,2

5.1.1 Suppose that u is a process of the space ILloc. Then

cr

n-, t~+, u~dW~ E \Jti i=0

)' /o' ~

u~ds,

(5.6)

in probability, as ]~1 "* O, where ~ runs over all finite partitions {0 = to < tl < " " <

tn = 1} of [0,1], and I~l = supo_~,
t~l. Moreover, the ~onvergence is i~

192

Proof." We will describe the details of the proof only for the case u C ILl,2. T h e general case would be deduced by a n easy a r g u m e n t of localization. For a n y process u in IL 1'2 a n d for any p a r t i t i o n 7r = {0 = to < tl < " ' < tn = 1} we define usdWs) 2 v~(~) = n-, Z ([q+~ ,~,, i=0

Suppose t h a t u a n d v are two processes in ]L 1'2. T h e n we have

(E ~ (ft]i+l(,~8-vs)dWs )2)2 1

E(fv~(u)-v'~(v)l)

<_

i=0 1_

,=o

(5.7)

_< I1~ - vll~.~ll~ + vlli~.~.

Therefore, it suffices to show the result for a class of processes u which is dense in 1L1'2. So we can a s s u m e t h a t m-1

Ut = ~ Fjl(sj,sj+d, j=0

where for each j , Fj is a s m o o t h a n d b o u n d e d r a n d o m variable, a n d 0 = So < ' " < s m = 1. We c a n assume t h a t the p a r t i t i o n 7r c o n t a i n s the points { s o , . . . , Sr,}, because lira E ( I V ~v{s~....... }(u) - V'~(u)]) = 0.

I~11o

sj we can write

If 7r c o n t a i n s the points

/

m--1

x 2

~

V~(u) j=0

{i:si
/

ati

m-1 [ = E

E

Fj2(w(ti+l)

{i:sj~_ti<sj+x}

j=0

-- W( ti))2

/,

-2(W(ti+l) - W(ti)) Jti TM PsFjds +

D

\Jr i

Using the properties of the q u a d r a t i c variation of the B r o w n i a n m o t i o n this converges in LI(•) to m-i

E i=0

f 1

Y (sj+l - sj) = )to u2sds,

as 17r] t e n d s to zero.

[] 1,p

As a consequence of these results, if u EILloc, P > 2, is a process such t h a t the Skorohod integral fd usdWs has b o u n d e d variation paths, t h e n u = 0. Let X E ILk'p, p > 1, a n d let q e [1,p]. We will denote by element of Lq([0, 1] x f~) defined by lira

/01

sup

s
D+X (resp. D - X ) the

E(IDsXt - (D+X),lq)ds = 0.

(5.8)

193

(resp. lim

/0

sup E(ID~Xt - (D-X)~lq)ds (~-~)vo
= 0)

(5.9)

provided that these limit exist. We will denote by ]L~4~ (resp. IL~[) the class of processes in ]Lk'p such that the limit (5.8) (resp. (5.9)) exist. For each p > 1, and q 9 [1, p] we set IL~'p = ]Lq+k'PAILq_ .k'P For u 9 lLqkm we will write ( V X ) t = (D+X)t + ( D - X ) t . Notice that a process X of the form X t ---~X o Jr-

/o u~dW~ + /o v~ds,

where Xo 9 ID 1'2, u 9 IL2'2, and v 9 IL1'2 belongs to the class IL~'2. Indeed, we have

DsXt = usl{s<_t} + DsXo +

/o D~v~dr + /: D~u~dW~,

and this implies that (D+X)t and (D X ) t exist and

/0 Dtvrdr -I- /0 DturdWr, ( D - X )t = Dt Xo -I- Dtvrdr + /o' /o DturdWr.

(D+X)t = ut Jr- DtXo +

In fact,

we

have 1

lim

L

sup

E(lDsXt - (D+X)sl2)ds

1/o'/<~+~'"' /o'/<"+',' E(iD"u~i~)drds + io' iol i/"+~)"'E(iDoD.~i')drdsdO, <_ -

E(lD.v,12)drds

n

+

J8

d8

and this converges to zero as n tends to infinity. One can show a change ogvariables formula similar to It5 formula for Skorohod integral processes. T h e o r e m 5.1.2 Consider a process of the form Xt = Xo + ft usdW~ + ft %ds, where 1,2c, u 9 (1L2,2 A IL1'4)1or and v 9 IL,o 1,2c. Let F : ]R ~ ]R be a twice continuously X0 9 ]Dlo differentiable function. Then we have Ft

1 ft

t

F ( X t ) = F(Xo)+7o F'(X~)dX~+-~ Jo F"(Xs)u~ds+ f

F"(X~)(D-X)~u~ds. (5.10)

JO

Notice that if the process X is adapted then ( D - X ) s vanishes, and we obtain the classical It5 formula.

Proof: Suppose that (fl n'l, X~), (fl ~'2, u n) and (fl n'3, v n) are localizing sequences for X0, u, and v, respectively. For each positive integer k let r be a smooth function such that 0 < Ck _< 1, Ck(x) = 0 if Ixl >_ k + 1, and r = 1 if Ixl <_ k. Define utn'k = ut Ck

(]01(ru n)2ds ).

194

Set X~ 'k = X~ + f~ u~,kdW8 + f~ v~ds, and consider the family of sets

Gn,k =

~n,1 N a n'2 71 a n'3 n { sup Ix~l ___ k} n { te[0,i]

(u2)2ds<_k}.

Then it suffices to show the result for the processes X~, u ~'k, and v ~, and for the function F n : = F O n . In this way we can assume t h a t X0 E ]D i'2, u E ILe'e A IL1'4, v E ]L1'2, f~ u2ds < k, and t h a t the functions F, F' and F" are bounded. Set t~ = ~ , 0 < i < 2~. Applying Taylor development up to the second order we obtain 2n--1 f(Zt)

= F ( Z o ) -[- E

F ' (X(t~))(X(t,+l) - x(tn)) + n n

2n--1 1

i=0

E

~F"(Xi)(Z(t~+l)

-

X(tt i l~2 l

,

i=0

(5.11) where Xi denotes a random intermediate point between X(t~) and X(t'~+l). Now the proof will be decomposed in several steps.

Step 1. Let us show t h a t 2n--1

t

F ( X i ) ( X ( t i + l ) - X(t~)) ~ --*

2 F " (Xs)usds,

(5.12)

i=O

in probability (actually in L I ( ~ ) ) , as n tends to infinity. The increment (X(t~+l) - X(tn)) 2 can be decomposed into t~+;

t~+1

Using the continuity of F " ( X t ) w e can show t h a t the contribution of the last two terms to the limit (5.12) is zero. Therefore, it suffices to show t h a t

~ff~01F/!E (~qi~i)~t/; ("t~+l~8dW8)2~~0t'FH(XS)~dS"

(5.13)

Suppose t h a t n >__m, and for any i = 1 , . . . , 2n let us denote by tl m) the point of the ruth partition which is closer to t~ from the left. Then we have 12~-1

-<

t?+~

,,

F (xi)(f7

2 u~dWs)-

t F'(X~)u~ds

fo

u~dWs) 2

2n--1 2~1

]

j=0

+

3=0

Jt~

i:tn<[ty,ty+,) t at?

r"(X(t?))

= bl + b2 + b3.

Jty~

u~ds -

f

Jo

F"(X~)u~ds

195

The term b3 can be bounded by sup

Js-rl<_t2-m

IF"(X~)

F"(X,.)I fo ~u~d,%

which converges to zero as m tends to infinity by the continuity of the process Xt. In the same way the term bl is bounded by sup

Is-rl <-tS-m

IF"(x~) - F"(X~)I

2n--1 tn ~, ([ '+' ~dW~) ~, i=0 Jt'~

which again tends to zero in probability as m tends to infinity uniformly in n. Finally, the term b2 converges to zero as n tends to infinity, for any fixed m, due to Theorem 5.1.1.

Step 2. Clearly the term

2n--I n t~+l F~ F (x(t,)(f~: v~ds) i=0 /

converges a.s. and in Ll(f~) to

f~ F'(Xs)v~ds, as n tends to infinity.

Step 3. From property (1.12) of the Skorohod integral we deduce

F'(X(t~)) ~i ?+~u~dW~ i

= =

+ f~,r f'r+~ F , (X(t~))%dW~ ~

Jt?

?

D~[F'(X(tp))]u~ds

[tLI DsX(tr)u~ds / 2 ?+1F'(X(tr))usdW" + F" ( X(" tnb~ i " Jt~

C1 -~-C2. Notice that the process F~(Xt)ut belongs to IL1'2 because u C ]L2'2 NIL 1'4, v C IL1'2, and the processes F'(Xt), F"(Xt), and fl u~ds are uniformly bounded. In fact, we

have

( F'(Xt)D~ut,

D~[F'(Xt)ut] = F"(Xt) u~l{~
/0 D~u,.dW~+ /0'P~v~dr) ut

and all the terms in the right-hand side of the above expression are square integrable. For the second term we use the duality relationship of the Skorohod integral:

~1~1 E ( ut fOr D~urdW.~2) dsdt : E { folfolfotDsur[2~ttDrut

(fotDsuodWo) -t-~tStDsur

<_cE ((llOul]~([0,11~) + ]]DSUllLS([o,1]a>)2) . Then, for any smooth random variable G E S we have l i m e ( G 2~-z

tr+l

196

This convergence is easily checked by duality and in this way o b t a i n the convergence of E(Gcl) to E(G ft F'(X~)u~dW~). One can also show the convergence in L 2 of cl to

f~ F'(X~)u~dW~. L ~ to f~ F"(Xs)(D-X)~u~ds.

On the o t h e r hand, the t e r m c2 converges in we have

In fact,

'2~1 (t~))i=,., F"(X Jt?/t?+'D~X(t~)u~ds- /ot F"(Xs)(D-X)~ufls 2~ - 1

/,t%

<- E F (x(t,n

1

[D~X(t~') -

,=0

(D-X),]ufls

t?

+ 2~1 f%~[F"(X(t~))- F"(X~)](D-X)~u~ds1. i=0

Jt~

Consequently, we o b t a i n ,, E ( 2~-' ~ F"(X(tr)) fjt?% , D'X(ti)u'ds-

f0t F"(X,)(D-X)~usds ) 1

_< IIF"II~

{j: E(u~)ds jo

E(IDsXt - (D-X),l~)ds

sup (s-'2-'~ )+ <_t<_s

which converges to zero as n t e n d s to infinity.

}, []

Remarks: 1. Notice t h a t for a process u E ]L 2'2, the condition E(f~ ]lDutll2dt) < oe is only required to insure t h a t X has a continuous version, and to show t h a t F(Xt)ut belongs to ]L 1'2. One can show the It6 formula under different t y p e of hypotheses. More precisely, one can impose some conditions on X and modify the a s s u m p t i o n s on 3/0, u, and v. For instance, one can a s s u m e either (&)

U

1,4

9 lLloc,

V

9 LI([0, 1]) a.s., a n d X C ( I L ~

(b) u 9 (]L 1'2 N L ~ ( [ 0 , 11 x fl))loc, a continuous version.

Y

N

]Ll'4)loc, or

9 LI([0, 1]) a.s., X 9 ILl'toe , and X possesses

2. Suppose t h a t X , u, v are stochastic processes verifying one of the h y p o t h e s e s (a) or (b). T h e n if there is a set G E -7- such t h a t for a l m o s t all w E G we have

Xt = Xo +

uflWs +

%ds, t C [0, 1],

(5.15)

then (5.10) holds a.s. on G. In fact, it suffices to check t h a t

F'(X(t i=O

,+1usdW~ + ~

2--1 1 F " X~) rt?+ 1 i=0

vsds) Jt~

~

rt%l

2

197 converges in probabiblity as [~r[ tends to zero to the right-hand side of (5.10). 3. Using the operator V we can write Eq. (5.10) in the following way:

F(Xt)

f

= F ( X o ) -I- Jo t f , ( X s ) d X s -.~ 21 fot F " ( X s ) ( V X ) s u s d s .

(5.16)

The following result contains a multidimensional and local version of the changeof-variables formula for the Skorohod integral. We will make the usual convention of summation over repeated indices. T h e o r e m 5.1.3 Let W = {Wt, t E [0, 1]} be an N-dimensional Wiener process. Suppose that X~ E 11)1'2, u ij E IL2'2ALL 1'4, and v i E IL t'2, 1 < i < M , 1 < j <_ N , are processes such that on a set G E jz we have a.s.

X~ = X~ + fot u s'J dW~i + fo t v;ds,

O < t < 1.

Let F : IR M ~ IR be a twice continuously differentiable function. Then we have on G a.s.

F(Xt)

=

F(Xo) + fa t (OiF)(X~)dX; + 21 fot(OiOjF)(Xs)tt~kujkds

+

fot(oiOjF)(Xs)((nk)-XJ)su~kd8,

where D k denotes the derivative with respect to the kth component of the Wiener process.

5.2

E x t e n d e d Stratonovich integral

In the sequel, 7r will denote a partition of the interval [0, 1] of the form 7r = {0 = to < tl < ... < t~ = 1}. The mesh of 7r is defined by ]TrI = sup~(t~+l - ti). Given a measurable process u = {ut, t E [0, 1]} such that f~ Iut]dt < oc a.s., we can introduce the following Riemann-type sums

~-11(f,+,)

STr = Z

i=0

t~+l----ti ,,at~

usds

( W ( t i + l ) - W(ti)).

Notice that S '~ =

/0'

usW(ds,

where

n-1 ( W ( t i + l ) - W ( t i ) ) W= = ~i=0 \ t,+l F ] l[t,.t,+,]. Definition 5.2.1 We say that a measurable process u = {ut,t E [0, 1]} such that f~ [utidt < oo a.s. is Stratonovieh integrable if the family S ~ converges in probability as ]Tr] $ 0, and in this case the limit will be denoted by f~ ut o dWt.

198

We know that if the process u is a continuous semimartingale, then the Stratonovich integral of u exists and it coincides with the It6 integral plus the correction term l(u, W)t. Let us now describe the relationship between the extended Stratonovich integral and the Skorohod integral. A basic result on the approximation of the Skorohod integral by Riemann-type sums is the following. For any process u E L2([0, 1] x ft) we define

= ~' E t , + : - t,

(w(t,+,) - w(t,)).

i=0

Notice each term in the summation is the product of two independent factors. Furthermore, from Lemma 1.5.2 the processes

"-11([

~,~(t) = ~

~+'

-

i=0

ti+l

--

)

s

ti \ d t i

l(t,,t,+,](t)

are Skorohod integrable, and

P r o p o s i t i o n 5.2.1 Let u C L2([0, 1] x ft). /f there exist a sequence of partitions {•(n)} of [0, 1] whose mesh tends to zero such that ~(n) converges in L2(r as n tends to infinity, then u is Skorohod integrable, and n

Moreover, this convergence always holds if u E ]L I'2. Proof: The first part of the proof follows from the fact that the operator 5 is closed and ~ converges to u in L2([0, 1] • f~). The second part follows from the convergence of ~ to u as 17rl tends to zero, in the norm of ]L1'2 (see [76] for the details of the proof). [] Concerning the extended Stratonovich integral we can show the following result: 1,2

T h e o r e m 5.2.1 Let u E lLl,loc. Then u is Stratonovich integrable and

/o'

ut

o dWt =

/o'

utdW, + -~

Vu)tdt.

(5.17)

Proof. By a localization argument we can assume that u E ]L~'2. We have, for any partition lr on [0, 1] 5(u ~) = S ~ - ~

i=O ti+l

--

ti

Jt,,

Jti

D,utdsdt,

where

n-l

u~(t) = ~

i=0

l_

ti+l - t,

( f t ' + l u ~ d s ) ) l(t~,t~+d(t). \~,,

(5.18)

199

The processes u ~ converge to u as Irrl tends to zero in the norm of ]L1'2. Hence, 5(u ~r) converges to 5(u) in L2(f~) as 17rI I 0. Therefore, it suffices to show the following convergences: lim E Ml0

moE Ii~lt

1 \t

ti+l

i=0

-

ft,+, dt ti Jt~

r

1

F

1/o1+ (D u)tdt )

(Dtu~)ds - ~

d t f (Dtu~)ds

\li--~ ti+l -- ti ,t,

Jr,

1 1

= 0, (5.19) O.

(5.20)

We will only show (5.19). We can write

E

\ti=o

ti+l

-

< E (~ -

\,

dtJt

ti 1

i=0

ti+l

-

(ntu~)ds - -~

ft,+l (ftt~+l[Dtus_

ti ati

(D u)tdt

(D+u)t]ds)dt )

+E ('-~=1of~'+1 t ~ + ' - t ( D + u ) t d t _ l f o 1(D+u)tdt ) Jti ti+l -- ti _<

/o 1

sup E(IDtu s - (D+u)tl)dt t<s<_(t+l~rl)hl

1,2

The first s u m m a n d in the above converges to zero due to the definition of the class ILl+. For the second term we will use the convergence of the functions ~-~i=0n--1 t::l-tit+l-tl(ti,ti+l][Itx) to the constant 71 in the weak topology of L2([0, 1]). This weak convergence implies that

1

n-1

~)

ti+l -- ti l(ti,ti+d (t) --

dt

converges a.s. to zero as I~1 ~ 0. Finally, the convergence in Ll(f~) follows by dominated convergence, using the definition of the space IL~'2. [] It6's formula for the Skorohod integral allows us to deduce a change-of-variables formula for the Stratonovich integral. Let us first introduce the following classes of processes: Set IL~'4 = {u E ]L~'4 : V u E ILl'e, u i s continuous in L2(a)}. 5.2.2 Let F be a real-valued, twice continuously differentiable function. Consider a process of the form Xt = Xo + J~ us o dWs + f~ vsds, where Xo E IDllo2c, 2,4 1,2 u E ILs, loc and v E lLloc. Then we have Theorem

F(Xt) = F(Xo) + fotF'(Xs)v~ds + fot[F(X~)u~] o dW~.

(5.21)

Proof: As in the proof of the change-of-variables formula for the Skorohod integral we can assume t h a t the functions F, F', and F " are bounded, f~ u~ds is bounded,

20O

X0 E D 1'2, u E IL~'4, and v E IL~'2. We know that the process Xt has the following decomposition: Xt = Xo +

u, dWs +

vsds + ~

(Vu)~ds.

This process verifies the assumptions of Theorem 5.1.2. Consequently, we can apply It6's formula to X and obtain F(Xt)

=

F(Xo) + t

/0

F'(X~)v~ds +

F'(X~)(Vu)~ds

t

1

t

The process {F'(Xt)ut, t G [0, 1} belongs to ]L~'2. In fact, notice first that as in the proof of Theorem 5.1.2, the boundedness of F', F", f l u~ds, and the fact that u E L z'4, v, Vu G lL m and X0 E ]D l'z imply that this process belongs to ]L 1'2 and Ds[F'(Xt)ut] = F ' ( X , ) D s u , + F"(Xt)utD~Xt.

On the other hand, using that u E I L 11,2 , U i8 continuous in L 2, and X E ]L~'2 deduce that {F'(Xt)ut, t E [0, 1} belongs to ]L~'2 and that (V(F'(X)u))~ = F'(X~)(Vu)~ +

we

F"(X~)ut(VX)~.

Hence, applying Theorem 5.2.1 we can write t [F'(Xs)u,] odW

= _=t F'(X )u,dW + 21 Joft(V(F'(X)u))~ds

t

1

t

Finally, notice that (VX)t = 2 ( D - X ) t + ut. This completes the proof of the theorem. [] The above theorem is still valid if we assume X e lL~, u e (ILI{4NC2([0, 1], L~(f~)), and v E L~([0, 1]) a.s. In the next theorem we state a multidimensional version of the change-of-variables formula for the Stratonovich integral, using these type of hypotheses. T h e o r e m 5.2.3 Let W = {Wt, t E [0, 1]} be an N-dimensional Wiener process. Suppose that we have processes X i e ILa2~, u'J e ]LI'4AC2([O, 1], L2(f)) ), and v' G LI([O, 11) a.s., 1 < i < M, 1 < j < N, such that on some set G E a~ a.s. we have

f

/:

Let F : IR M ~ IR be a twice continuously differentiable function. Then we have on G

JO

JO

201

5.3

Substitution

formulas

In this section we will consider the following problem. Suppose t h a t u = {ut(x), 0 ~ t _~ 1} is a stochastic process parametrized by x E IRm, which is square integrable and a d a p t e d for each x C IRm. For each x we can define the It6 integral

~0'ut(x)dWt. Assume now t h a t the resulting random field is a.s. continuous in x, and let F be an m-dimensional r a n d o m variable. Then we can evaluate the stochastic integral at x = F , t h a t is, we can define the random variable

~1 ut(x)dWtlx=F.

(5.22)

A natural question is under which conditions is the n o n a d a p t e d process {ut(F), 0 < t < 1} Skorohod integrable, and what is the relationship between the Skorohod integral of this process and the random variable defined by (5.22). A similar question can be asked for the extended Stratonovich integral. Let us first consider the following preliminary result. L e I n m a 5.3.1 Suppose that {Yn(0), 0 C ]Rm}, n > 1 is a sequence of processes which converges in probability to a random field {Y(0), 0 C ]RTM}for each 0 C ]Rm. Suppose that

E(IY.(O) - Yn(O')l ,) _< e,,/do - 0'1%

(5.23)

for all 101, 10'1 < K, n > 1, K > 0 and for some constants p > 0 and ce > m. Then, for any m-dimensional random variable F one has

l ~ r n ( F ) = Y(F), in probability. Moreover, the convergence is in L v if F is bounded. Proof: Fix K > 0. Replacing F by FK := FI{IFI<_K} we can assume t h a t F is bounded by K . F i x e > 0 and consider a random variable F~ wich takes finitely many values and such t h a t IFd _< K and IIF - F~II~ <_ e. We can write IYn(F) - Y ( F ) I < IYn(F) - Yn(Fr TakeO<m'
+ [Y~(F~) - Y(Fr

+ IY(FJ - Y(F)I.

By (5.5) there exist constants C1, 6'2, and random variables

Irn(0) -- Y.(0')I' _< C , IO - O'lm'rn, E(Pn) _< C2cp,~, and

ly(e) - r ( e ' ) l , _< c i t e - e'l='r, E(F) < C2cp,K.

202

Notice that Eq, (5.23) is also satisfied by the random field Y(O). Hence, E(IYn(F ) -- Y ( F ) I p) <_ Cp (2C1C2cp,Kem' + E(IYn(F~ ) - Y(F~)IP)), and the desired convergence follows by taking first the limit as n --* oo and then the limit as e ~ 0. [] Consider a random field u conditions:

{ut(x), 0

=

< t < 1,x C IRm} satisfying the following

( h l ) For each x C IRTM, and t C [0, 1], ut(x) is G-measurable. (h2) There exist constants p _> 2 and a > m such that -

u,(x')[p)

s

c,,

rx -

for all Ixl, Ix'I _< K, K > 0, where f~ Ct,Kdt < oc. Moreover f~ E(lut(O)12)dt < OO.

Notice that under the above conditions for each t E [0, 1] the random field {ut(x), x C IRm} possesses a continuous version, and the It6 integral f~ u t ( x ) d W t p o s s e s s a continuous version in (t, x). In fact, for all Ixl, Ix'l _< K, K > 0, we have

<_ %( fo 1 G,Kdt)lx

-

-

x'l

The following theorem provides the relationship between the evaluated integral f~ ut(:c)dWtl==F and the Skorohod integral 5(u(F)). We will need the following hypothesis which is stronger that (h2): (h3) For each (t,w) the mapping x ----* ut(x) is continuously differentiable, and for each K > 0.

-/1 E( I=l_
where u' denotes the gradient o f u and q > 4, q > m. Moreover f3 E(lut(O)12)dt < (X).

T h e o r e m 5.3.1 Suppose that u = {ut(x),O < t < 1,x E IRm} is a random field satisfying conditions (hl) and (h3). Let F : f~ --* IR m be a bounded random variable such that F ~ E ID 1'4 for 1 < i < m. Then the composition u(F) = {u~(F), 0 < t < 1} belongs to the domain of 5 and (5.24) j=l

203

Proof." Consider the approximation of the process u given by

ur(x) = ~ n

~s(x)ds 1(~,~1(t).

(5.25)

i=l

Notice that ut(F) belongs to L2([0, 1] x f~) and the sequence of processes urn(F) converges to ut(F) in L2([0, 1] x ft). Indeed, if F is bounded by K we have

E (fo 1 ut(F)~dt)

<_ fore \lxl_
2 fo 1 E(l~t(0)l ~)dt + 2K~ /o ~E(sup I~'~(~)l~)dt < Ixl<_K

~.

The convergence of u'~(F) to u(F) is obtained by first approximating u(F) by an element of C([0, 1]; L2(f~)). From L e m m a 1.5.3 we deduce that u'~(F) belongs to Dom 5 and

5(un(r))

= y-~ n

u,(F)ds

(W(

) - W( ))

i=1

-~-~n

us)(F)ds(~DrFJdr

(5.26)

i=i j=l

The It6 stochastic integrals fo1 u~(x)dWt satisfiy

E

(/:

~':(~)dW~-

/o1~r(x')dW~

<_ c & - ~'l~.

Hence, by L e m m a 5.3.1 the sequence fo1 u'~(x)dWtl~= y converges in L q to the random variable f~ ~t(~)dW~l~=.

On the other hand, the second summand in the right-

hand side of (5.26) converges to ~jm__1 f~(Ojut)(F)DtFJdt in L 2, as it follows from the estimate

fo 1(Ojut)(F)DtFJdt - fa' (OJUr)(F)DtF~dt 1

<_ ItDFJlIH

I(aj~d(F)

-

(Oy:)(Y)l~dt

The operator 5 being closed the result follows. The preceding theorem can be localized as follows:

~. []

T h e o r e m 5.3.2 Suppose that u = {ut(x), 0 < t < 1, x C IRTM} is a random field that

satisfies (hl) and (h3). Let F : f~ ~ IRTM be a random variable such that F ~ C lDlls for 1 < i < m. Then the composition u(F) is locally in the domain of 5 and (5.2~) holds. Notice that in the above theorem the Skorohod integral 5(u(F)) may depend on the particular localizing sequence, because we do not know if the operator 5 is local in Dom 5.

204

The Stratonovich integral also satisfies a commutativity relationship but in this case we do not need a complementary term. This fact is coherent with the the general behavior of the Stratonovich integral as an ordinary integral. Let us consider the following condition which is stronger that (h2): (h4) There exist constants p > 2 and c~ > m such that Z ( l u t ( x ) - ut(x')l~) _< c,dx - x'l ~, ) - u t ( x ' ) - u , ( x ) + u,(x')F

E(M(x

) _< c/~lx - x'l~ s - tl[,

for all Ixl, Ix'l _< K, K > 0, and s, t 9 [0, 1]. Moreover f@ E(lut(O)l~)dt and t --~ ut(x) is continuous in L 2 for each x.

< ~,

5 . 3 . 3 Let u = {~,(z), 0 < t < 1, = e ~ = } be a random field satisfying hypothesis (hl) and (h4). Suppose that for each x E IR m the Stratonovich integral f~ ut(x) o dWt exists. Consider an arbitrary m-dimensional random variable F. Then u( F) is Stratonovich integrable, and we have Theorem

fo 1 ut( F) o dWt = fo 1 ut(x) ~ dWt x = F . Proof:

Fixapartition~r={0=t0
sTr =

(5.27)

We can write

n-1 __1 ( ft,+, us(F)ds) (W(ti+l) - W(ti)) E ti+l __ ti \ati i=0

n--i

=

~ v~,(F)(W(t,+,) - W(t,)) i=0

+ n-=ll

W(ti))

--

i=0 ti+l --- ti

= al -I- a2. T h e continuity of t ---* ut(x) in L 2 implies t h a t for each x C IR "~ the R i e m a n n sums n--1

u,,(x)(W(t,+~)

-

w(t,))

i=0

converge in L 2, as 17:1 t e n d s to zero, to the It5 integral f3 ut(x)dWt. have for Ixl, Ix'l _< K ,

( "-'

))

P)

E \ ~=o[ut,(x) - ut,(x')](W(h+l)- W ( h

Moreover, we

< c, cKIx - XI] ~.

Hence, by L e m m a 5.3.1 we deduce t h a t al converges in probability to

fo ut( z )dWt ==F " Now we will s t u d y t h e convergence of the t e r m a2. For each fixed x E ]R m, t h e sums -1 1 R~(x) := ,Ei=0 ti+l -- ti .

(s: .

"+'[u~(x) .

ut,(x)]ds .

)

(W(t,+l)

W(h))

205

converge in probability, as lTrI tends to zero, to the difference

V ( x ) := fol Ut(x) o dWt - fol Ut(x)dWt. So it suffices to show that R~(F) converges in probability, as 17el tends to zero, to V ( F ) . This convergence follows from Lemma 5.3.1 and the following estimations where 0 < e < 1 verifies ec~ > m, and Izl, Ix'l _< K,

n-1

1

Q+I

< ~ - f i=0 ti+l - ti Jt~

II[us(X) - ut,(x) - Us(X') + ut~(x')l(w(ti+l) - W(h))Hp~ds

ft--1

_< ep,~ ~

,

[t~+~ - t~189 sup

[E(lu~(x) - u~,(z) - u~(x') + u~,(x

!

)1')]"

s6 [ti,ti+ ll

/=0

<_ e,,~e~lx - z'l~. [] It is also possible to establish substitution formulas similar to those appearing in theorems 5.3.1 and 5.3.3 for random fields u~(x) which arc not adapted. In this case additional regularity assumptions are required. More precisely we need regularity in x of Dsut(x) (for the Skorohod integral) and of (Vu(x))t for the Stratonovich integral. One can also compute the differentials of processes of the form Ft(Xt) where {Xt, t 6 [0, 1]} and {Ft(x), t e [0, 1], x e ]RTM}are generalized continuous semimartingales, i.e., they have a bounded variation component and an indefinite Skorohod (or Stratonovich) integral component. This type of change-of-variables formula axe known as It6-Ventzell formulas. Below we show a change-of-variables of this type which will be used in the next chapter (see [85]). Let us first state the following differentiation rule (see Ocone and Pardoux [85, Lemma 2.3]) . L e m m a 5.3.2 Let F = (F 1. . . . . F m) be a random vector whose components belong to ID l'v, p > 1 and suppose that IF] <_ K. Consider a measurable random field u = {u(x), x E ]R"~} with continuously differentiable paths, such that for any x E ]RTM, u(x) E ]D 1, and the derivative Du(x) has a continuous version as an H-valued process. Suppose that we have E

(sup II<x)l + \IxI_
/

<

< oo,

where q1 + ~1 = ;1 Then the composition G : u( F) belongs to ID 1'~, and we have k

D G = ~ ( O ( a ) ( F ) D F { + (Du)(F). i=1

206

Sketch of the proof." Approximate the composition u(F) by the integral

f~m u ( z ) ~ ( F - x)dx, where r

is an approximation of the identity.

[]

Let W be an N-dimensional Wiener process. Consider two stochastic processes of the form: X~= X; + u~J odW~ + v;ds, 1 < i < m, (5.28)

/:

and

/0

/:

Ft(x) = Fo(x) +

H~(x) o dW; +

We introduce the following set of hypotheses:

/0

G~(x)ds,

(5.29)

(A1) For all 1 < i < N, 1 < j < m, u~J,X j E ]L~'4, v j E LI([0,1]) a.s., and the processes X and u are continuous and bounded by constants K and K1 respectively. (12) x ~ Ft(x) is of class C 2 for all (t,w); the processes Ft(x), F[(x), and F{'(x) are continuous in (t, x); F(x) E]L 1'2 and there exists a version of DsFt(x) which is differentiable in x and for each fixed s E [0, 1] the functions DsFt(x), (D~Ft)'(x) are continuous in the region {s < t < 1, Ixl _< K} (resp. {0 < t < s, Ixl _< K} ). (13) For all Ix] <_ K, 1 < i < N, H~(x) E ILl'2; the processes Hs(x), and H'~(z) are continuous in (s, x); and there exists a version of DsHt(x) which is differentiable in x and for each fixed s C [0, 1] the functions D, Ht(z), (D~Ht)'(x) are continuous in the region {s < t < 1, Ixl _< t(} (resp. {0 < t < s, Ixl -< K} ). (A4) x ~ Gt(x) is continuous. (15) The following estimates hold:

E(

sup

\ Ixl
(IFt(x)l 4+lf/(x)l 4+ IFg'(x)l 4 \

+]Ht(x)l 4 +

IHg(~)l4 + la,(~)4)) < ~ ,

Izl
+lDsH~(~)l 4 + I(O~Hd'(x)l~))ds <

(X3.

T h e o r e m 5.3.4 Assume that the processes Xt and Ft(x) satisfy the above hypotheses. Then (F[(Xt),u~} and H[(Xt) are elements oflLl '2, i = 1,... ,m, and

F~(Xt) = r0(X0) +

/o t

,

X,), ~)' o d W

+

/o>:(

X~),vs)

Moreover if (5.28) and (5.29) hold a.s. for all t E [0, 1] on some set G C ~ then (5.so) holds for all t 9 [0, 11 a.s. on a.

207

Sketch of the proof: To simplify we will assume N = 1. The proof will be done in several steps.

Step 1: Consider an approximation of the identity r on lR TM. For each x C ]Rm we can apply the multidimensional change-of-variables for the Stratonovich integral (Theorem 5.2.3) to the process Ft(x)r - x) and we obtain Ft(x)Or

- X) = Fo(x)Or

- x)

+ . ~ t g ~ ( x ) O , ( X ~ - x) odWs + .~tG~(x)r162 + fot F,(x)(O~Or

x)ds - x)v~ds.

- x)u'~ o dW~ + fot Fs(x)(O,r162

If we express the above Stratonovich integrals in terms of the corresponding Skorohod integrals we obtain

Ft(x)O~(Xt - x) = Fo(X)r

.], H~(x)(c3zOr

+ i t F~(x)(O,r 1

- x)

- x)(VX~)~ds

- x)~'sdW,

t

t

+/~ F~(x)(Oja, O~)(Xs - x)(VX0j~d~ + fot F~(x)(&C~)(X~ - x)v~ds. Notice t h a t all the terms appearing in the above expression have continuous versions in (t, x). Grouping the Skorohod integral terms and the Lebesgue integral terms we can write

JO

JO

Step 2: Consider the processes Hs(Xs) and (F~(Xs),u~}. We claim t h a t these processes belong to the space IL1'2. In fact, first notice t h a t for eazh x, F'(x) belongs to ]L 1'2 and D~(Ft(x)) = (D~Ft)'(x). This follows from hypotheses (A2) and (Ah). On the other hand, X, u C IL 1'4. Then we can apply Lemma 5.3.2 (suitably extended to stochastic processes) to Ht(x) and Xt, and also to (F[(x),ut) and Xt taking into account the following estimates t h a t

/o1E(IH~(Xs)[~) ds <_ E /o1 sup E

/o 1 sup

IIDH,(x)ll~ds < oc,

Ixl<_K

E

sup Izl
IH'(x)l~ds < o~,

IH,(x)12ds < oc,

208

fo1E(l(<(XJ,uJl2)ds

<_ II~I[~E s sup l<(x)12ds < oo, Izl
fo 1F_,(lF'~(xAl211Dusll~)ds <_ E

sup

Izl<_g

[F~s(x)14dsE

HDusll~ds

< oo,

fo 1 E,(IF:'(Xs)1211DXsll~u~)d3 <_ IlullL x {/: E

1

sup If'l(xDI4d~E /o1IIDXsll~ds }. < oo, Ixl
/o1E(ll(OF.),(XjllSu~)ds

<_ IblILE

//

sup bl
II(nfA'(x)ll~ds < ~,

The derivatives of these processes are then given by Ds[Ht(Xt)] :

(H~(X,), DsXt)

+

(Dsttt)(Xt),

and

Ds[(F[(XJ, ut)] = (F['(Xt), DsXt | ut) + (( DsFt)'(Xt), u,) + (F[(X~), Dsut). Then from our hypotheses it follows that the processes to L{ '2, and (V[H,(X0])t (V[(F' - (X.), u.)])t

= =

Hs(XJ and (F~'(XJ, us) belong

(H~(Xt), (VX)t) + (VH)t(Xt), (Ff'(XO, (VX)t | ut) + ((VF)',(Xt), ut) § (F[(X,), (Vu)t).

Step 3." We claim that

/_

i ,11

converges in L l(f~) to the process

9 t = Ft(Xt)-Fo(Xo)-fot[Gs(Xs)+~(VH)s(Xs)

+(O~F)~(X~)(W% + (09,F),(X~)(VXOM

+(OiF)s(Xs)v;] ds.

In fact, the convergence for each (s, w) follows from the continuity assumption on x of the processes Gs(x), ( v g ) s ( x ) , O~(VF)s(x), and the fact that Hs(x) is of class C 1, and Fs(X) is of class C 2. The Ll(f~) convergence follows by dominated convergence because we have

209

From Step 2 it follows that

r

=

-

F,(X,)

-

-

~0t [as(xs)

+
H~(Xs) + (F'~(Xs), us)] o dW~

+ f0t[Hs(Xs) +

Step 4:

Fo(Xo)

{F~(Xs), us>]dWs.

Finally, we have

f ~ ~(s, z)dx = f~m Hs(x)r + f.,~ Fs(~)(a~r

fa,~ a,(s, x)dx converge as e tends (F~(Xs), Us}. Hence, the Skorohod integral

The integrals

fot (/~,~ar

- x)dx -

x)~'sd~

to zero in the norm o f L 1'2 to

dWs

Hs(Xs)+ (5.32)

converges in L2(~) to

r/ot[Hs(Xs) + {F'~(Xs), us)]dWs. By Fubini's theorem (5.32) coincides with expression (5.31). Consequently, we get

Ct = which completes the proof.

S0'H~(X~) [ + (F~(Xs), us)]dWs, []

B i b l i o g r a p h i c a l n o t e s : There are different methods to define stochastic integrals of non adapted integrands with respect to the Wiener process. One approach, developed by Ogawa [87, 88, 89] and Rosinski [94] is based on the orthogonal development of the integrand in L2([0, 1]). Another possibility is to use Riemann sums that converge to the so-called forward integral, which is equal to 6(u) + f~ Dtutdt. This stochastic integral is an extension of the It6 integral, and it follows the same change-of-variable rule. Berger and Mizel [13] introduced this integral in order to solve stochastic Volterra equations. Russo and Vallois [95] introduced the forward integral considering the approximation of the Brownian path by its convolution with a rectangular function. In [56] Kuo and Russ@ study anticipating stochastic integrals in the context of the white noise analysis. Different versions of the change-of-variables formula for the Skorohod integral can be found in Sevljakov [97], Hitsuda [43], [Istiinel [105], and Sekiguchi and Shiota [96]. The indefinite Skorohod integral has properties similar to that of a semimartingale. In particular, one has established the existence of a continuous local time and a Tanaka's type formula in [47] and [48].

Chapter 6 Anticipating stochastic differential equations The anticipating stochastic calculus developed in Chapter 5 can be used to formulate stochastic differential equations where the solutions are nonadapted processes. Such type of equations appear, for instance, when the initial condition is not independent of the Wiener t)rocess, or when we impose a condition relating the values of the process at the initial and final times.

6.1

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 in t h e S t r a t o n o vich sense

In this section we will present examples of stochastic differential equations formulated using the extended Stratonovich integral. These equations can be solved taking into account that this integral satisfies the usual differentiation rules. S t o c h a s t i c differential e q u a t i o n s w i t h r a n d o m initial c o n d i t i o n Suppose that W = {W(t), t E [0, 1]} is a d-dimensional Brownian motion on the canonical probability space (t2, 9r, P). Let Ai : IRm --* IRm, 0 < i < d, be continuous functions. Consider the stochastic differential equation X t = Xo +

A~(Xs) o d W j + i=1

/0

Ao(X~)ds,

t 9 [0, 1].

(6.1)

The initial condition is an arbitrary random variable )2o E L~ ]Rm). Suppose that the coefficients Ai, 1 < i < d, are continuously differentiable with 1,--.m bounded derivatives. Define B = Ao + -~ ~k=Z ~.d ~ = 1 AkO i k A ~, and suppose moreover that B is Lipschitz. Let {Tt(x), t 9 [0, 1]} be the solution to the following stochastic differential equation starting at point x 9 lRm: (pt(x) = x +

Jo' A i ( ~ s ( x ) ) d W 2/+o tB ( ~ s ( x ) ) d s .

i=1

(6.2)

211

In terms of the Stratonovich stochastic integral we can write

i=l

We know (see, for instance, Kunita [54]) that there exists a version of ~t(x) such that (t, x) ~ ~t(x) is continuous, and we have E(l~s(x)

- ~(x')l')

_<

C,,,~(It

- sl~ + Ix - x'lP).

for all s, t ~ [0, 1], N , [x'l -< K, p > 2, K > 0. Then we can establish the following result T h e o r e m 6.1.1 Assume that the coefficients Ai, 1 < i < d are of class C 3, and Ai,

1 < i < d, B have bounded partial derivatives of first order. Then for any random vector Xo, the process X = {~t(X0), t e [0, 1]} satisfies the anticipating stochastic differential equation (6.1). Proof: Under the assumptions of the theorem we know that for any x E ]Rm Eq. (6.2) has a unique solution Sot(x). Set X = {Sot(Xo), 0 < t < 1}. By a localization argument we can assume that the coefficients Ai, 1 < i < d, and B have compact support. It suffices to show that for any i = 1 , . . . , d, the process u~(t, x) = Ai(~t(x)) satisfies the hypotheses of Theorem 5.3.3, suitably extended to a d-dimensional Brownian motion. Condition (hl) is obvious. On the other hand, using It6's formula one can easily check condition (h4). This completes the proof. [] The uniqueness is a more difficult problem. We are going to show that there 1,4 m is a unique solution in the class of processes 1L2.1oc(]R ), assuming some additional regularity conditions on the initial condition X0. We will denote by L(]R m) the space of m-dimensional processes (or random variables) whose components belong to the class L. 1,p L e m m a 6.1.1 Suppose that Xo C ]Dlor

m

) for some p > q > 2, and assume that the coefficients Ai, 1 < i < d, and B are of class C 2 with bounded partial derivatives 1,q m of first order, then we claim that {Sot(X0), 0 < t < 1} belongs to IL2,1or ).

Proof:

1,p m Let (~n,X~) be a localizing sequence for X0 in IDiot(JR ). Set an,k.M = ~ n n { [ X 3 l 'Q ]~} N {

sup

I~,(X)I ~ M } .

]xl
On the set G n'k'M the process 99t(Xo) coincides with SotM (Xol3k(Xo)), n n where /3k is a smooth function such that 0 < /3k ~ 1, /3k(x) = 1 if Ixl < k, and /3k(x) = 0 if Ixl > k + 1, and ~M(x) is the stochastic flow associated with the coefficients ~MAi, 1 < i < d, and 13MB. Hence, we can assume that the coefficients A~, 1 < i < d, and B are bounded and have bounded derivatives up to the second order, X0 E lDl'p(lRm), and X0 is bounded by k. Now we can apply Lamina 5.3.2. The following estimates hold: E (

sup

\l~l_
[IW,(x)l ~ + IID[So,(x)lll~] ~ /

E(

sup [~;(x)[ ~) kl~l
<

~,

(6.3)

<

c~,

(6.4)

212 for any r _> 2. These estimate (69 and (6.4) follow from our assumptions on the coefficients A~, 1 < i < d, and B, taking into account that

= A j ( ~ ( x ) ) + Lt(OkAt)(~r(x))D~[~k(x)]dW~

D~[~t(x)]

t

"

k

+ L (OkB)(cpT(x))D~[~p~(x)]dr, for 1 <_ j _< d and 0 _< s _< t _< 1. Hence, from Lemma 5.3.2 we obtain that {~t(X0), 0 < t < 1} belongs to ILI'q(IRm) and that Ds[~t(Xo)] = ~'t(Xo)D, Xo + [D,~t](Xo). Finally, from the above expression for the derivative of ~t(Xo) it is not difficult to show that {~t(X0), 0 < t < 1} belongs to L~'q(IRm), and (D[~(Xo)])~+'- = ~'~(Xo)D~Xo + (D~) +' (Xo).

[] T h e o r e m 6.1.2 Assume that the coefficients Ai, 1 < i < d and B are of class C a,

with bounded partial derivatives of first order. The initial condition Xo belongs to

1,p m IDIot(JR ) for some p > 4. Then the process X = {~t(X0), t e [0, 1]} is the unique

1,4 m IL2,,o~(]R )

solution to Eq. (6.1) in

which is continuous.

1,4 m Proof.9 By Lemma 69 we already know that X belongs t o IL2,1oc(]R ). Let Y be another continuous solution in this space. Suppose that (f~, X~') is a localizing 1,p rn 1 4 m sequence for X0 m9 ]Dloc(]R ), and (f~,l, rn) is a localizing sequence for Y m L 2 ( I R ) . Set

Hn,k.M = ~n n a n'l n {IXgl _< k} n {

sup I~,(x)l _< M} r-I { sup I~1 <- M}. Ixl
The p r o c e s s e s Y t n ~ M ( Y t n ) , (~MA~)(Ytn), and ( ~ M A o ) ( Y t n) satisfy hypothesis (A1). On the set H n'kM w e have the following equality

Yt = Xo +

A~(Y~) o dW~ +

/0'Ao(Y~)ds,

t C [0, 1].

Let us denote by ~ t l ( x ) the inverse of the mapping x ---* ~t(x). By the classical It6's formula we can write d

t

~ t l ( x ) = x -- E fo (~'s(q~

t

dW: - fo (qJs(qf~l(x)))-lA~

i=1

We claim that the processes Ft(x) = ~ t l ( x ) , H i ( x ) = (~'s(~tl(z)))-XAi(x), and Gt(x) = (T's(~tl(x)))-lAo(x) with values in the space of m x m matrices satisfy hypotheses (A2) to (A5). This follows from the properties of the stochastic flow associated with the coefficients Aj.

213 Consequently, we can apply Theorem 5.3.4 and we obtain, on the set H n'k'M,

~-i(~) -- Xo + fnt(~;1)'(Y~)Ai(Y~)

o dW; + .[t(~;')'(Y~)Ao(Y~)ds

Notice t h a t

(r

= _ ( ~ ( ~'

-~ ( x ) ) ) - l .

Hence, we get ~ - l ( Y t ) = X0, t h a t is, Yt = ~r uniqueness.

which completes the proof of the []

One-dimensional stochastic differential equations with random coefficients Consider the one-dimensional equation

Xt -- Xo +

/0a(Xs) o dW~ + f b(Xs}ds,

(6.5)

where the coefficients r and b and the initial condition Xo are random. If the coefficients and the initial condition are deterministic and we assume t h a t the function a is of class C 2 with b o u n d e d first and second derivaties, and t h a t b is Lipsehitz, then there is a unique solution Xt. This solution (see Doss [26]) can be written as Xt = u(Wt, Yt), where u : ]R2 --~ IR is the solution of the ordinary differential equation 0u

a-~ = ~ ( ~ ( z ) ) ,

~(0, ~) = y,

and Y~ is the solution to the ordinary differential equation with random parameters

Yt = Xo +

f f(Ws, Ys)ds,

where f(x, y) = b(u(x, y)) exp ( - f~ ~'(u(z, y))dz). Notice that

oy

exp

~'(~(~,

y))~z).

Suppose t h a t now the coefficients and the initial condition are random. We claim t h a t in t h a t case the process Xt = u(Wt, Yt) solves Eq. (6.5). Consider a partition 7r = {0 = to < t~ < . . . < tn = 1} of the interval [0,1]. We can introduce the polygonal approximation of the Brownian motion defined by

vc

/o~'~-'E (w(t~+~)-wm))

1N.t~+d (s)ds.

Set X~ = u(W~', 5). This process satisfies

X [ = Xo +

/: a ( X : ) W9: d s + z exp (/j

)

a'(u(y, Y~))dy b(X~)ds = A~ + A~.

The term A~ can be written as A~ = [ [~(X~~ ) - ~(X~)]l/V'~ds JO

+

= ~:+B~. fo~O(x~)W:ds

214

Note t h a t B~ is the Riemann sum that approximates the Stratonovich integral of a ( X , ) . We know t h a t At1 converges in probability to X t - Xo - fd b ( X , ) d s as [rr I tends to zero. As a consequence, if we show t h a t B 1 converges in probability to zero as Irr] tends to zero this will imply a ( X t ) is Stratonovich integrable on any interval [0, t] and (6.5) holds. We have 1

- ws) + ~[~(~')~ + ~"](r

~(x:) - ~(x~) = o~'(x~)(w:

- w ~ ) ~,

where { is a r a n d o m intermediate point between W [ and W,. The last s u m m a n d of the above expression does not contribute to the limit of B~. Finally the convergence to zero of B~ follows from Lemma 6.19 L e m m a 6.1.2 Let q~t be a continuous process9 Then, the integral fO 1 ~ w ;" 7r(w~7r - W~)ds

converge in probability to zero as ]Tr[ tends to zero, where 7r = {0 = to < tl < ' " < tn = 1} ~ m s over all partitions of the interval [0, 1]. Proof." Fix a partition 7r = {0 = to < t1 < " . ' < tn = 1}. Consider another partition of the interval [0, 1] of the form {sj := ~ , 0 < j < m}. We can write 9 7r

7r

fo ' CsW~ ( W j - W , ) d s <_ m-1 Cs, i~j+ ~ W'~ . ~ ( W j - W~)ds J + ~-s c*,+~ ( ~ j=O J

~b~, )W;" '~

~'(W;- W s ) d s = C 1 -~ C 2.

The term C 1 converges to zero in probability as Irrl tends to zero, for any fixed value of m. In fact, if the points sa and sj+l belong to the partition 7r, we have

i

9 7r (W~7r - W s ) d s = s'+I W~ E w(ti+l) - w(ti) J i:tie[s~,sj+l) ti+l 7 ~ i

-

1 u

( ft'+l(w~r _ Ws)ds \Jti

1 rti+l Vd E j,, [( ,,+, - w , ) = - ( w , - W,,)=]d~, i:tiE[sj,sj+l) ti+l -- ti

1 1 a n d this converges to ~(Sj+l - 8j) - ~(Sj+l - 8j) = O. The term 6'2 can be bounded as follows: C 2 ~

m-1 i

I{~S -- ~ ' l E

sup

I*-*1_<{

j=o

"r~

re

Sj+I IW8 ( W ~ -- W * ) l d s

J

_< sup I ~ , - ~ / ' t l I~-tl_<{ m-1

• ~

j=O

E

{:ti~[Sj,Sj+l )

1

2(t~+~- t~)

~t2i +1

[(w~,+l-

w~) ~ + (w~ - WJ]ds.

)

215

As n tends to infinity the right-hand side of the above inequality converges in probability to 89 Hence, we obtain 1 sup [~s-- qSt[, limsup C2 _< ~ 18-tJ_<~ and this expression converges to zero in probability as m tends to zero, due to the continuity of the process ~St. [] The uniqueness requires the use of the It6-Ventzell formula, and we have to impose regularity assumptions on the coefficients (cf. [51]).

6.2

S t o c h a s t i c differential e q u a t i o n s w i t h b o u n d ary c o n d i t i o n s

Consider the following Stratonovich differential equation on the time interval [0, 1], where instead of giving the value X0, we impose a linear relationship between the values of X0 and XI:

dXt = ~d= 1Ai(Xt)

o

dW~ + Ao(Xt)dt

(6.6)

h(Xo,Xl) = 0 We are interested in proving the existence and uniqueness of a solution for this type of equations. We will discuss two particular cases: (a) The case where the coefficients Ai, 0 < i < d and the function h are affine (see Ocone and Pardoux [86]). (b) The one-dimensional case (see Donati-Martin [25]).

Linear stochastic differential equations with boundary conditions Consider the following stochastic boundary value problem:

Xt = Xo + ~i=1 d f~ A~Xs o dW~ + Ao f~ Xsds,

(6.7)

HoXo + H1X1 = h. We assume that Ai, i = 0 , . . . , d, H0, and H1 are m • m deterministic matrices and h C ]Rm. We will also assume that the m • 2m matrix H0 : HI has rank m. Concerning the boundary condition, two particular cases are interesting:

Two-point boundary-value problem: Let l E IN be such that 0 < 1 < m. Suppose that H0 =

, H1 =

H~'

, where H~ is an l x m matrix and H i' is an (m - l) x m

matrix. Condition rank (H0 : H1) = m implies that H~ has rank l and that H~' has rank m - l .

If we write h =

h 0 ) , where h0 C IRl and hi C IR"~-z, then the boundary hi /

condition becomes g ; X o = ho,

H'IX1 = hi.

216

Periodic solutions to SDE's: Suppose H0 = - H 1 = I and h = 0. Then the boundary condition becomes X0 = X1. With (6.7) we can associate an m x m adapted and continuous matrix-valued process de solution of debt = ~}-~i=l d Ai~t o dW~ + BcStdt, (I)0 = I.

(6.8)

Using the properties of the extended Stratonovich integral, one can obtain an explicit solution to Eq. (6.7). By a solution we mean a continuous stochastic process X such that A{(Xs) is Stratonovich integrable with respect to W i on any interval [0, t] and such that (6.7) holds. T h e o r e m 6.2.1 Suppose that the random matrix Ho + HI(I)I is a.s. invertibIe. Then there exist a solution to the stochastic differential equation (6.7), which is unique 1,4 among those continuous processes whose components belong to the space ]L2,1o c. Proof." Define Xt : ~tXo,

(6.9)

where X0 is given by

xo : %

+

(6.10) 1,4

Then it follows from this expression that X i belongs to ]L2,1oc,for all i = 1,..., m, and this process satisfies Eq. (6.7). 1,4 m In order to show the uniqueness, we proceed as follows. Let Y E IL2,1oc(lR ) be a solution to (6.7). Then we have d

t

1

t

By the change-of-variables formula for the Stratonovich integral (Theorem 5.2.2), we see that q)tlYt = Y0, namely, Yt : q)tY0. Therefore, Y satisfies (6.9). But (6.10) follows from (6.9) and the boundary condition HoYo + H1Y1 = h. Consequently, Y satisfies (6.9) and (6.10), and it must be equal to X. [] Notice that, in general, the solution will not belong to IL~'4(IR"). One can also treat more general linear equations. For instance, suppose that we have the equation: d t i t Xt = Xo + ~i=1 f~) AiXs o dW~ + f~) AoXsds + Vt,

(6.11)

HoXo + HIX1 : h, where Vt is a continuous semimartingale. In that case,

1,4 is a solution to equation (6.11). The uniqueness in the class of processes IL2,1or can be established provided the process V, belongs also to this space.

m

)

217

One-dimensional stochastic differential equations with b o u n d a r y

condi-

tions Consider the one-dimensional stochastic boundary value problem

Xt = Xo + ft o-(X~) o dW~ + ft b(X~)ds aoXo + alX1 = a2.

(6.12)

Applying the techniques of the anticipating stochastic calculus we can show the following result.

Theorem 6.2.2 Suppose that the functions a and bl := b+ 89

are of class C 2 with bounded derivatives and aoal > O. Then there exists a solution to (6.12). Furthermore if the functions a and bl are of class C 4 with bounded derivatives then the solution is 1,4 unique in the class ]L2,1o c.

Proof: Let ~t(x) be the stochastic flow associated with the coefficients a and bl. By Theorem 6.1.1 for any random variable X0 the process ~t(Xo) satisfies Xt = Xo +

/0

o-(X~) o dW~ +

/0

b(X~)ds.

Hence, in order to show the existence of a solution it suffices to prove t h a t there is a unique random variable X0 such t h a t

~l(Xo) - a2 - aoXo

(6.13)

al

The mapping g(x) = ~ l ( x ) is strictly increasing and this implies the existence of a unique solution to equation (6.13). Let us now turn to the proof of the uniqueness. It suffices to show t h a t the unique solution X0 to equation (6.13) belongs to IDllof for some p > 4. Taking into account that the equation is one dimensional, one can represent the solution 9~t(x) by means of the m e t h o d of Doss. Hence, pt(x) is Fr~chet differentiable in w. Using this fact, and the implicit function theorem it is not difficult to show t h a t X0 belongs to the 1,p space IDloc. []

6.3

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 in t h e S k o r o hod sense

Let W = {Wt, t E [0, 1]} be a one-dimensional Brownian motion defined on the canonical probability space (f~, ~-, P). Consider the stochastic differential equation

Xt = Xo +

f0t cr(s, Xs)dW~ + f0' b(s, X~)ds,

(6.14)

0 < t < 1, where X0 is .Tl-measurable and a and b are deterministic functions. The stochastic integral f t a(s, Xs)dW~ is interpreted in the Skorohod sense. First notice t h a t the usual Picard iteration procedure cannot be applied to show t h a t this equation has a unique solution. Indeed, the Sokorohod integral is not a bounded operator in

218

D ~ In some sense Eq. (6.14) is an infinite dimensional hyperbolic partial differential equation. In fact, this equation can be formally written as

Xt=Xo+

~ot c~(s,X~)odW~+-~l~ot a ' ( s , X ~ ) [ ( D + X ) ~ + ( D - X ~ ) ] d s + j~ot b(s, Xs)ds.

If the diffusion coefficient is linear one can show that there exist a unique global solution. In fact, let us consider the following particular case:

X t = Xo + a

~0t X~dWs.

(6.15)

When X0 is deterministic, the solution is given by the martingale

Xt = XodW~-89176 We have seen in (5.4) that if Xo = signW1, then, a solution to Eq. (6.15) is

Xt = sign(W1

at)e aw~-89

-

More generally, one can show (see Theorem 6.3.1 below) that if X0 E LP(f~) for some p > 2, then X t = Xo(At)eawt- 89~% is a solution to Eq. (6.15) where At(w)s = us - ~ ( t A s). In terms of the Wick product (see [22]) one can write Xt

= Y~ 0 vA e

aWt- 89

1 2t = Xo(At)e~w~-~o

Consider the following equation:

X t = Xo +

/o'

a~X~dW~ +

/o'

b(s, Xs)ds,

0 < t < 1,

(6.16)

where a E L2([0, 1]), X0 is a random variable, and b is a random function satisfying the following condition: (H.1) b : [0, 1] • ]R • ~t --* ]R is a measurable function such that there exist an nonnegative function % on [0,1], Vt -> 0, a c o n s t a n t L > 0, and a s e t N1 C ~- of probability one, verifying I b ( t , x , w ) - b(t, y,~)l Ib(t, 0,~)l

_< %1x - y[, <

~0t 7tdt

<_ L,

L,

for atlx, y C l R , t E [ 0 , 1 ] and ~ E N 1 . Let us introduce some notation. Consider the family of transformations Tt, At : ~ f~, t G [0, 1], given by tA$

Tt(~)8 = ~ , +

~du, JO tAs

At(za)s = Us --

fJ0

a,~du.

219

Note that

TtAt = AtTt

= Identity. Define et = exp(f0 t a~dW~-~l

fot(r:ds)

Then, by Girsanov's theorem E[F(At)st] = E[F] for any random variable F E L l ( a ) . For each x E 1R and w E f~ we denote by Zt(w, x) the solution of the integral equation

Zt(w,x) = x + fote21(Tt(w)) b(s, es(Tt(w))Zs(ca, x),T~(w))ds.

(6.17)

Notice that for s _< t we have e,(Tt) = exp( f~ a~dW,, + 89]~ a2~du) = e~(T~). Henceforth we will omit the dependence on w in order to simplify the notation.

Fix an initial condition Xo E I_2(•) for some p > 2, and define Xt = etZt (At, Xo(At) ) . (6.18) Then the process X = {Xt, 0 < t < 1} satisfies l[0,t]aX E Dom5 for all t E [0, 1], X E L2([0, 1] • ~), and X is the unique solution of Eq. (6.16) verifying these conditions. Proof: We will only prove the existence. The uniqueness can be shown using similar

T h e o r e m 6.3.1

arguments. Let us prove first that the process X given by (6.18) satisfies the desired conditions. By Gronwall's lemma and using hypothesis (H.1), we have

IX, I < etetn(IXo(A,)l + L

e;l(rs)ds),

(6.19)

which implies suPte[0.1] E(IX, Iq) < oo, for all 2 <_ q < p, as it follows easily from Girsanov's theorem and HSlder's inequality. Indeed, we have

~ CqeqL{E (gq-l(Tt),Xo,qq-Lq~q-l(Tt)~otgsq(T:)ds)} _<

c {E (IXolp) + 1}.

Now fix t E [0, 1] and let us prove that l[o,tlaX E Dom~ and that (6.16) holds. Let G E ,S be a smooth random variable. Using (6.18) and Girsanov's theorem, we obtain

= E[ fo'a, esZ~(As,Xo(A~))D~Gds ]

E[ fota~X, DsGds]

E[ fota~Z,(Xo)[D~G](Ts)ds].

(6.20)

Notice that ~ G ( T s ) = a s [ D ~ a ] ( T ~ ) . Therefore, integrating by parts in (6.20) and again applying Girsanov's theorem yield =

- fo' e:'(T,)b(s, es(Ts)Z,(Xo), Ts)G(T,)ds]

=

E[XtG] -

E[Xoa] - fot E[b(s, X~)a]ds.

220

Because the random variable Xt - Xo - f~ b(s, X~)ds is square integrable, we deduce that l[0,tl(rX belongs to the domain of ~ and that (6.16) holds. [] When the diffusion coefficient is not linear one can show that there exist a solution up to a random time. Existence and uniqueness results for one-dimensional Skorohod equations with random initial condition are established by Buckdahn in [20]. Let us state the main result of this paper. Its proof is based on Doss representation of the solution to a one-dimensional stochastic differential equation and on the change-ofvariable formula for the Skorohod integral. We recall that i denotes the mapping from H = L2([0, 1]) into t2 = C0([0, 1]), defined by i(h)t = f~ hsds. T h e o r e m 6.3.2 Consider the stochastic differential equation in the Skorohod sense

Xt = Xo + fot(r(X~)dWs + fotb(Xs)ds.

(6.21)

Suppose that ~ and b are functions of class C 2 with bounded derivatives of first and second order. Suppose that the initial condition Xo is bounded and there exist a process D~Xo bounded by M such that Xo(w + i(h)) = Xo(w) +

/o 1DsXoh~ds

+ o(llhllg),

IDsXo(w + i(h)) - DsXo(w)l < MlthllH. Then for any bounded ball Br(h) of radius r > 0 around i(h) there exist a t > 0 such that Eq. (6.21) has a unique solution X on [O,t] x Br_3tllo, ll~(h) which belongs to LI([O, t] x Br(h)) and it possesses and extension 2 in the space 1D~2 (set of elements in ID 1'2 with a bounded derivative).

B i b l i o g r a p h i c a l n o t e s : There are several papers devoted to the study of the anticipating process ~t(X0), composition of a stochastic flow with a random initial condition. The large deviation principle has been investigated in [68] and [69], the characterization of the support of the law of the solution in [67], the absolute continuity of the law was studied in [65], and the regularity of the density was treated in [23]. Stochastic differential equations in the Skorohod sense were analyzed by Buckdahn, in the linear case, in connection with the anticipating Girsanov theorem (see [19] and [21]). Other works on the Skorohod stochastic differential equations are [78] (multidimensional Brownian motion) and [22] (linear multidimensional case).

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LISTE

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FLORIT Carme

DES AUDITEURS

UFR Math6matiques et Informatique, Universit~ Ren~ Descartes, PARIS V Laboratolre de Probabilit~s, Universit~ PARIS VI URA CNRS 1378 Math~matiques, Univeruit~ de ROUEN l,aboratoire de ProbabilitY, Universit~ PARIS VI Laboratoire de Probabilit~s, Universit~ PARIS VI D6partement de Math~matiques, Unlverslt~ de ROME (Italie) Laboratoire de Probabilit~s, Universit~ PARIS VI D~partement de Math~matlques, Universit~ de BARCELONE (Espagne) Laboratoire de Math~matlques Appliqug~s, Universit~ Blaise Pascal, CLERMONT-FD D6partcment de Mathdmatiqucs, Universit6 Paris-Sud (OILSAY) Laboratoire de Math~matiques Appliques, Universit(~ Blaise Pascal (CLERMONT-FD) URA CNRS 1321, Universit~ PAI)dS VII l,aboratoire de Probabilitds, Universit~ PARIS VI Laboratoire de Probabilit~s, Universit~ PARIS VI Laboratoire de Mathdmatiques Appliqu~,es, Universit~ Blaise Pascal, CLERMONT-FD UFR Mathdmatiques, Universit~ PARIS VII D~partement de Math(~matiques, Unlversit~ de BREST Unitd de Recherche INRIA, SOPHIA ANTIPOLIS D6parternent de Mathdmatiques, Universitd de ROME (Italic) Ddpartement of Math~matiques, Univeruit(~ Paris-Sud (OILSAY) Laboratoire de Math~matiques, Ecole Normale Sup~rieure, PARIS lastitut de Math~matiques, Universit~ de FIRENZE (Italie) D@artement de Math~matiques, Universit~ Paris-Sud (ORSAY) Laboratoire de Probabilitds, Universitd PARIS VI Laboratoire de Math6matiques Appliqu6es, Universit~ Blaise Pascal, CLERMONT-FD D6,partement de Math~matiques, Unlversit~ de VERSAILLES URA CNRS 1378,Univcrsit6 de ROUEN Institut Elie Cartan, VANDOEUVRFrLES-NANCY l~ole Supdrieure des Tdl&ommunications Ddpartement R~seaux, PARIS CERMICS - ENPC, La Courtine Ddpartment de Mathdmatiques, Universitd du MANS Laboratoire de Probabilit~s, Universitd PARIS VI Laboratoire de Probabilit~s, Universitd PARIS VI CPHT, Ecole Polytechnique, PALAISEAU D6partcment de Mathdmatiques, Universit6 Cadi Ayyad MAItRAKECII (Maroc) D(~partement de Math~matiques, Facult~ des Sciences de RABAT (Maroc) D~partement de Math~matiques Universitd de BARCELONE (Espagne) D~partement de Math~matiques Universitd de PADOVA (Italie) D@artcment de Math~matiques Universit~ de BARCELONE (Espagne)

231 Mlle Mr Mr Mile Mr Mr Mr

FRADON Myriam FRANZ Uwe GALLARDO Ldonard GANTERT Nina GARNIER Josselin GRADINARU Mihai GRISIIIN Stas

Mr

G R O R U I ) Axel

Mr

G U I M I E R Alain

Mr Mr

GUIOT'rO Paolo HU Ying

Mile Mr Mme

JACQUOT Sophie JAKUBOWSKi Jacek JOLIS Maria

Mr Mile Mr

I,ACIIAL Aired LAGAIZE Sandrino LEDOUX Michel

Mr Mr Mr Mlle Mr

LE GALL Jean-Franc;ois LEURIDAN Christophe L O R A N G Gdrard M A L I T A S C A R I A T Elena M A R Q U E Z David

Mile Mine Mr Mile Mr Mr Mr

MARSAI,I,E Laurence MASTROENI Lorctta MATHIEU Pierre MAVIRA MANCINO MAZLIAK Laurent MESNAGER Laurent MICLO Laurent

Mine Mme Mile Mme

MILLET Annie MULINACCI Sabrina NAJA Dania NEGRI Ilia

Mr Mr

OUZINA Mostafa PARDOUX Etienne

Mile Mr Mr Mr

PAROUX Katy PESZAT Szymon PIAU Didier P[CARD Jean

Mme Mine Mr

PIETRUSKA-PALUBA Katarzyna PONTIER Monique RACHAD Abdelhak

Mr

ROUX Daniel

Mr

ROVIRA Caries

Ddpartement de Mathdmatiques, Universitd Paris-Sud (ORSAY) Ddpartement de Mathdmatiques, Universitd de N A N C Y I Ddpartement de Mathdmatiques, Universit~ de B R E S T Fachbereich Mathematik, BERLIN (Allemagne) CMAP, Ecole Polytechnique, P A L A I S E A U Ddpartement de Mathdmatiques, Universit6 Paris-Sud (ORSAY) Department of Mathematics, University of California IIWINE (USA) Centre de Mathdmatiqucs et Informatique, Universitd de Provence, MAIKqEILLE Ddpartement de Mathdmatiques, Unlversitd de Y A O U N D E (Cameroun) Ecole Normale Supdrieure, PISE (Italie) Laboratoire de Mathdmatiques Appliqudes Universltd Blaise Pascal ( C L E R M O N T - F D ) Ddpartement de Mathdmatiques, Universitd d'ORLEANS Instituteof Mathematics, University of W A R S A W (Polognc) Ddpartement de Mathdmatiques, Univcrsitd de B A R C E L O N E (F~spagne) INSA de I N O N l)dpartement do Mathdmatiqucs, Universitd d'ORLEANS Laboratoire de Statistique ct Probabilitds, Universitd de T O U L O U S E Laboratoire de Probabilitds, Universitd PARIS VI ]nstitut Fourier, SAINT M A R T I N D'IIERES Ddpartement de Mathdmatiques, Universit6 de N A N C Y I Laboratoire de Mathdmatiques, Universitd de N A N C Y I Ddpartement de Mathdmatiques, Universltd de BARCELONE (F~pagne) Laboratoire de Probabilitds, Universitd PARIS VI Facultd d'Economie, Universitd de ROME (italie) CMI-LATP, Universitd de Provence, MARSEILLE Institut de Mathdmatiques, Univemitd de FIRENZE (Italic) Laboratoire de Probabilitds, Universitd PARRS VI Ddpartement de Mathdmatiques, Universitd Paris-Sud (ORSAY) Laboratoire de Statistiques et Probabilitds, Universitd de TOULOUSE Laboratoirn de Probabilitds, Univcrsitd PARIS VI Ddpartement de Mathdmatiques, Universitd de PISE (Italie) Institut Elie Caftan, VANDOEUVRE-LES-NANCY l)dpartement do Statistique et Mathdmatiques, Universitd de MILAN (Italic) URA CNRS 1378, Universtd de ROUEN Ddpartement de Mathdmatiques, Universitd de Provence, MARSEILLE Ecole Normale Supdrieure, LYON I Institut de Mathdmatiques, KRAKOW (Pologne) Laboratoire de Probabilitds, Universitd de LYON [ Laboratoire de Mathdmatiqucs Appliqudes, Universltd Blaise Pascal, CLERMONT-FD Institut de Mathdmatiques, Universitd de WARSAW (Pologne) Ddpartement de Mathdmatiques, Universitd d'ORLEANS Laboratoire de Mathdmatiques Appliqudes, Universitd Blaise Pascal, CLERMONT-FD Laboratoirc de Mathdmatiques Appliqudes, Universitd Blaise Pascal, CLERMONT-FD Ddpartement de Mathdmatiques, Universit~ de BARCELONE (Espagne)

232 Mlle Mr Mr

SAADA Diane SABOT Christophe SAINT LOUBERT BIE Erwan

Mile

SARRA Monica

Mlle

SAVONA Catherine

Mr

SERLET Laurent

Mine Mr

THIEULLEN Mich~le TINDEL Samy

Mr Mr

TOUBOL Alain TROUVE Alain

Mme

TROUVE Isabelle

Mr

VIENS FYederi

Mile Mr

WANTZ Sophie WOLF Jochen

Mr Mr

YAKOVLEV Andrei ZEITOUNI Ofer

U F R Mathdmatlques de la Ddclsion, Universltd PARIS IX Laboratolre de Probabilitds, Universitd PARIS VI Laboratoire de Mathdmatiques Appliqudes, Universitd Blaise Pascal, C L E R M O N T - F D Ddpartement de Mathdmatiques, Universit~ de B A R C E L O N E (Espagne) Laboratoire de Mathdmatiques Appliqu6m, Universitd Blaise Pascal, CLERMONT-FD Ddpartement de Mathdmatiquea, YAMOUSSOUKRO (C6te d'Ivoire) Laboratoire de Probabilitds, Universitd PARIS VI Ddpartement de Statistique, Universitd de BARCELONE (Espagne) ENPC - CERMICS, La Courtine Ddpartement de Mathdmatiques et Informatique, Ecole Normale Supdrieure, PARIS UFR Sciences Economiques, Universitd Paris-Nord, VILLETANEUSE Department of Mathematics, University of California, IRVINE (USA) Institut Elie Cartan, V A N D O E U V R F ~ L F ~ S - N A N C Y Institut fiirStocha.stik Universifiit J E N A (Allemagne) Ddpartement de Mathdmatiques, Universlt6 d ' O R L E A N S Department of Electrical Engineering, Teehnion Institute, IIAIFA (Israel)

LIST OF PREVIOUS VOLUMES OF THE "Ecole d'Etd de Probabilitds"

1971 -

1973 -

1974 -

1975 -

1976 -

J.L. Bretagnollc (LNM "Proccsslm h accroi~cments in(ldpcn(lants" S.D. Chatterji "Les martingales et leurs applications analytiques" P.A. MEYER "Presentation des processus de Markov" P.A. MEYER (LNM "Transformation des processus dc Markov" P. PRIOURET "Proccssus dc diffusion ct 6quations (tiff(',rcntiellcs stochastiqucs" F. SPITZER "Introduction aux processus de Markov ~ param~tres dans Z," X. FERNIQUE (LNM "Rdgularit5 des trajectoires des fonctions aldatoires gaussiennes" J.P. CONZE "Syst5mcs topologiqucs ct mdtriques cn thdoric crgodiqlm" J. GANI "Processus stochastiques de population" A. BADRIKIAN (LNM "Proldgom~nes au calcul des probabilitds dans les Banach" J.F.C. KINGMAN "Subadditive proccsscs" J. KUELBS "The law of the iterated logarithm and related strong convcrgcncc thcorcms for Banach space valued random variablcs" (LNM J. HOFFMANN-JORGENSEN "Probability in Banach space" T.M. LIGGETT "The stochastic evolution of infinite systems of interacting particles" J. NEVEU "Processns ponctuels"

307)

390)

480)

539)

598)

234

1977-

1978 -

1979 -

1980-

1981-

D. DACUNHA-CASTELLE "Vitesse de convergence pour certains probl~mes statistiques" H. HEYER "Semiogroupes de convolution sur un groupe localement compact et applications ~ la th~orie des probabilit~s" B. I.tOYNETTE "Marches al~atoircs sur les groupes de Lie" R. AZENCOTT "Grandes d~viations et applications" Y. GUIVARC'H "Quelques propri~t~s asymptotiques des produits de matrices aldatoires" R.F. GUNDY "Indgalitds pour martingales & un et dcux indices : l'espace H p'' J.P. BICKEL "Quelques aspects de la statistiquc robuste" N. EL KAROUI "Les aspects probabilistes du contr61c stochastique" M. YOR "Sur la thdorie du filtrage" J.M. BISMUT "Mdcanique aldatoire" L. GROSS "Thermodynamics, statistical mechanics and random fields" K. KI'tICKEBEI'tG "Processus ponctuels en statistique" X. FERNIQUE "R~gularit6 de fonctions al6atoircs non gaussiennes" P.W. MILLAR "The minimax principle in asymptotic statistical thcory" D.W. STROOCK "Some application of stochastic calculus to partial differential equations" M. WEBER "Analyse infinit~simale de fonctions al6atoires"

(LNM 678)

(LNM 774)

(LNM 876)

(LNM 929)

(LNM 976)

235

1982 -

R.M. DUDLEY "A course on empirical processes" H. KUNITA "Stochastic differential equations and stochastic flow of diffeomorphisms" F. LEDRAPPIER "Quelqucs propri~tds des expo~nts caractdristiques" 1983D.J. ALDOUS "Exchangeability and related topics" I.A. IBRAGIMOV "Th~or~mes limites pour les marches al~atoires" J. JACOD "Th~r~mes limites pour lcs proccssus" 1984 R. CARMONA "Random SchrSdingcr opcrators" H. KESTEN "Aspects of first passage percolation" J.B. WALSH "An introduction to stochastic partial differential equations" 1985-87- S.R.S. VARADHAN "Large deviations" P. DIACONIS "Applications of non-commutative Fourier analysis to probability theorems" H. FOLLMER "Random fields and diffusion processes" G.C. PAPANICOLAOU "Waves in one-dimensional random media" D. ELWORTHY "Gconmtric aspects of diffusions on manifolds" E. NELSON "Stochastic mechanics and random fields" 1986 O.E. BARNDORFF-NIELSEN "Parametric statistical models and likelihood"

(LNM 1097)

(LNM 1117)

(LNM 1180)

(LNM1362)

(LNS 5O)

236

1988-

1989-

1 9 9 0

-

1991 -

1992 -

1993 -

A. ANCONA (LNM "Thc%rie du potentiel sur los graphes et les varidt~s" D. GEMAN "Random fields and inverse problems in imaging" N. IKEDA "Probabilistic methods in thc study of asymptotics" D.L. BURKHOLDER (LNM "Explorations in martingale theory and its applications" E. PARDOUX "Filtrage non lindaire et dquations aux d6rivdes partielles stochastiques associ~es" A.S. SZNITMAN "Topics in propagation of chaos" M.I. FREIDLIN (LNM "Semi-linear PDE's and limit thcorcms for large dcviations" J.F. LE GALL "Some properties of planar Brownian motion" D.A. DAWSON (LNM "Measure-valued Markov processes" B. MAISONNEUVE "Processus de Markov : Naissance, Retourncment, R6gdnSration" J. SPENCER "Nine Lectures on Random Graphs" D. BAKRY (LNM "L'hypercontractivitd et son utilisation en th6orie des semigroupes" R.D. GILL "Lectures on Survival Analysis" S.A. MOLCHANOV "Lectures on t~m Random Media" P. BIANE (LNM "Calcul stochastique non-commutatif" R. DURRETT "Ten Lectures on Particle Systems"

1427)

1464)

1527)

1541)

1581)

1608)

237

1994 -

1995 -

1996 -

R. DOBRUSHIN (LNM 1648) "Perturbation methods of the theory of Gibbsian fields" P. G R O E N E B O O M "Lectures on inverse problems" M. LEDOUX "Isoperimetry and gaussian analysis" (LNM 1690) M.T. B A R L O W "Diffusions on fractals" D. NUALART "Analysis on W ner space and anticipating stochastic calculus" (LNM 1665) E. GINE "Decoupling anf limit theorems for U-statistics and U-processes" "Lectures on some aspects theory of the bootstrap" G. G R I M M E T T "Percolation and disordered systems" L. S A L O F F - C O S T E "Lectures on finite Markov chains"

Printing: Weihert-Druck GmbH, Darmstadt Binding: Buchbinderei Sch~iffer, Gr/instadt