Stochastic Analysis and Related Topics (Lecture Notes in Mathematics) 3540193154, 9783540193159

Citation preview

Lecture Notes in Mathematics For information about Vols.1-1090 please contact your bookseller or Springer-Verlag. Vol. 1091: Multifunctions and Integ rands. Proceedings, 1983. Edited by G. Salinetti. V, 234 pages. 1984. Vol.1092: Complete Intersections. Seminar, 1983. Edited byS. Greco and R. Strano. VII, 299 pages. 1984. Vol. 1093: A. Prestel, Lectures on Formally Real Fields. XI, 125 pages. 1984. Vol. 1094: Analyse Complexe. Proceedings, 1983. Edite par E. Amar, R. Gay et Nguyen Thanh Van. IX, 184 pages. 1984.

Vol. 1117: D.J. Aldous, J.A. lbragimov, J. Jacod, Ecole d'Ete de Probabilites de Saint-Flour XIII- 1983. Edite par P.L. Hennequin. IX, 409 pages. 1985. Vol. 111'8: Grossissements de filtrations: exemples et applications.

Seminaire, 1982/83. Edite parT h. Jeulin etM. Yor. V, 315 pages.1985. Vol. 1119: Recent Mathematical Methods in Dynamic Programming. Proceedings, 1984. Edited by I. Capuzzo Dolcetta, W.H. Fleming and T. Zolezzi. VI, 202 pages. 1985. Vol. 1120: K. Jarosz, Perturbations of Banach Algebras. V, 118 pages. 1985.

Vol. 1095: Stochastic Analysis and Applications. Proceedings, 1983. Edited by A. Truman and D. Williams. V, 199 pages. 1984.

Vol. 1121: Singularities and Constructive Methods for Their Treatment. Proceedings, 1983. Edited by P. Grisvard, W. Wendland and J. R. Whiteman. IX, 346 pages. 1985.

Vol. 1096: Theorie du Potential. Proceedings, 1983. Edite par G. Mokobodzki et D. Pinchon. IX, 601 pages. 1984.

Vol. 1122: Number Theory. Proceedings, 1984. Edited by K. Alladi. VII, 217 pages. 1985.

Vol. 1097: R. M. Dudley, H. Kunita, F. Ledrappier, Ecole d'Ete de Probabilites de Saint-Flour XII - 1982. Edite par P. L. Hennequin. X, 396 pages. 1984.

Vo1.1123: Seminaire de Probabilites XIX 1983/84. Proceedings. Edite par J. Azema et M. Yor. IV, 504 pages. 1985.

Vol. 1098: Groups - Korea 1983. Proceedings. Edited by A. C. Kim and B. H. Neumann. VII, 183 pages. 1984. Vol. 1099: C. M. Ringel, Tame Algebras and Integral Quadratic Forms. XIII, 376 pages. 1984. Vol. 1100: V. lvrii, Precise Spectral Asymptotics for Elliptic Operators Acting in Fiberings over Manifolds with Boundary. V, 237 pages. 1984. Vol. 1101: V. Cossart, J. Giraud, U. Orbanz, Resolution of Surface Singularities. Seminar. VII, 132 pages. 1984. Vol. 1102: A. Verona, Stratified Mappings - Structure and Triangulability. IX, 160 pages. 1984. Vol. 1103: Models and Sets. Proceedings, cogic Colloquium, 1983, Part I. Edited by G. H. MUller and M. M. Richter. VIII, 484 pages. 1984. Vol. 1104: Computation and Proof Theory. Proceedings, Logic Colloquium, 1983, Part II. Edited by M. M. Richter, E. Borger, W. Oberschelp, B. Schinzel and W. Thomas. VIII, 4 75 pages. 1984. Vol. 1105: Rational Approximation and Interpolation. Proceedings,

1983. Edited by P.R. Graves-Morris, E. B. Saff and R. S. Varga. XII, 528 pages. 1984. Vol. 1106: C. T. Chong, Techniques of Admissible Recursion Theory. IX, 214 pages. 1984. Vol. 1107: Nonlinear Analysis and Optimization. Proceedings, 1982. Edited by C. Vinti. V, 224 pages. 1984.

Vol. 1124: Algebraic Geometry, Sitges (Barcelona) 1983. Proceedings. Edited by E. Casas-Aivero, G. E. Welters and S. Xamb6Descamps. XI, 416 pages. 1985. Vol. 1125: Dynam1cal Systems and Bifurcations. Proceedings, 1984. Ed1ted by B. L. J. Braaksma, H. W. Broer and F. Takens. V, 129 pages. 1985. Vol. 1126: Algebraic and Geometric Topology. Proceedings, 1983. Edited by A. Ranicki, N. Levitt and F. Quinn. V, 423 pages. 1985. Vol. 1127: Numerical Metltods in Fluid Dynamics. Seminar. Edited by F. Brezzi, VII, 333 pages. 1985. Vol. 1128: J. Elschner, Singular Ordinary Differential Operators and Pseudodlfferential Equations. 200 pages. 1985. Vol. 1129: Numerical Analysis, Lancaster 1984. Proceedings. Edited by P.R. Turner. XIV, 179 pages. 1985. Vol.1130: Methods in Mathematical Logic. Proceedings, 1983. Edited by C. A. Di Prisco. VII, 407 pages. 1985. Vol.1131: K. Sundaresan, S. Swaminathan, Geometry and Nonlinear Analysis in Banach Spaces. Ill, 116 pages. 1985. Vol. 1132: Operator Algebras and the1r Connections with Topology and Ergodic Theory. Proceedings, 1983. Edited by H.Araki, C.C. Moore, $.Stratila and C. Voiculescu. VI, 594 pages. 1985. Vol. 1133: K. C. Kiwiel, Methods of Descentfor NondifferentiableOptimization. VI, 362 pages. 1985. Vol. 1134: G. P. Galdi, S. Rio nero, Weighted Energy Methods in Fluid Dynamics and Elasticity. VII, 126 pages. 1985.

Vol. 1108: Global Analysis - Studies and Applications I. Edited by Yu.G. Borisovich and Yu.E. Gliklikh. V, 301 pages. 1984.

Vol. 1135: Number Theory, New York 1983-84. Seminar. Edited by D.V. Chudnovsky, G.V. Chudnovsky, H. Cohn and M.B. Nathanson. V, 283 pages. 1985.

Vol. 1109: Stochastic Aspects of Classical and Quantum Systems. Proceedings, 1983. Edited by S. Albeverio, P. Combe and M. SirugueCollin. IX, 227 pages. 1985.

Vol. 1136: Quantum Probability and Applications II. Proceedings, 1984. Edited by L. Accardi and W. von Waldenfels. VI, 534 pages. 1985.

Vol. 1110: R. Jatte, Strong Limit Theorems in Non-commutative Probability. VI, 152 pages. 1985.

Vol. 1137: Xiao G., Surfaces fibrees en courbes de genre deux. IX, 103 pages. 1985.

Vol. 1111: Arbeitstagung Bonn 1984. Proceedings. Edited by F. Hirzebruch, J. Schwermer and S. Suter. V, 481 pages. 1985. Vol. 1112: Products of Conjugacy Classes in Groups. Edited by Z. Arad and M. Herzog. V, 244 pages. 1985. Vol. 1113: P. Antosik, C. Swartz, Matrix Methods in Analysis. IV, 114 pages. 1985. Vol. 1114: Zahlentheoretische Analysis. Seminar. Herausgegeben von E. Hlawka. V, 157 Seiten. 1985. Vol. 1115: J. Moulin Ollagnier, Ergodic Theory and Statistical Meehan· ics. VI, 147 pages. 1985. Vol. 1116: S. Stolz, Hochzusammenhangende Mannigfalligkeiten und ihre Rander. XXIII, 134 Se1ten. 1985.

Vol. 1138: A. Ocneanu, Actions of Discrete Amenable Groups on von

Neumann Algebras. V, 115 pages. 1985. Vol. 1139: Differential Geometric Methods in Mathematical Physics. Proceedings, 1983. Edited by H. D. Doebner and J.D. Hennig. VI, 337 pages. 1985. Vol. 1140: S. Donkin, Rational Representations of Algebraic Groups. VII, 254 pages. 1985. Vol. 1141: Recursion Theory Week. Proceedings, 1984. Edited by H.-D. Ebbinghaus, G. H. Muller and G. E. Sacks. IX, 418 pages. 1985. Vol. 1142: Orders and their Applications. Proceedings, 1984. Edited by I. Reiner and K. W. Roggenkamp. X, 306 pages. 1985. Vol. 1143: A. Krieg, Modular Forms on Half-Spaces of Quaternions. XIII, 203 pages. 1985. Vol. 1144: Knot Theory and Manifolds. Proceedings, 1983. Edited by D. Rolfsen. V, 163 pages. 1985.

continued on page 377

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Institut de Mathernatiques, Universite de Strasbourg P. A. Meyer Adviser:

1316 H. Korezlioglu A.S. Ustunel (Eds.)

Stochastic Analysis and Related Topics Proceedings of a Workshop held in Silivri, Turkey, July 7-9, 1986

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Editors

Hayri Korezlioglu Ali Suleyrnan Ustunel Ecole Nationale Superieure des Telecommunications 46, rue Barrault, 75634 Paris Cedex 13, France

Mathematics Subject Classification (1980): 60BXX, 60GXX, 60HXX, 60JXX ISBN 3-540-19315-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387 -19315-4 Springer-Verlag New York Berlin Heidelberg

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 other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in Its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210

FOREWORD

This a meeting from July Center of

volume contains the lecture notes and papers presented at on Stochastic Analysis and Related Topics, held in Silivri, 7 to July 19, 1986, at the Naz i.m Terzioglu Graduat.e Research the University of Istanbul.

The first week of the meeting was

devoted to the following lectu-

- Brownian notion, by H.H Kuo.

Infinite

res. Diffusion

and

Dimensional

Calculus,

- Stochastic Calculus of Variations, by D. Ocone. - Stochastic - Noncaus a l

Partial

Diffeeential

Stochastic

Ir.t.egr-a Ls

Equations, and

Calculus,

by

P. Kotelenez. by

D. Nualart.

The lecture notes are presented at the beginning of the volume. We regret the absence of the Lec t ur-e notes by P. Kotelenez who, being overloaded at the time, was unable to send in his contribution.

The second week of the meeting was devoted to contributed papers. Their presentation in the volume goes from the techniques of analysis on the Wiener space to vaeious topics.

We would herewith like to thank the ENST and the CNET foe their mat.erLa I support in the preparation of the meeting and the present volume. Our thanks go particularly to Dr. G.Mazziotto. We likewise thank Prof. T. Teezioglu and all other colleagues of the Depaetment of Mathematics of the Middle East Technical University in Ankar a , without whose invaluable help the local or-gan i z.a t i on could never have taken place. Last but not least, we warmly thank the vice pr-es i dent of the Interbank (International Bank for Industry and Commer-ce ) Dr. V. Aki s i k for a material con t r i bu ti on which allowed us to visit the histoeical places of Istanbul.

H. KOREZLIOGLU

A.S. USTUNEL

TABLE of CONTENTS

D. OCONE

A guide to the stochastic calculus variations .

D. NUALART

Noncausal stochastic integrals and calculus

H. H. KUO

Brownian motion, diffusions and infinite dimensional calculus

P. KREE

.

La theorie des distributions en dimension quelconque et l'integration stochastique ..

80 130 170

H. KOREZLIOGLU and A.S. USTUNEL: An Ito formula for processes with values in an abstract Wiener space..................

234

A.S. USTUNEL: Some comments on the filtering of diffusions and the Malliavin Calculus...........

247

J. PICARD

Approximation of stochastic differential equations and application of the stochastic calculus of variations to the rate of convergence .

267

D. ELWORTHY : Brownian motion and harmonic forms..........

288

P. BALDI and M. CHALEYAT-MAUREL: An extension of Ventsel-Freidlin estimates......

305

M. PONTIER and J. SZPIRGLAS: Uniqueness of solutions of the filtering equation with observations on a Riemannian symmetric space .

328

B. LAPEYRE

R. COHEN

Majoration a priori des solutions d'equations differentielles stochastiques stables .

340

A filtering formula for a nonlinear system having a continuous observation and a discrete observation at random times . . . .

352

A GUIDE TO THE STOCHASTIC CALCULUS OF VARIATIONS

Daniel L. Ocone Mathematics Department Rutgers University New Brunswick, NJ 08903 USA

Chapter 1

A.

Introduction In his path-breaking articles,

[43] and [44], P. Malliavin

succeeded in obtaining a probabilistic theory of hypoellipticity for second order elliptic and parabolic partial differential equations. Following his lead, Stroock [64,64,66,67], Kusuoka and Stroock [39], Bismut [6], Ikeda and Watanabe [30], Shigekawa [62], and others have developed Malliavin's ideas into an extensive theory for examining existence and regularity of densities for the probability distributions of Wiener functionals, and this theory is now finding an ever increasing arena of applications.

Today, the subject is

known informally as the 'Malliavin calculus', and formally as the 'stochastic calculus of variations'

(hereafter, SCV).

The name

'stochastic calculus of variations' is perhaps misleading at first; it refers not to calculus of variations in the traditional sense of minimizing functionals of paths, but to the variations in path space used to define certain derivatives of Wiener functionals.

Such

derivatives, together with a corresponding differential calculus, playa basic role in the theory. The following lectures are intended as a beginner's guide to SCV as presented in the work of Kusuoka, Stroock, and Watanabe, and to the applications based thereon.

No claim for originality of treat-

ment is made here; my debt to the various papers of Kusuoka, Stroock, and Watanabe is obvious and gratefully acknowledged.

My purpose is

to carefully motivate and explain the central definitions and results, to supply background material, to develop intuition, and to survey applications, all in a way accessible to the novice.

The

style is expository; I shall not often present complete and detailed proofs, which can be found in the literature, or try to give the most elegant approach.

Instead, I shall just try to convey what appears

2

to be the important idea.

To read these notes, one should need just

a basic understanding of Wiener space, stochastic integrals and stochastic differential equations involving Brownian motion, and some distribution theory and Sobolev space theory.

A not inconsiderable

amount of mathematical sophistication would also help! The lectures are organized as follows.

In Chapter I, we briefly

sketch the central idea of Malliavin for studying densities of Wiener functionals and we motivate the need for a theory of generalized functionals on Wiener space.

To motivate the eventual construction

of these generalized functions, we then study the finite dimensional case. By finite dimensional case, I mean the case of functions on RM with the Gauss measure (2")-M/2exP[-ilxI2]dX, that is functions of unit normal,

RM-valued, random vectors.

By way of contrast,

Wiener space consists of a Gaussian measure on the infinite dimensional space of continuous paths.

Studying the finite dimensional

situation has both practical and pedagogic value; the constructions are easier to understand and to derive, and, it turns out, the differential calculus on Wiener space is a direct generalization of -M/2 2 that on (R M,(2") exp[-1/2Ixl ]dx). In fact, the basic identities of the Wiener space calculus are just consequences of the corresponding finite dimensional identities.

This illustrates an im-

portant principle for learning SCV recommended to all newcomers: work out the finite dimensional case! In Chapter 2, the full differential calculus and distribution theory is presented for Wiener space.

As an illustration, we show

how the various operators apply to functionals arising as solutions to stochastic d.e. 's and we calculate the Malliavin covariance matrix for these solutions.

Then we discuss how the calculus can be used to

define Skorohod's integral and to derive representations of Wiener functionals as stochastic integrals. In Chapter 3, we first establish the basic criterion of Malliavin for the existence and smoothness of densities of Wiener functionals.

We then apply this to deriving regularity of densities

for solutions to stochastic differential equations and indicate how this relates to Hormander's hypoellipticity theorem for second-order operators. of SCV.

Finally, we briefly mention a range of other applications

No attempt at a history of the subject is made here, and I apologize in advance for any omissions of references or errors of attribution.

Also, there are nice alternative approaches to SCV for

3

stochastic differential equations in Bismut [6], Bichteler and Fonken [3,4], Varsan [76] and others, which I shall regretfully not cover at

all. I wish to thank Drs. Alpay, Altuna, Korezlioglu, Terzioglu, and Ustunel for organizing this workshop and for their invitation to talk on the stochastic calculus of variations.

B.

Central problem. motivation. and the finite dimensional case.

1.

The setting for all our work shall be the Wiener space

defined as follows.

=

i)

is a continuous,

= O} = Borel sets of

ii)

ed ·

0d

with respect to the sup-norm topology on

= Wiener measure on

iii)

•

We shall often suppress the e

to denote points in

function on [0,1] with

e,

'd'

in

denote the canonical process,

(eft)}

generates the filtrations

iv)

= a (e (s) I0

v) 2.

t = a (e (s) -6 (t) I t

In addition, we shall use

and, at the same time, we shall let

(eft)}

s

ed .

t},

and

s

l}.

= wit)

e(t,w)

for

weB.

= Borel sets) shall

Any measurable function

be called a Wiener functional or Brownian functional. really mean the equivalence class of maps

- a.s.

defines a random vector on the probability space

(By

F,

equal to

we F).

F

and its

distribution is P

that is,

PF(A) =

F

=

0

F

for every

-1

,

A c B(R d).

The central prob-

lem of the stochastic calculus of variations is to determine i)

when

PF

is absolutely continuous w.r.t. Lebesgue measure,

and ii)

the regularity of the density

dPF/dx

when it exists.

4

3.

Stochastic differential equations.

Example: Let

= x +

x-I

Io o ( t

(.)

t xt (e ) ) ds +

X

'ods'

In (3.1),

iJott

d I

i=1

I

t

0

X. ( t xt (s))

p(t,x,y)

P(t,x, '),

0

de. (s) .

( 3. I )

1

1

denotes Stratonovich integration.

P(t,x,·) =

Let

denote the distribution of the functional

are interested in determining when sity

and consi-

to

der the solution

t xt (s)

n

IR ,

be smooth vector fields on

XO,X I,·· "X d

such that

P(t,x,')

We

admits a smooth den-

P(t,x,dy) = p(t,x,y)dy.

Note that

which is the Markov transition probability for

is

a fundamental solution in the weak, or distributional sense, of the Fokker-Planck equation u(t,y) = L;U(t,y),

and

where

L* y

(3.2)

is its formal adjoint.

We

are thus asking whether (3.2) has a smooth, classical, fundamental solution

p(t,x,y).

Previous to Malliavin, probabilists were in the

embarrassing situation of having to rely on p.d.e. theory to answer this question, and, in particular, on Hormander's criterion for hypoellipticity of the second order operation differential operator 00

u e C.

If

A

=

d I

i=1

A

is hypoelliptic if

2

Yi + YO

L*. A

for vector fields

U

E

Roughly, a implies

YO,Y I, ... ,Y p'

Hormander's theorem establishes a full rank criterion on the Lie algebra generated by

YO,Y

I,

see Chapter 3, Theorem 12.1.

... ,Y

that implies hypoellipticity

p

For

- L*,

Malliavin realized how

to bypass p.d.e. theory and obtain existence and regularity of p(t,x,y)

under Lie algebraic rank conditions by a direct stochastic

method -- see Chapter 3, Theorem 11.5.

That this should be possible

had been an intuitive conviction of probabilists because the Lie algebra of the vector fields

XO,X

I,

... 'X

d

can be used to describe

the infinitesmal directions in which the solution to the stochastic d.e.

(3.1) can move.

Malliavin thus laid the basis for turning this

intuition into rigorous mathematics.

More importantly, probabilists

5

soon discovered how to extend his technique to other problems inaccessible to p.d.e. theory

see Chapter 3, paragraphs 18, 19 and

20 -- and hence, the interest in the subject today. 4.

Malliavin-Stroock Approach.

Malliavin based his approach on the

following type of analytical result.

In the statement of this

theorem, which we take from Stroock [64],

a = (a , ... ,a 1 n)

multi-index of non-negative integers,

=

n X

1

and

Q. 1

denotes a aaY' (x) =

a ax n. n

4.1

Theorem.

Let

that for some

v

N > n,

be a finite Borel measure on

x

Y'

E

C

n O(IR),

R

Suppose

N.

Then

I

exists and

dv/dx

(R n ) .

dv/dx E

I

Va,

n.

such that

3 C

C sup 1Y'(x)

V

R

A simple proof may be found in [64]. n Suppose that F:e IR is a Wiener functional. aaY'(x)PF(dx) =

=

I

Note that

Therefore, by Theorem

n

4.1,

P

F

will have a smooth density if we can prove estimates of the

form (4.2)

Let us show how (4.2) can be obtained in terms of a gradient operator and an integration by parts formula on Wiener space by making some purely formal calculations

in the case

Assume that the gradient operator R

and the gradient inner product

lal v

= 1,

R

M

to

can be generalized to an

D

(DF,DG>.

Then, application of the chain rule,

aY'(F) ax

= 1.

on functions from

operator

implies that

n

acting on Brownian functionals with inner product = 0

,

is full rank

n

S.

then,

x,

looks essentially like a projection from

F

Consider such a projection

duced on

has a simple intui-

A(x) = dF(x)(dF(x))

Recalling that

onto

The measure

in-

is clearly absolutely continuous w.r.t. Lebesgue mea-

Indeed if

B

is a set of Lebesgue measure zero, an applica-

tion of Fubini's theorem shows

vM(S

-1

(B)) = 0

also.

This explains,

at least locally, the reason for the full rank hypothesis.

In fact

in finite dimensions we may go much further, and we do not really need the Malliavin approach.

The co-area formula used in geometric

measure theory says that, under appropriate hypotheses

=

where

(211)-M/2

x

M-n

is Hausdorff measure of dimension

Federer [87). by

PF(U)

E

From this we see that the measure VM[1(F

Lebesgue measure.

E

M-n P

IR M;

in F

on

n R

see defined

is absolutely continuous w.r.t.

It follows immediately that condition (14.5) alone 0 F -1 M Bouleau and Hirsch [12] actually exploit

is sufficient to guarantee the absolute continuity of w.r.t. Lebesgue measure.

v

the co-area formula to show that a.s. positivity of the Malliavin covariance matrix implies existence of probability densities in the Wiener space context.

19

Chapter 2:

Wiener Space Analysis

In this chapter we construct a differential calculus for Wiener functionals and define generalized Wiener functionals, and we describe applications to stochastic d.e. 's, Skorohod integrals and stochastic integral representation of functionals.

Part C,

paragraphs 21­24, of this chapter are not required for Chapter 3, which treats the application to density questions.

In this treat-

ment we follow Watanabe [77] and Sugita [69]. The origins of the ideas presented here go back to early work of Wiener, Gross, Segal [68], and Cameron and Martin, and the theory was developed by various authors.

In particular, we note the contribu-

tions of P. Kree [32,34], M. Kree [33,34] and B.

Lascar [41].

Hida

[27,28] has also developed a theory of generalized Wiener functionals.

Professor Kuo's lectures at this conference show how

Malliavin's work may be interpreted via Hida's calculus. A.

The Ornstein­Uhlenbeck semi­group and Sobolev spaces over Wiener

space.

1.

A basic idea underlying this theory is first to study all

operations on 'cylinder' functionals of the special form F(e) = where RM, R. Noting that v M'

such an

F

may be viewed through

0

as a function on

and this we know how to handle with the methods of the previous chapter.

All definitions and formulas will thus be set out for

cylinder functionals.

Then we will extend the analysis to more

general functionals by completion and continuity arguments. 2.

Let

be the dual of

ed'

e*d

is the collection of

d­vectors of bounded Borel measures on [0,1].

* d' Note that

tie)

tion by parts:

l(e) =

1

fa

That is if d

e(s)·de(s) =

'\L f1

ei(s)dei(s).

i=l 0

may be written as a stochastic integral by integra-

20

e (a) =

'I.\ I d

1

i=l

a)

2.2 Definition.

Let

c)

ll, .•• ,ln

1.

1.

and norm

III

denote the completion of

mal w. r , t

(2.1)

(s) .

define the inner product

For

p:=E/(a)h(8) b)

e . ( ( s , 1 ) ) da.

0

w.r.t

II· lip'

are said to be orthonormal if they are orthonor-

< . , . >. p

(l)

c0*d

a

n ne l

-* orthonormal basis of ®d'

is an orthonormal basis if it is an

Note that from (2.1), d

It e

-*d

d

I f.

'\ oI = (I. 1

F(a) =

... ,In(a))

(ll, ... ,ln) c

e*

If.

(s ) da. (s) 1.

From now on we suppress the

F

and

(2.3)

i=l 0

1.

3 Definition.

2 i ( ( s , 1 ) ) ds ,

d

E

1.

2 L ( [0,1) ) ,1 1 ,k e JR} •

with semi-norms X

iii) ID -00 = When

X

=

is a complete countably normed space

IDp,k'

X

u u IDp,k' p>l -ooH

is positive definite and

is a complete Hilbert space with respect to

H'

e

H and the space -* defined above in paragraph 2. Namely, if

There is a natural isomorphism between with inner product t

e0*

set

j(t)

jJ

to be the function in

H

given by

29

t

Jot (s ,

=

j(t)(t)

(10.2)

1] ds

Because of (2.3)

II Ie (

I I j ( e ) I IH2 = and hence

0

I 2 ds = I I e I IP2 ,

s , 1]

( 10 . 3 )

e*

defines an isometry between and H. j is onto 2 because the space c of functions in H with two continuous derivatives is dense in H, and if y e C2, t ( ( s , I ] ) = y' (s) defines a j

7

bounded Borel measure such that From now on we identify

tee *.

when in which

=

The significance of

and

For

p.

= p(A-'i');

p.,.(A)

with

= 7. H

for

and write

This leads to the embeddings,

H

cB,

H

c 7

e H,

P.,.

for us is contained in the well­known

Given two measures

measure space, we write v

)

(H

If

K

q ..

J,1

2 2 H + H'

=

DH

(e (e) ) e.

1

i&l

e ., J

define

Note that

is called the divergence operator.

14. Basic formulae connecting

D,,,

and

The following formulae

L.

are all direct consequences of the corresponding formulae in finite dimensions treated in Chapter 1, paragraph 14.

Let

F,G

be in

J e (14. 1)

(14.2)

EHS = Et"'s

(19.1a)

acts as a state transition matrix for equations of the form

(19.2)

where

That is, the variation of constants formula,

Zt = "'tZO +

Jo"'t"'t

s

1

f

s

solves (19.2).

ds,

The point of introducing x

f

s

=

e R

n

is that the equation (18.2) for

is precisely of the form (19.2) with

d

X k=1

4>t

Zo = O.

and

s

It follows that

d

2

=

k=1

This means that

e H

is given by its time derivative

n xd

(19.3)

In particular, if

x (t,i

denotes the ith component of

(19.3.a)

defines the time derivative of 19.4 Proposition.

=. J

d

t

t

.-1 0 s

L

X ( (:) k

( (:)

k=1

x x s a (t s ) (4) s ) ds

4> t

T.

4> t

Thus the diffusion

matrix enters neatly into the object which we shall analyze to determine the existence and regularity of probability densities. 20 Exercise.

Let

tt

solve

Jo t

tt = x +

where

tt

E

n

R

ssian process.

and

A

and

Show that

Bare

J t

At ds + s

0

Bt de s s

nxn - matrices.

is a Gau-

is the covariance matrix of

At this point, the reader is prepared to go on to Chapter 3, which describes a general criterion for existence and smoothness of densities and applies this to stochastic differential equations using formula (19.4). C. Multiple integrals, Skorohod's integral and representation of Brownian functionals. 21. Action of

D

and

6

on multiple stochastic integrals.

For simplicity of notation we consider only the case

d = 1

of

one-dimensional Brownian motion. If f E the symmetric 2 L functions on [0,1)k, recall from paragraph 7 the definition of k. f 0 e We shall compute D(f 0 e k). Let us proceed formally. If k F(e) = f 0 e and E H

43

F(s+e"l) =

Io ... I

1

1

f (t 1, · · O

.. , [ds(t

+ e7(t

1)

1)dt]

k

=

f

0

l

Sk + e

j=l "I '

(t ,') d t , ... da ( t 1 ) J

J

+ higher order terms. of

f, the second term can be written

k-1 "I' (s )ds, o a

where

represents the element of

f( , " s)

obtained by holding the last variable of at

fixed

f

It follows that

s.

d

(ll

so that

f(sHe.)

D[f

= H =

0

J 1

Okf( .. ,s)

0

a

k-1

"I'

(s)ds,

It

Okf( .. ,s)O ak-l ds.

To make this calculation rigorous, one first verifies it for that

f

f

such

is a simple function taking constant values on rectangles

and then approximates an arbitrary

f

by simple functions and passes

to the limit. 2 GeL (f';H)

Next consider

6(G).

We shall calculate

v

such that

fJ

e H.

To do this, it is convenient to work with

the process

O(a)(s):6Jx [0,1) ... JR. Note that 2(ex O(s)(s) e L [0,1], f' x Leb) because > E

J° 1

O(a)

2

For this reason it is possible to find a determinis-

(s )ds.

tic function

g(t

g(t 1,· .. ,tk,s)

1,

.•. ,tk,s) e L 2([0,1]k+1)

is symmetric in

t

1,

•.. ,t

O(a)(s) = g( ... ,s) 0 a k for a.e. s. geL 2 ( [0,1] k+ 1) because

co

>

E

Jo0

1 2

EIGI H2 =

(a) (s) ds

= k!

II II °

g ( ... ,s)

o

II

and

k s

2ds L

such that for every

1.

Note that

= k!ll g l 1 2 k+1' L ([0,1] )

s

44

g

itself is not necessarily symmetric.

However

kt1

l

1

= KIT

j=l

g(t 1,··· ,tj_1,tjt1'"

.,t k t 1,t j)

is the projection of g onto the symmetric subspace of L 2([O,l]kt1). We claim that

= goa kt1

l

we

E D

det A(e)

will follow from the general result;

00

G- l e

(Gh)-l e

n e

4. step 3 - Deriving the inequalities (2.6). We will show how (2.6) is derived for

k = 1.

This is already

quite a bit of work, but it will be clear from the details that the argument extends to arbitrary

k.

Hence, by Corollary 2.5, the proof

of Theorem 1.1 will be complete.

The precise statement we prove in

this section is For any

p > 1 and any integer C < P

00

II(I-K n )"'(F)II P,-J.

s

there is a

(Given (4.1) , set 1I1'(F)

II P,-J.

.,.

= (I-K n )-11'

S Cpll(I-Kn)-I1'lloo'

j

2 (4.1)

such that Cp for

11"'1100 l'

'Y .,. e

n

e :r .

:rn .

Then

and this is (2.6) for

k = 1.)

To prove (4.1) , first observe that it is equivalent to (4.2)

55

where

p

-1

dense in

+ q [J

-1

=

I,

because

[J

0

Now recall that

o'

q,J

is dual to

q,J K

2

n

n

=X

i=1

30

[x

-

,2

[J

.,

P,­J

+ n/2.

and

[J

00

is

Thus to

1

prove (4.2) it is really only necessary to prove

2 IE(3 1°1'(F)G)I for any

1 SiS n,

s

111'11 00 IIGII q,J0'

Cp

(4.3)

because the contribution of the remaining terms

is

Let

?

E

n

and for

G

Il 00

E

set

Now

repeat the exact calculation of paragraph 14, Chapter 1, leading up to equation (14.6), but in the context of Wiener space. n is, if f1 E :r ,

2 n

3 if1(F) =

(=1

and, hence, n

E[3 i

f1 ( F )G]

= E[

2H] (=1

n

= E[f1(Fl

26(GAi:DFtl]

n

2Ki,t(Gl].

= E[f1(F)

t=1

t=1

Applying this twice yields

I

E1' ( F l

Therefore,

n

'\ L

t ,j =1

K

1,J 0

0

(

K. • ( Gl 1,_

l

I.

The result

56 n

'\ L

t • j =1 Now,

(4.4)

ElK 1, . .J (K.1, e(G))I·

K . • (G) 1. c

implying that (4.5)

Furthermore, from Lemma 12.5, Chapter 2 and a minor amount of calculation DK.

1.

where +2

e (G)

-1

-1

+1 = H + AieLF e

and

+2 = D[+1)'

Now

by Lemma 3.1 and assumptions (1.2) and (1.3).

E

where

is a polynomial in

and

1+ 11,

1+ 21,

­1 IDAi e

Assumptions (1.2) and (1.3) imply

I,

\DF

+3

E

+1 E

and

This implies

e I H,

2 ID FlIHS'

n LP(p). p>l

If

we now combine (4.5) and (4.6)

(4.7) where

+4

E

n

p>l

LP(p).

Using Meyer's inequality (Chapter 2, Theorem

15.5), now apply (4.7) to (4.4) to obtain, for

j

+

2,

+ ID

2GI:(2)]

• 5. Remarks.

If one is only interested in obtaining

dPF/dx e Cm(R n)

57

for some

m

hypotheses (1.2) and (1.3) may be replaced by

F e [lq(m) ,kIm) p(m)

E[det A(e)]-p(m)
0

7

> 1,

9.

E 1{'t"

Indeed, suppose

-1

=

1:

1:

a. s ,

for every

V e A.

In

To complete the proof it suffices to show that if q:S R is x x q Tv (tV(tq = 0 a.s. then q = 0 a.s.

*

any random vector such that But

*

T x x q v (tvtt q =

that

12.

J.

q e VO+

=

{O}

-1

for

w ec1J· (V), w e Coo{V).

(where

a. s , , which implies

t

• Let G S IR n.

A

be a linear differential

A

is said to be

if for any open

/J'cU)

12.1 Hormander's Theorem.

n(X O""

2 ds,

a.S.

operator defined on an open set G

x

s

Connection to hypoellipticity.

hypoelliptic in

Let

2

a 1 15 N- 1

there exists an

N

such that

2

ds .

69 d

2

2

inf{

k=l

2111 z II =

(x) , z >

l}

= .,N >

O.

It then follows easily that

y]

Lemma 14.1 then implies that positive

for all 16.

C

and

for some

•

p > 1.

Localization.

The analysis sketched above required global condiX ..• ,X O,X 1, d.

calize the technique.

G

=

y

and hence that

tions on the vector fields

.x

ce-

C

e G

} .

Let

It is also possible to lo-

be an open ball in

R

n

and let

Define

and consider the problem of the existence and regularity of Clearly conditions on

X

pend on the character of

O

' " "X

d

dpG/ d x•

for such regularity should only de-

in G. The Malliavin calculus d can in fact be localized to prove theorems using only local conditions.

C.

XO"",X

For details consult Stroock [64] or Kusuoka and Stroock [40].

Other applications. The analysis above recaptured a result already known by p.d.e.

methods. However there are many problems to which the stochastic calculus of variations is now being applied and which have not been solved by other means.

In the remaining sections we will briefly

mention some of these applications, without giving details, to orient the reader to the current state of the field.

70

17. Existence and regularity of densities in nonlinear filtering. Suppose

d(x t

=

x XO«(t)dt +

d

2Xk«(t) x

o dB

k,

1

d

dY

t

=

+

2gk r ext)

o dB

k

+ dW,

1

where F

Y

t

=

Band a{Y(s) 1 0

given

Y F t

Ware independent Brownian motions and s

t).

Then the conditional distribution of

=

is n

one may ask whether

U t

dU(w)/dx t

regular is

YO = O.

Let X t

is a random measure and

exists almost-surely and if so, how

This can be answered by thinking of

Ut(w)

as

Ut(Y(w)), i.e. a measure valued functional of V-paths, representing Ut(Y(w))

by an expectation over Wiener space in which

Y(w)

appears

as a fixed parameter, and developing a conditional Malliavin calculus for each fixed

Y

path.

Relevant references are Michel [51], Bismut

and Michel [11], Kusuoka and Stroock [40], Ferreyra [21] and Sussmann [85] . A somewhat different application to nonlinear filtering concerns the differentiability of the map calculus of variations sense.

Y e e

in the stochastic

Chaleyat-Maurel [15] gives conditions

for the differentiability of this map.

Ocone [58-60] recaptures some

of these results by applying stochastic calculus of variations to stochastic p.d.e. 's, and in particular to Zakai's equation of nonlinear filtering.

Also, a weak generalization of Theorem 11.5 is

obtained for the stochastic p.d.e. case. 18. Existence of densities for other functionals. Several authors have considered existence of densities for functionals other than solutions of s.d.e. 'so

For example, Shigekawa [62] shows that

multiple integrals, as defined in chapter 2, admit densities, Cattiaux [14] treats diffusions in domains with boundaries and various boundary conditions, and Stroock [66] discusses func-

71

tionals that arise from solutions to integral equations such as

y(t) =

fo t

s

a[fop(s-r)Y(r)dr]dS(S).

Holley and Stroock [Z9J study infinite dimensional diffusions of the Z Z following type. For k e z let 0k:R Rand bk:R R be functions with a uniformly finite range. such that all

bk(x) = bk(y)

with

It-kl $ L

for all for all

k.

(. "'(_I(t)'£O(t)'(I(t)'£Z(t), ... ) e

where

That is, there exists an

x,y e

R

Z

such that

L

x t = Yt

for

t(t) =

Let R

Z

be a solution to

is a collection of independent, Brownian motions.

Holley and Stroock [59] then use stochastic calculus of variations to derive conditions implying, for example, that

((I(t)""'£N(t))

admits a density. 19. Existence of densities for stochastic d.e.'s with only Lipschitz coefficients.

Bouleau and Hirsh [12] develop a theory of

Dirichlet forms over Wiener space and build a theory related to the Malliavin calculus.

In particular they define the Malliavin covari-

ance matrix under weaker differentiability assumptions than have been imposed in these notes and so succeed in treating s.d.e.'s with only Lipschitzian type assumptions on the coefficients. 20. Miscellaneous.

Other applications include,

1)

Small time asymptotics of heat kernels; see Watanabe [78] and

2)

Regularity of the Poisson kernel for degenerate elliptic opera-

Leandre [42]. tors; see Ben Arous, Kusuoka and Stroock [2]. 3)

the study of diffusions on Wiener space corresponding to generators of the form L

k

L +

are vector fields on

E L 9

k

where

L

is the

o-u

operator and

(Cruzeiro [18]).

4)

analysis of stochastic oscillating functionals

5)

extensions to analyze functionals of jump processes (Bismut [7],

as

A

(Moulinier (53] and references therein).

72

Gravereaux and Jacod [23], Bichteler and Jacod [5], Bass and Cranston [1]). 6)

relations to large deviations and the Atiyah-Singer index theorem (Bismut [9,10]).

I hope this is enough to convince the reader of the use fullness of this new theory!

21. Acknowledgement:

Work on this paper was partially supported by

NSF grant No. MCS-8301880.

Appendix Proof of Lemma 10.6.

Let

show that for every

p > 1,

a

be a Brownian motion.

a for any stopping time

=

t

and

Brownian motion and For

k e Z,

Fk,j = Gk n {2

where

let

k- j- 1


zk).

and on

Therefore

ec

2 [ {Gk'U,F k=-oo E 1

.(t)

is a

-1

• (t) .(1)-1/2 = N*I1/ 2•

2 k+ 1j

and

[a:]P+l /•P/Z

p+l

00

Gk' U Fk,j' j=O

Gk


H coincides with the directional derivative

f

d . F (0) + E h(s) d s) I de a

The mapping injection of H into

h 1-7 T h , Q =

defined by

(T h) (t) =

r a

.

h (s) d s ,

(3.2)

provides a continuous

C ([0,1]), and D h F is the Frechet derivative of

F :Q

IR

in the

direction T h . The derivative generally, the

N-t h

can also be regarded as oa random variable taking values in

DF

derivative of

F,

DNF

will be the

H. More

HL2([O,I]")

2

L ([0,1]")

#

The proposition is now immediate.

Remarks: ( 1) As a consequence of the preceding result, 0 is a closed unbounded operator with domain Dom 0 dense in L 2 ([0, 1] x Q). (2) Here we have proved the duality relation (3.8) using the expression of the operators D and

0

on the Wiener chaos decomposition. On the other hand, the equality (3.8) can also be considered as an integration by parts formula (see Watanabe [50] and Bismut [5]). In fact, suppose that F and G

are smooth functionals of the form (3.4),

follows from

Lemma 3.2

and

u t = h (t) G

(see also the expression (4.3)

multiplied by a random variable) that

s (u)

u

where

h e H. In that case, it

for the Skorohod integral of a process

is Skorohod integrable, and

G

1

I

o

0

f h d W - f h (t) DtG d t

So, the duality relation (3.8) coincide in this case with equality (3.3) which is an integration by parts formula in finite dimension. We recall the following basic properties of the derivation operator that will be used in the sequel. (A) Chain rule: Let derivatives. 11 2 ,1 ' Then

lp:

IRn

Suppose that lp (F) E 11 2,1

IR be a continuously differentiable function with bounded partial F

=

(F I ,,, ., P')

is a random vector whose components belong to

and n

D lp (F)

L. «(Iilp) (F) D r' i=1

This property is immediate approximating F by smooth functionals.

(3.9)

91

(B) The derivative of a conditional expectation:

For any Borel set

be the a-algebra generated by the family of random variables

{ W (C)

A =

r C

o

[0, 1]

let ':f A

Ie d W, C E $,

Cc A}.

If F E

E (F / ':f A) E

then

and (3.10)

a.e. in [0, 1] x

Q .

Indeed, it suffices to assume that

F = 1m (fm)

and, in this case, the result follows easily from

(3.6) and the relation (3.11)

Integral representation of Wiener functionals by means of the operator

(C)

square integrable functional

F

D:

Any

of the Brownian motion can be represented by a stochastic integral

of the form 1

=

F

where

u

Clark showed that if

regularity conditions, then u l

=

f

o

ut d W t

L2 ([0, 1] x Q).

is an adapted process of

In [7],

E (F) +

F

is Frechet differentiable and satisfies certain technical

E (AF «t, 1]) / ':ft) , for each t E [0, 1] where AF (d t) denotes

the signed measure associated to the Frechet derivative of F.

AF «t, 1]) = D t F.

hE L 2 ([0, 1])

In fact, for any

With our notations it holds that

we have, taking into account formula (3.2),

that l

I

f D l F h (t) d t

o

= Dh F =

f

t F

A (d t)

0

(f

1

f AF «t,

h (s) d s) d t =

0

1]) h (t) d t

0

Therefore we can write Clark's formula as follows 1

F = E (F) +

f E ro, F / ':f

o

t)

Ocone has proved in [29] that this formula is true for any F in E (D, F / ':ft)

F

=

L

(3.12)

d Wt .

Notice that the process

is square integrable and adapted. Then we can give a simple proof of (3.12) for any

1 (f ) E m=O m m

'

i :

using (3.10) and the definition of the Skorohod integral:

92

F-E(F) .

As a consequence, and applying Burkholder, Holder and Jensen inequalities we deduce the following inequality !

P

E ( IF I for any p 2 2 and F

4.

) ::;

Cp ( I E (F) IP + E

f ID F I

P

o

(3.13)

d t) ,

t

:1)2.1 .

E

Some properties of the Skorohod integral

We are going to establish the main properties of Skorohod's integral. First we remark that the set of Skorohod integrable processes, Dom 0, is too large for our purposes, and we will introduce a smaller set of processes. that u t E :1)2,1 E

rf o

0

!

(D,

We denote by

for almost all

ui d s

dt
e j (t)

is a nuclear operator, and its Carleman-Fredholm

determinant is equal to 1

( 1-

f (D F) h (t) d t)

o

l

1

exp {

f (D F)

0

t

h (t) d t }

124

f (x)

IRn defined by

fl a

ej d W -

f g (

c}
0, there exists a

such that

E

P

such that Let

B

P 1 P. E

It can be

be the completion of

IIxll = ITxl.

(H,B)

(H,B)

Then

Let

B

H

an abstract Wiener space.

JRI

is an abstract Wiener space.

be any real separable Hilbert

Hilbert­Schmidt operator of

into

is well­

B = the Banach space of continuous functions from [0, I] into 2dt 0. Let H = {x E B; IX'(t)1 < 00) with the inner product

satisfying the above conditions, we have

\I(dy)

=

h! H.

This is the integration by parts formula for the standard Gaussian measure B.

For the details of the proof, see [22].

14

on

More generally, we have the formula

I B f(y)[g(y)(y,h)

I B ]

h ! H.

INFINITE DIMENSIONAL CALCULUS There are

three kinds of

infinite dimensional calculus:

(1)

calculus of

differentiable measures; (2) Malliavin calculus; and (3) white noise calculus.

In

this section, we will give the motivation and brief description to these calculi. From the next section on, we will concentrate on the white noise calculus.

4.A

Calculus of differentiable measures This calculus was introduced by Fomin

[4]

in 1968.

The motivation is to

study the regularity properties of weak solutions of infinite dimensional partial differential equations, to develop an integration theory on infinite dimensional manifolds,

and

stochastic

differential

solution

to

of

study

P(DH =

by a smooth function

g

(t,t> = If(x)g(x)dx. measure does

not

exist.

that

weak

Let

(H,B)

°

In

probabilities the

finite

in

the

infinite

infinite

P(D)

dimensional

One way to get around is solution

P(D)

of

dimensional

case,

the weak

can be represented

p(DH

=

°

Lebesgue

an

infinite

dimensional

can be represented by a smooth measure.

be an abstract Wiener space. A

the

case,

to develop a calculus for

for

A Borel measure

to be H-differentiable if there exists a map, denoted by H, such that for any

of

dimensional

for a hypoe Ll.Lpt Lc operator

However,

hypoelliptic operator

transition

by integrating with respect to the Lebesgue measure:

measures

so

the

equations.

B (B)

and

h

H

v

on

v' ('), from

B

is said (B)

into

140

lim E-1(V(A+Eh) - v(A» E+O h

in

H-derivatives of

and for each

v

H,

is a signed measure on

can be defined inductively.

called H-smooth if it has H-derivatives of all orders. Gaussian

measure

is

(H,B)

on

[6,

H-smooth

The higher order

B.

v

A Borel measure

on

B

is

For instance, the standard

28]

with

the

first

two

H-

derivatives given by v(dx)

/',G$ ; 0 and

/',B

[3J and are

[27].

/',B$; 0

the Gross

H.

It

has been

[25] are represented

and Beltrami Laplacians,

respectively, given as follows:

traceHf"(x) - (x,f'(x». The abstract

differentiability of measures associated with parabolic equations on an Wiener

Kolmogorov's

space

forward

has

been

studied

equation can be

in

[37J.

formulated

The

in terms

infinite

dimensional

of differentiable mea-

sures [26].

4.B

Kalliavin calculus Consider the following stochastic differential equation mentioned in §2

dX(t)

o(X(t»dB(t) + b(X(t»dt,

The transition probabilities of

X(t)

t > O.

are given by

p{X(t) E dyIX(O)

x},

t

> 0, x

E

lR •

141

In 1976, Malliavin [36] invented a probabilistic method, which is called Malliavin calculus, with

to

show that

respect

to

the

for fixed

Lebesgue

t

and

measure.

x,

Then

Pt(x, one

has a density

dy )

can

show

fundamental solution of the Kolmogorov forward equation. provides

a

probabilistic

method

to

obtain

that

Pt(x,y)

Pt(x,y)

is

a

Thus Malliavin calculus

results

on

partial

differential

equations. Note

that

for

l\c( t )

variable

fixed

t

and

x, Pt(x,')

Xx( t ) stochastic differential equation starting at Q

IR ,where

-7

is

the

distribution

denotes

x.

the

of

solution

the

of

the

Therefore, Q

can be

Pt(x,')

space setup

taken to be this abstract Wiener space.

has a density can be

[401.

Let

Gaussian measure on ask

the

question

(H,B)

B. of

be

Suppose

when

the

above

Note also that we can take a

Brownian motion defined on the abstract Wiener space in Example 3.1 with

whether

random

d = 1.

Thus the problem of

stated in the following abstract Wiener

an abstract Wiener space and

(L 2 ) - , we can define its U-functional by

in

e

where

«', .»

-II II2 12

e( •

denotes the pairing between

,0» (L 2 ) -

,

E

is in

,j

(L 2)+.

and

The analysis of

the generalized Brownian functionals is very often expressed in terms of their Ufunctionals.

6.A

B(t)-dlfferentiation We have

Thus

B(t)

seen that for each

t, B(t)

is a generalized Brownian functional.

is a well-defined mathematical object.

calculus, {B(t); tEm}

is

In fact,

regarded as a coordinate system.

in the white noise This is very useful

since the time propagation can be taken into account in a natural way. coordinate sytem

coordinate differentiation, i.e. Let

U

B(t)-differentiation.

be the U-functional of a generalized Brownian functional

for each fixed

in

,j,

U

has a functional derivative

(6.1)

where o(ll)

is with respect to any Sobolev HP(UO-norm of

i.e.

(6.2)

to be

B(t)-derivative

Suppose

Uk(')' i.e.

II E

is a U-functional, then we define the in short,

With this

{B( t); ts m}, we can define the infinite dimensional analogue of

,j,

ll, p

> O.

If

of

the generalized Brownian functional with U-functional

U(.)(t) or

at

U(.)(t),

154

Note that if the function derivative

a

of

$

Uk(')

t$ U' ( • ) is in t; and • then a(.)$

hand, i f

*

(L Z)-, i.e. for any functional.

11

and

pairing (a(.)$, 11)

a11 $

=

Let

f(t).

$

f

A-

n+1

(a(.)$, 11)

is a generalized Brownian as follows:

Note that

=

all

f lR ll( t ) at $

f lR f(u)il(u)du,

f

H-

1

is an average of

(L Z ) -

of

at' i.e.

dt ,

( llO .

Then

f lR f( u ) t;(u)du

U(t;)

and

Therefore, we have

a Example 6.2.

=

B(t)-

On the other

lR

are differential operators acting on the space

generalized Brownian functionals.

Example 6.1.

t

is defined as a tempered distribution with values in

in

11

LZ(llO-function, then the

so that the integral in (6.1) is the pairing between

In this case, we will rewrite

Therefore, at

Uk(t)

in (6.1) is an

is only defined for almost all

f

l1$

lR

f(t)l1(t)dt

=

(f, 11),

11

Let

-Z-(nf).

Then its U-functional is given by

U(t;)

f nf

=

f(u

1,

.. ·,u )t;(u n

1)

.. ·t;(u )du .. ·du. 1 n n

t;

It is easy to check that Uk(t)

=

n

f

nf

- 1 f(t,u1,···,un_l)t;(u1)···t;(un_1)du1 ••• du n_ 1•

Therefore, we have

where the integral with respect to dt is actually the pairing between

*

and

•

155

Example 6.3.

Let

• 2] :exp [I c T B(u) du :,

4>

c1'.!.. 2

Then the U-functional of

4>

is given by

Obviously, we have

On the other hand, by Example 5.9, the U-functional of is given by

f;(t) - exp [C -1-2c 1-2c

I

T

:B(t) exp[c

IT

B(u)2du]:

f;(u) 2 du] ,

Therefore, we have

In fact, just as

B(t)

does not require renormalization, we have

• . B(u) 2] :B(t) exp [ c IT·B(u) 2 du ] : = .B(t) :exp[ c IT du : Thus we have

Moreover, for

Reaark.

T"\ IE:

'/,

It is natural to conclude from the above examples that

os (t). This relation is interpreted as in the distribution sense, i.e. for fixed is a distribution in the t-war t ab l e , notation, then

0t(s)

=

0

if

t

*s

and

{

0 (t) t

dt'

t

t

=

1dt

Thus

*s s.

This is the infinite dimensional analogue ofaxi/ax coordinate system

{B( t); t

lRl

s , it

However, if we use the nonstandard analysis

= 0ij with respect to the j and can be used to differentiate a generalized

Brownian functional directly without going through its U-functional.

156

6.B

Adjoint and multiplication operators

a*

The adjoint operator

«', .»

pairing

of the

t

between

(L 2)-

*

suppose

Then for any

1ji E Kn + 1

where

f

E

f

n+l -2-(JR!),

where

g

E

f

-Z- (JR!+1), we have

&I

as follows:

K(-n)

E

n

is given by

(n+l)

n+2

nl]

where

(L 2 ) +

and

To find the U-functional of

is defined by the

B(t)-differentiation

JR!

f(u1,···,u )(n+l)g(t,u1, .. ·,u )du1···du n n n

denotes the symmetric tensor product.

*

Therefore,

is given by

(6,3)

A straightforward computation shows that

II

where

r

0t

&f

I1

Z

f

Z

lI f ll

n+Z

f

-Z-(JR!+l)

is the gamma function.

Therefore,

r (n+ 1) 2 (n+l)I1r--

(6.4)

2 Note

that

r(n;1)/r(n;2) -!2/n-l

operator on the space

(L Z)-

n+l -Z-(JR!)

for

large

2

n,

II IKZ (_)' n n

Thus

a:

of generalized Brownian functionals.

is

an

unbounded

157

*

It follows from (6.3) that the U-functional of

is given by

But

Therefore, we have

(6.5) In fact,

it follows from the linearity of U-functional that this identity holds for any in (L 2)- so that 0* is defined. • 2 I t is easy to check from (6.5) that ilC t ) = ot* 1 , :B( t ) : (0:)2 1• In general, we have

* a generalized multiple Wiener integral 0t'

Hence, by using the adjoint operator

can be rewritten as

The commutation relations for

0t

and

* 0t

are given by

o

where

cs(t)

nonstandard operator and

can analysis

* at

be

interpreted point

of

as

view.

in

the

Thus

distribution

0t

the pointwise creation operator. o* n

=

1lR n(t)o *t 1.

space,

we have

the

is

see

*P

,J

for

Laplacian

b

any

C

It has been shown in [33] that

terms of the oeprators

Here

Cross

to

an

at

* at:

and

ordinary

*p

,J

Let

be

the

completion of

(L 2 0 R) ,

that

is

,J*) p

an

Hence the standard Caussian measure

is supported in the space

defined in 4.A.

Define a weaker norm

p > 1.

and

With this abstract

Beltrami

b and C

b

B

Laplacian b as B can be expressed in

Brownian

functional

satisfying

certain

regularity

generalized

Brownian

functionals,

we

in

Suppose the second

conditions_ In

the

Laplacians.

analysis Let

of

U

be the U- functional of

functional derivative of

U

= If

4>2(·'·)

4>

is in the domain of

f lR 4>1 (t)dt. L 2 ( JR) ,

is a trace class operator of

Volterra Laplacian

Then

and

6

then

4>

is in the domain of

and

V

6 4> = tr 4>2' V

On the other hand, if we use the nonstandard analysis notation, then

6 4> L

can be

rewritten as

An interesting relation between the Levy Laplacian and Ito's formula has been obtained in [39].

f JRl

Example 6.5.

4> =

4>

6 4> = - n4>

K

and

B class operator of n

Example 6.6.

f(u 1,··· ,un) and

6 4>

O.

L such that

L 2 ( JR)

f lR feu)

•

G(t, s ) " fCt,s,·, ... ,.)

If

tr G

t2(JRI-2), then

n

:B(u) : du, where

satisfies the following condition

f

dA

n+l (l

Then

6

4>

C

does not exist, 6 4> B 6

Example 6.7. such that

L4>

=-

is a trace

< "'.

+ 11.1 2 )- 2n4>, 6 4>

= n(n-l)

V

=

f lR feu)

and

0

•

:B(u)

n-2

• • 2 f(u,v) :B(u)B(v) : dudv, where

: duo

f

is a continuous function

161

Then

lI

G

does not exist, LIB = -3, and

'\

2f Ii

lI V

• 2 4f JRf(t,t) :B(t) : dt.

f(u,v) B(u) dudv

The

the

following table gives a comparison on the various Laplaicans acting on space L 2(,/ *) of ordinary Brownian functionals and the space (L 2)of

generalized Brownian functionals:

II

L 2(,/ *)

(L 2)-

lIG

defined

does not exist

LIB

lIBI K n

17

LIB

lI V

V

(-n) = - n K n

I

defined

0

lIL I1

- n

defined.

"'C

SOME APPLICATIONS OF WHITE NOISE CALCULUS In

this

final

section,

we give

some applications

of white noise

calculus.

For further applications, see the forthcoming hook [12]-

7.A

Halliavin calculus via white noise calculus Suppose

and let

lJ

calculus

to

*

is a real-valued random variable on the probability space (,/ ,lJ)

be the distribution of study

the

absolute

measure as discussed in 4.B.

.

Potthoff

continuity

of

lJ

[38]

has used the white noise

with respect

Again, we need to show that

(7.1)

for all

f

in

JJ OR), where

C

is a constant independent of

Observe that by the chain rule, we have

(7.2)

f.

to

the Lebesgue

162

As in 4.B, we need to solve the above equation for the integral in (7.1).

in order to estimate

To do this, take the inner product of (7.2) with

3t

in the t-variable so that

Hence

JIR J *3 t

t

Now, apply the following product formula for

3* t

to the last integral

Then we obtain (7.3)

=

where

J

-

N is the number operator, i.e.

Note that the inner product

«', .»

«'(x».

Thus we have derived exactly the same condition as in 4.B, i.e. if we assume that 1 (N 0,

replace

band

f(B(t»

c

by

meaning as an operator. (7.4)

are constants.

*t).

fca

When

f

In view of 7. B,

it is reasonable to

*

is a polynomial, f (d t)

has an obvious

Therefore, we can instead consider the following equation

*

-AX(t) + f(dt)(bX(t) + c).

X(t)

To solve this equation,

let Ut be the U-functional of fies the following ordinary differential equation

X(t).

Then

Ut

satis-

which

follows

(7.5)

Note

that

we

have

readily from (6.5).

used

the

fact

The stationary solution of (7.5) is given by c e-

At

ft......

+ bft u

166

It is easy to check that (7.6)

e

-At

can be rewritten as the following series bn- 1

I

.. ·J:'

n=l

e

A(u1A••• Aun )

Finally, note that

Therefore, i t follows from (7.6) that the solution of the equation (7.4) is given by X(t)

c e

-At

I

n=l

For an application of this solution in prediction, see [10].

l.E

Fourier transform

·Consider the finite dimensional Fourier transform on

For the infinite dimensional analogue of

interval in

JR.

1 00 ( - ) =0

Although

this Fourier transform on

natural to replace the inner product

,,;*, it is

JT b(t)il(t)dt, where

by

and exp[-iJ

;z,;-

nf

T

b(u)B(u)du]

and

T dB

is an do not

make sense, it is shown in [29] that their product makes sense and is given by :exp[-iJ

where

b

is

the

T

=

J* ,,;

A

ib(t), i.e.

A •

in the

b-variable.

Therefore,

of a generalized Brownian functional :exp[ -if

This Fourier transform takes the by

b

renormalization

define the Fourier transform

•

b(t)B(t)dt]:.

•

T

b(t)B(t)dt]:.

b

B(t)-differentiation A.

(b) = ib(tH(b).

we can

by

at

For the details,

to the multiplicaton see

[29].

Recently,

Hida and Saito [13] have shown that

This is an interesting relation between the LE;vy Laplacian transform.

C>L

and the Fourier

It follows from this relation that the Gaussian Brownian functionals

167 1 . 2] :exp [ c I T B(t) dt :, c '" "2

Laplacian.

'

given in Example 5.7 are eigenfunctionals of the Uivy

The Fourier-Mehler transform [32] is defined by

( 11 04>)(b) where

K (b, B) 0

4>n

K 0(b,

in

4>(B) d u,

is the following kernel function

K 0(b, Let

I,/ *

B)

i = :exp [ sin 0

•• b(t)B(t)dt

IT

i 2 tan

o IT

be the Hermite Brownian functional of degree

Example 5.9).

• 2 ] b(u) du :.' b n

with

(see

c

Then we have e

in0

4>n'

n=O,I, ••••

This is an infinite dimensional analogue of Wiener's result [47].

Acknowledgements. to BiBoS, Workshop

The original version of this paper was prepared during my visit

Un Lver s I t at; Bielefeld in the summer of 1986. on

Stochastic Analysis

at

Silivri,

July

1986.

It was presented in the

The

present

expanded

version was delivered in the Workshop on Infinite Dimensional Calculus and its Applications at the Mathematics Research Center, Taiwan University, July-August 1986.

I would like to express my deepest appreciation to Professors L. Streit, H.

Korezlioglu and N.R. Shieh for their hospitality and to BiBoS and National Science Council for their financial support.

REFERENCES

1.

Arnold, L.:

Stochastic Differential Equations, Wiley-Interscience, 1974.

2.

Donsker, M.D. and Lions, J .L.: Volterra variational equations, boundary value problems and function space integrals, Acta Math. 108 (1962), 147-228.

3.

Elson, C.: An extension of Weyl's lemma to infinite dimensions, Trans. Amer. Math. Soc. 194 (1974), 301-324.

4.

Fomin, S.V.: Differential measures in linear spaces, Uspehi Mat. Nauk 23 (1968), 221-222.

5.

Gross, L.: Abstract Wiener spaces, Prob. 2 (1965), 31-42.

6.

Gross, L.: 181.

7.

Hida, T.: Analysis of Brownian Functionals, Carleton Math. Lecture Notes, no. 13, Carleton Univ., Ottowa, 1975.

Pr oc ,

5th Berkeley Symp ,

Math.

Stat.

Potential theory on Hilbert space, J. Func , Anal. 1 (1967), 123-

168 8.

Hida, T.: Generalized (1978), 55-58.

multiple

Wiener

integrals,

Proc.

Japan Acad.

54A

9.

Hida, T.: 1980.

10.

Hida, T.: Causal analysis and an application to prediction theory, in Masani Volume (1981).

11.

Hida, T.: Brownian motion and its functionals, Ricerche di Matematica, vol , 34 (1985), 183-222.

12.

Hida, T., Kuo, H.-H., Potthoff, J. and Streit, L.: Dimensional Calculus.

13.

Hida, T. and Saito, K.: print (1986).

14.

Ikeda, N. and Watanabe, S.: An introduction to Malliavin's Taniguchi Symp. Stochastic Analysis, Katata (1983), 1-52.

15.

Ito, K.:

16.

Ito, K.: Extension of stochastic integrals, Pr oc , Intern. Symp. (1976), 95-109.

17.

Ito, K. and Nisio, M.: On the convergence of sums of independent Banach space valued random variables, Osaka J. Math. 5 (1968), 35-48.

18.

Kallianpur, G. and Kuo, H.-H.: Regularity function, Appl. Math. Optim. 12 (1974), 89-95.

19.

Krl!e, P.: Solutions faibles d'l!quations aux fonctionnelles, I: Lecture Notes in Math., vol. 410 (1974), 142-181, Springer-Verlag, II: ibid. vo l , 474 (1975), 16-47.

20.

Kubo, I.: Ito formula for generalized Brownian functionals, Lecture Notes in Control and Information Sciences, vol. 49 (1983), 156-166, Springer-Verlag.

21.

Kubo, I. and Takenaka, S.: Calculus on Gaussian white noise, I: Proc. Japan Acad , 56 A (1980), 376-380, II: ibid. 56 A (1980), 411-416, III: ibid. 57 A (1981), 433-436, IV: ibid. 58 A (1982), 186-189.

22.

Kuo, H.-H.: Integration by parts for abstract Wiener measures, Duke Math. J. 41 (1974), 373-379.

23.

Kuo, H.-H.:

24.

Kuo, H.-H.: Gaussian Measures in Banach Spaces, Lecture Notes in Math. vol. 463, 1975, Springer-Verlag.

25.

Kuo, H.-H.: Distribution theory on Banach space, Lecture Notes in Math., vol. 526 (1976), 143-156, Springer-Verlag.

26.

Kuo, H.-H.: Differential calculus for measures on Banach spaces, Lecture Notes in Math. vol. 644 (1978), 270-285, Springer-Verlag.

27.

Kuo, H.-H.: The chain rule for differentiable measures, Studia Math. vo L, LXIII (1978), 145-155.

28.

Kuo, H.-H.: Integration in Banach spaces, Notes in Banach Spaces, Edited by H. Elton Lacey, Dniv. Texas Press (1980), 1-38.

Brownian Motion, Springer-Verlag, Heidelberg, Berlin, New York,

White Noise:

An Infinite

White noise analysis and the Levy Laplacian, precalculus,

Multiple Wiener integral, J. Math. Soc. Japan 3 (1951), 157-169.

property

of

SDE, Kyoto

Donsker's

delta

Differentiable measures, Chinese J. Math. 2 (1974), 188-198.

169 29.

Kuo, H.-H.: On Fourier transform of generalized Multivariate Analysis 12 (1982), 415-431.

30.

Kuo, H.-H.: Donsker' s delta function as a generalized Brownian functional and its application, Lecture Notes in Control and Information Sciences, vol. 49 (1983), 167-178, Springer- Verlag.

31.

Kuo, H.-H.: Mathematicae

32.

Kuo, H.-H.: Fourier-Mehler transforms for generalized Brownian functionals, Proc. Japan Acad. 59A (1983), 312-314.

33.

Kuo, H.-H.: On Laplacian operators of generalized Brownian functionals, Lecture Notes in Math. vol. 1203 (1986), 119-128, Springer-Verlag.

34.

Kuo, H.-H. and Russek, A.: White noise approach to stochastic integration, to appear in J. Multivariate Analysis.

35.

P.:

Brownian functionals (1983), 175-188.

Processus

and

Brownian functionals,

applications,

Acta

J.

Applicandae

stochastiques et mouvement brownien, Gauthier-Villars,

1948. 36.

Malliavin, P.: Stochastic calculus of variation and hypoelliptic operators, Proc. Intern. Symp. SDE, Kyoto (1976), 195-263.

37.

Piech, M. Ann: Differentiability of measures associated with parabolic equations on infinite dimensional spaces, Trans. Amer , Math. Soc. 253 (1979), 191-209.

38.

Potthoff, J.: On the connection of the white noise and Malliavin calculi, Proc. Japan Acad. 62 (1986), 43-45.

39.

Saito, K.:

40.

Shigekawa, I.: Derivatives of Wiener functionals and absolute continuity of induced measures, J. Math. Kyoto Univ. 20 (1980), 263-289.

41.

Streit, L.: Quantum theory and stochastic processes - some contact points, Lecture Notes in Math. vol. 1203 (1986), 197-213, Springer-Verlag.

42.

Streit, L. and Hida, T.: Generalized Brownian functionals and the Feynman integral, Stochastic Processes and their Applications 16 (1983), 55-69.

43.

Stroock, D.W.: The Malliavin calculus and its applications to second order partial differential equations, I, II, Math. System Theory 14 (1981), 25-65, 141-171.

44.

Sugita, H.: Sobolev spaces of Wiener functionals and Malliavin' s calculus, J. Math. Kyoto Univ. 25 (1985), 31-48.

45.

Watanabe, S.: Malliavin' s calculus in terms of generalized Wiener functionals, Lecture Notes in Control and Information Sciences, vo l., 49 (1983), 284-290, Springer-Verlag.

46.

Wiener, N.:

47.

Wiener, N.: Hermite polynomials and Fourier analysis, J. Math. and Phys. 8 (1928-29), 70-73.

Ito's formula and

Laplacian, preprint (1986).

Differential space, J. Math. Phys. 2 (1923), 131-174.

La Theorie des distributions en dimension quelconque et l' Integration Stochastique par Paul KREE

La physique quantique, la physique statistique conduisent certaines perturbations demesures Gaussiennes analogues super symetriques avec d

a

3, 2 au 1

=

v' (JRd+l)

=

if'

et des

Les series de perturbation

ou n est quelconque et ou f est valeurs opere teurs I i nea i res non bornes de L2 (;fI ) . Les pro-

comportent des integrales n-uples une fonction

P sur

a rechercher

blemes poses aujourd'hui par le cas

Int n f d

=

0 ant moins d'interet physique; on

sait aussi que la difficulte de ces problemes augmente avec la dimension Cependant 1 'approche directe de ces probl emes est

t

r es difficile. En consequence

conme i ndt que par P.A. Meyer au seminaire Bourbaki (P.A. ME 86 .) Hudson et Parthasarathy (HP) etudient des modeles de ces problemes ou systematiquement d

=

0 et ou 1 'on se place presque exclusivement (le cas Poisson est aussi

etudie) en coordonnees Browniennes de

a pouvoir appliquer 1 'enorme arsenal

probabiliste correspondant. Autrement dit on travaille presque toujours avec l'espace de Wiener (HccXw' Pw) mcde l i sant le Brownien (B t) a une dimension sur JR+ = JO, + 00 [ • Selon (P.A. Me 86) [P.A. ME 87] le coeur des probabi l i tes quantiques est un calcul stochastique fonde sur une integrale stochastique simple non commutative imposees

f _) IntI f , mais des conditions t res strictes sont

a f, et le pont avec le calcul stochastique usuel n'est pas etabli.

Ainsi les probab i l i tes quantiques utilisent jusqu'ici un calcul symbolique consistant a ecrire les operateurs de L2(X w) comme combinaisons lineaires formelles de produits d'annihilateurs a(u) = a-(u) et de createurs

= a+(u) et une notion de noyau qui n1existe pas toujours, ou il n'y a rien d'analogue aux noyaux de L. Schwartz, et donnant

a la

formule du produit

des noyaux de Berezin une forme tres esthetique. La reference utilisee est ou calcul chaotique (u) Voir aussi P.K. + C. Soize, Mecanique Al eato i re , fin du chap. 11.

171

LF.A. BE 85] ou bien Maassen. Cette maniere d'ecrire les operateurs lineaires de L2(Xw) apporte une aide limitee

a 1 'integration stochastique.

peu d'espoir d'arriver ainsi au traitement pour d = 3 et tout Int n f

avec f

a valeurs

11 y a donc

n d'integrales

operateurs non barnes. Le but du present travail est

de proposer un tel traitement et aussi comme sous produit un debut de theorie des integrales multiples usuelles du type Ito. Pour expliquer la methode, signalons

la theorie 1972-1977 des distri-

butions en dimension infinie et en particulier la these de B. Lascar [B. LA 78J n'imposait pas d

=0

et introduisait pour les operateurs lineaires un point

de vue tres different de celui utilise en plysique : lineaires de L2(Xw) mais

a) on y etudie non seulement des

des operateurs lineaires changeant d'espace; de valeurs comme par exemple i nforme 11ement

et meme les prolongements de ces operateurs

a des espaces de distributions

b) On ne travaille plus comme F.A. Berezin (resp. L. Gross) seulement

Xw) ' mais on travaille simultanement dJ cote Fock et du cote L w)' Ceci a conJuit a plonger Fock dans co l' espace pal (X') = TT Pol (X') des ser-ies formelles sur a prolonger w n=o n w la decomposition chaotique aux series f)rmelles puis a constater que V du cote Fock (resp. seulement du cote 2(X

s imp lemente en la derivation t

D des ser ie s formelles,

s'implementant en

1'operation suivante

"*

3

F(z) h

F(z) z

l:

Pol (X')

Puis l'on devel oppe ainsi un "calcul chaoti que" en cons i derant la transformation chaotique

TC = I

-1

comme une sorte de TF;

172

c) Le noyau de L. Schwartz de tout operateur dans les distributions est d'abord defini ; On montre que c'est une distribution sur 1'espace double et 1 'on constate que le noyau de Berezin est simplement la transformee chaotique du noyau de L. Schwartz [Po KR + R. RA 78J . Question symbole, comme 1'expliquait L. Schwartz dans [L. SCH 77J les operations fondamentales ne sont plus annihilateurs et createurs comme chez Berezin mais les iteres vk et du gradient et de la divergence au sens des distributions. Dans la these de B. Lascar, tout 2(X operateur lineaire L de L w) est ecrit

ou les coefficients

s'expriment en fonction du symbole de L; cette

notion de symbole n'apparaissant pas d'ailleurs chez Maassen. B. Lascar reussit meme a developper la theorie L2 des EDP en dimension infinie. Ces travaux non seulement definissaient mais developpaient le Calcul dans les Sobolev Gaussiens [M. KR 74 + 77] [P. KR 74 + 75) [B. LA 76J [B. LA 78] [EDP

00

74J .

d) 11 y a, un analogue anticommutatif pour tout ceci mais aucune applica­ tion probabiliste n'y est traitee e) Ph. Paclet dans [EDP 00 77J etudie 1e cas non Gaussien et redefinit 1es classes WZ,1 en dehors d1ensembles de capacite nulle. Le contenu de ces references est utilise par P. Malliavin et son groupe

a partir

de 1976. L'app1ication

"

prol onqeant 1'operation du Fock definissant

1'integrale de Skohorood [SK 75J par transformation chaotique, i1 en resulte sans ca1cul que 1'integrale de Skohorod est prolongee continuement par la divergence J au sens des distributions. La publ ication [B. GA + P. TR 82J utilise le calcu1 dans les Sobolev mais

referant au Calcul des Variations

Stochastiques [CVS] , elle montre seulement que l'integrale de Skohorod est le transpose formel de la restriction aux polynomes cylindriques du gradient au

173

sens du CV5 (c'est-a-dire du gradient usuel). Cette remarque de Gaveau-Trauber s'est revelee tres utile car elle a entraine recemment des prolongements distributions du calcul de K. Ito [5. US 86) [D. DC 84J . Un resultat fondamental [S. US 86J est que l'integrale de Ito est induite par la divergence. Le present travail prolonge ce type de methodes aux integrales stochastiques d'ordre quelconque de processus a valeurs operateurs pour des espaces plus generaux que Xw' Le travail etant redige en termes de "calcul chaotique" je tiens a ce propos a remercier P.A. Meyer qui nous a signale que la (n-1)e redaction de ce travail ne fait

pas apparaitre la necessite d'un (n+1)e

calcul et la necessite de sortir du cas Brownien ou Poisson. Ceci nous a conduit a changer le titre de ce travail et

a expliquer

sa motivation

a) Le calcul chaotique est avant tout une reformulation extremement simple et directe d'une partie de la theorie 1972-1977 deja evoquee qui n'etait pas Brownienne. La lecture de cette theorie n'etant pas facile, des reformulations locales et Browniennes plus simples (tenant compte aussi de progres ulterieurs) ont ete effectuees. Mais ces formulations oublient la theorie de L. Schwartz des distributions (pas de noyaux de L. Schwartz, pas de distributions meme, definition des Sobolev comme des completes, etc) et ne font pas apparaitre la decomposition en chaos comme le moteur de la theorie (mise en avant de la quasiinvariance de Pw par ex.). 0'00 la necessite d'une presentation encore plus simple centree sur les distributions et la decomposition en chaos cf. chap. I, II et IV et aussi [Po KR 81) ; b) L'interet et 1'efficacite de cette reformulation dans le cas particulier Brownien sont demontres par plusieurs resultats entierement nouveaux: un debut de theorie des integrales stochastiques multiples, la preuve que le couple

(Int+, Int-)

d'integrations stochastiques de Hudson-Parthasarathy

est prolongee par une certaine divergence, la preuve en corollaire d'un resultat de surjectivite pour ces integrales.

174

c) On a constate en novembre 1986 (manuscrit communique a L. Schwartz, S. Us tune 1 ... exposes a Cl ermont et au 1aboratoi re de probabil i tes ) que 1e calcul dans les Sobolev Gaussiens, les travaux de S. Ustunel, D. Ocone s'etendaient aux processus ponctuels de Poisson et partiellement aux PAIS a condition d'avoir des redactions analogues pour les cas Gauss et Poisson

voir chapitre VI.

d} Apres l'expose de P.A. Meyer a Bourbaki il est apparu que le calcul symbolique en "Y et

permettait d'expliciter et de generaliser le theo reme HP

reliant (B t) au Poisson (yt)a condition d'etendre la theorie des noyaux et symboles aussi a Y. 0'00 le chapitre V qui a aussi des applications aux inte­ grales

HP. e) La definition des decompositions en chaos par Gram­Schmidt introduite

au chap. I a l' inconvenient de ne pas coincider dans le cas particulier des processus sur ffi+ avec la definition probabiliste implicite usuelle [K. IT 55J[P.A. ME 76J qui est formulee en termes d'integrales stochastiques multiples et qui exclue les espaces de dimension finie. Notre definition par Gram­Schmidt des decompositions en chaos permet d'etablir deux resultats generaux : le fait que le produit de deux chaotiques est chaotique, qui eclaire le resultat de [K. IT 56) et le theoreme de decomposition chaotique induite (chap. III) qui eclaire les phenomenes cylindriques. Donc cette definition peut etre utile pour etudier les problemes ouverts concernant la decomposition en chaos. A ce propos la non exclusion a priori des espaces chaotiques de dimension finie permet d'abord de poser des problemes en dim. finie et laisse la perspective d'employer des "Bebes Fock" de plus en plus grand pour atteindre le cas general f} Application a la physique quanti que ou statistique. Signalons aussi un travail a paraitre en collaboration avec P. Bernard appliquant le calcul chaotique aux problemes de statistiques de trajectoires (modeles de Slepian,

175

passage

a

a un

niveau fixe ... ) pour tous 1es processus et 1es champs Gaussiens trajectoire C1. Il est clair que ces probl enes sont de nature t res di f fa-

rentes pour le Brownien. L'Auteur remercie P. Kerezlioglu et S. Ustunel pour leur invitation Si1ivri et aussi pour leur encouragement

a

a rediger un papier qui soit 1isib1e

par 1es probabi 1is tes. Dans cette opti que l' objectif vi se es t d' i niti er une theorie genera1e des integra1es stochastiques; on pourra se limiter en premiere lecture au cas Brownien. L'Auteur remercie aussi M. Yor qui lui a signale deux references

[M.

YO 761 CK. IT 56J qui sont

col l eques qui l'ont aide

a l'origine

de ce travail et 1es

a redi qer ce travail: P.A. Meyer pour son interet et

1e rappel de 1'importance du repere Brownien, R.L. Hudson, A. Dermoune et

J.M. Lindsay dont 1es observations ont permis d'e1iminer deux erreurs, A. Badrikian, S. Chevet, P.L. Hennequin et S. Ustune1.

176

I - Les espaces chaotiques Les 1ettres H, K, H', K' ... s ont re se rvees pour desi gner des es paces munis de structure Hilbertienne. Donc tout espace vectoriel non Hilbertien est note avec d' autres lettres X, Y, X' ... Cette convention permet d' evi te r d'ecrire des chapeaux pour distinguer les produits tensoriels Hilbertiens des produits tensoriels usuels ou d'utiliser un signe particulier pour distinguer une somme Hilbertienne d'une somme algebrique ... Par ex. on pose (1.1 )

ces symboles designant 1'espace complete si

le produit scalaire de Hn etant note @

Xn

somme al qebr i que des Xn;

Pour tout es pace Hi 1berti en K et tout

Xn

H

>n

somme Hilbertien des Hn 0,

f) K desi gne l' es pace

Hilbertien deduit de K en multipliant sa norme par

. Pour traiter parfois

simultanement les questions relatives deux Focks, on prend un indice £ valant +

si

ou - , on convient que c-symetrique signifie symetrique (resp. antisymetrique) £. = - (resp + ), on note

l ' al gebre des tenseurs E. -symetr i ques sur X•••

XC

(1.2) 1ntensite d'un processus lineaire. Soit Hilbert separable M : X'

H' . On dit que H'

X' un sous espace dense d'un

est l'intensite d'un processus l inea i re

si le processus l i neat re centre as soc i e est isometrique : W ;:, X' '" u

II

Mu - E [MuJ

2 L (SL)

On ne travaille ci-apres qu'avec des processus lineaires de tous les ordres i.e. (Mu)n e L2(SL) pour tout u e X' ; ce qu'on ecr i t

.

M ayant des moments et pour tout entier

177

(1.3) Processus Lineaires Chaotiques

a) Soit o-(M)

la sous tribu de la tribu

cJ

des evenements (supposes

complete) engendree aux evenements negligeables pres par les classes UEX' . Soit

le sous-espace ferme de

rables par rapport

on Yl

a 2

-

2

O = n! Q( u1 , •.. , un) = n !

n=o n! < Fn ' Gn > n

Ce produit scalaire se prolonge en une dual i te entre T' al qebre symHrique et les series forme11es. Le produi t sca 1aire dans

H'

(i denti fi e

a

son dual) etant note

de < l,l'> l'exponentie11e exp u = e U as soc i ee

et elle veri fie pour tout

F e Fock H'

a

tout

ueH '

II

est

I

a

au 1i eu

X'

180

< F , eZ

( LIZ)

:>

pour tout

F(z)

1ntroduisant pour tout

ZE

H'

ZE

H'

le vecteur exponentiel

E:Xp

Z

= 1(exp z)

il vient la formule explicite suivante pour la transformation chaotique ( L13)

Comme pour tout lR

(L14 )

fixe

Z

3

E

Hila fonction suivante est analytique

s -----'----,- exp sz

=

sn zlan

2: --ri""!

co Fock H'

il en est de meme de ( 1.15)

Done

( 1. it)

s -7 £xp

IR

SZ

1(exp

SZ)6

2

LM(51 )

sxp sz

Noter que

1((

z) n)

(1.17) Dans 1e cas part i cu1i e r 00 M est decompose par une variable aleatoire p

a

val eurs dans un espace vectoriel

va1eurs dans une partie

X de X ,

X en dualite avec

qui s epare

p.

X et que

X'

X . Ceci signifie que tout

contient une partie denombr-ab l e

D

Vu [L. SC 73J ceci entraine que la tribu Bo re l i enne ey,B de

est engendree par loi

pa

X muni d' une topologie souslinienne

Borel - compatible avec la dual i te avec l'espace ueX' est Borel i enne sur

X' , on suppose

D. On munit

de la completion de

1B

11 apparait ainsi la structure d'espace chaotique

qui permet d'oublier

par rapport

a

X'c H' , 10i

r)

et de developper une analyse stochastique autonome.

La structure d'espace chaotique generalise la notion d' espace de Wiener abstrait et aussi celle-ci : (LI8) Un espace probabilise Gaussien

espace chaotique tel que

(X - X'c H' ,P)

est defini comme un

X = X , top X est vectorie11e, X'

est contenu dans

181

P a pour transformee de Fourier exp(- lIu/i2/2). On suppose

le dual de X et

aussi que l'espace reproduisant de P est un sous espace Hilbertien Par ex. l'espace de Wiener (Xw -

, Pw)

00

X de f i ni t l'espace probabil i te Gaussien w est le dual de l'espace de Cameron Martin

00

designe l'espace vectoriel engendre par les mesures de Dirac

H de X .

Hc et at

pour

Hc -; et comme ces deux espaces ne servent pas en calcul d ' Ito, + C(R)o par la derivation d/dt au sens on transporte la structure de X w des distributions sur ffi+ . Ainsi d/dt induit une isometrie D de Hc sur 2 H = L (ffi+ ' dt) et un i somorphisme de Xw sur 1"es pace X des deri vees distributions sur ffi+ des fonctions continues sur fa, +0)[ , 0 appl iquant Pw " /\ -1 1 sur une mesure Gaussienne P. En transposant J = J) et J = D- il vient t

. Corrme

0

alors sur la 2eme ligne du diagramme commutatif suivant un espace probab i l i s e Gaussien plus simple que X'W

--t

X w H'C

'V

-

(1.19 ) X'

Comme

J

T

mode l i s e

iT

H _i_> X

H'

= kt ' X' est engendre par les fonctions

a tout

instant

t

par la forme l inea i re

(1.::10) Distributions. Pour tout espace chaotique

kt et le Brownien est let sur X

- X' c H' , P), l'espace

des polynomes chaotiques sur X est defini par

Vue l'identite de polarisation vant

O'(X)

est enqeridre par les

X' . Dans le cas particulier (1.18) on sait que

n'est pas toujours vrai.

I(zn), z dec r i-

CY(X) = P(X)

mais ceci

182

a l'injection

En ecrivant le triplet assoc ie espace

on obtient un

de "distributions" et un prolongement de la transfonnation chao-

tique

I­I

aux distributions. 0'00 deux triplets isomorphes

t9 (X):i: (1.21) (X'f c­­;>Fock H'

Pol(X')

(1.22) Les distributions vectorielles relatives a x et a valeurs dans un (K.c Km = K'mc 1 se definissent en tensorisant 1 1'isomorphisme (1.21) de triplets avec 1'application identique de K. triplet algebrique donne

t::=

K•

=

L2(X.K)< m

CP (X)

1

(1. 23)

Tout element

1

1

(X'r II Ki

e...-.-..+

f de

=

• L(cP(X),

8

Pol(X'.

Fock(H' •

est appele une distribution

K

1

K.­vectorielle.

Comme en theorie des distributions vectorielles. f

peut etre interpretee

indifferemment soit comme une forme lineaire sur

• soit comme une applica­

tion l i neaf re Q­l(X)

. Donc < f. Q>

soit un element de

si

1

Q

E

des iqne soit un reel si

Q

'C"

•

cP(X)

(1.23) Produit d'espaces chaotiques a) Consi derons

p processus 1i neai res chaotiques

Mj Pj

Xj ­7" LP( Sl.)

partie

Hj • Mj etant decompose par une variable a valeurs dans une d'un espace Xj en dualite avec Xj ; j = 1 ... P . 0'00 P espaces

chaotiques

(X .• X'.cH'. • P . ) .

d'intensite

­J

J

J

J

183

M = L Mj

Le processus l ineai re H' =

@

base sur

Hj . 11 est chaotique [Po KR 87J et

valeurs dans

X' = II" Xj a pour i ntens i te

p = (p) a

I(M) = III I(M j ) . Comme

X = II" X. decompose M il vient l'espace chaotique -J

- X' c H' , P)

appel e produit des espaces chaotiques

Xj

b) Structure des series formelles sur un produit. On a une bijection p V = III

(1.24 )

(X'.)' ::>11 t . _ t

j=l

J -I

J

... -pt = Sym(1I t . )

00 'v'j,l'injection(X')j 3t j l'injection

_>

Polf(X')

est notee

lk"

.

Qk(x

)= Qr(x

k1

est l'espace

J

, ... ,

symetriques par rapport aux k arguments

X

kp p )

k. sur 1T(Xj) J

Xj . En transposant

il vient une bijection: ,.,

->

V

1\

Qr (Xk) sur le polyn6me x = L xj _

qui appl ique

c) Triplet centre sur qui est ponderee en

Vk! (H')k

->

k! = k ... kp 1 -> polyn6me Z = L Zj -7 Qr (zk). Faisons injections

(H')k. J

(1. 25)

de tout multi-indice

Le dual de II (X')k. -?>k

des formes lkl-lineaires

qui sont (1.24)

est la tensor i see de

Xj --,?X' . La longueur Lk j

k= (k 1 , ... , kp)

(X')

J

(X'

)k

\

avec

Qk'

Utilisant la dualite entre

V et

, cette bijection applique

Qk sur le fixeJ le produit tensoriel des

If 1 = k

0'00 une injection

J

-'>kGl, = k (II (X'.) J

j

)

-7

J0

(H ') k =@ 11"1= k

VkJ. !

J

j

qui est utilisee pour construire le triplet centre sur Fock H' . d) Pour 1'etude de distributions

a valeurs

dans un triplet centre sur

(H')k ' il est en general plus commode d'utiliser la dualite sans aucune ponderation entre

V et V'" . 0'00 le triplet

184

(1. 27)

( X' ) -

(H' ) -

k

k

=

\"C,

(B =

k

(

H'. ) kJ

Po 1k(X' )

J

Illustration en coordonnees Browniennes (P.A.

w se casse en deux pour tout temps t . Par le transport de structure defini par d/dt 86J rappelle l ' importance en calcul stochastique du fait que

l'espace probabil i se Gaussien

X de

X

(I. 19) se casse aussi en deux. Pour fai re

'in

1a theorie des integrales stochastiques n-uples, nous utiliserons le fait que pour toute subdivision o-=£O=ao.ca1< .oo

Ct Cy l (X)

SC 77] . On dit que

est

v

Pear il prolonge

f P(X). C'etait en 1974 un aspect nouveau

car jusqu'alors seule la derivation absolue [H. KU 74] . la notion de de r i vee relative

fP -7V(fP)

avait ete e tud iee

V et son compagnon d

ont immedia-

tement permis d'etudier et de resoudre des EDP non lineaires [Po KR 74 + 75J et l i nea i re s [B. lA Th. 78]

en dimension infinie. Ces notions sont aussi indispensa-

bles en probabilites. En effet, sans cette notion, Skohorod a pu definir son i nteqr a l e stochastique

f = J f seulement en supposant a priori que s done pas montrer que Int induit 1 'integrale de Ito s

f ---7 Int

OT F e Fock. Il ne pouvait

[So US 86J . Par ailleurs le Calcul des Variations Stochastiques

(CVS - 1976-78)

a propose ulterieurement une autre formulation (sans Sobolev, sans le Fock, sans la divergence

I

aujourd' hui abandonnee ) de la der-i vation relative. Pour certaines

189

f sur X le gradient stochastique V f ESt def i rri en CVS comme une w classe de fonctions sur X ' Malheureusement il n'est pas clair avec cette definiw .., tion que V definit une application entre classes et que cette application est fonctions

lineaire, en dehors du cas trivial 00 fest polynomiale cylindrique. C'est une des raisons pour lesquelles B. Gaveau et T. Trauber ont seulement rappe l e dans leur article que Int

est la trdnsposee formelle de la derivation au sens du s CVS (autrement dit usuelle ) des polyn6mes cylindriques. La theo r i e des distributions donnait sans aucun travail beaucoup plus. Nous verrons en V que

stochastique non commutative par rapport

au Brownien necess i te de travail le r avec des distributions sur dans

a valeurs

a ce que 1 'operation de "divergence partielle

(X')*. On peut donc s'attenire

sur un produit" soit de quelque

Xx X

ut i l i

te .

(11.7) Oerivees iterees et divergences partielles : pour les distributions sur un produit. Soit

X le produit ds -> tout multi-indice j = (jr ... jp)

X.. Oefinissons pour

p espaces chaotiques

J

la derivation partielle

.."

vj f des

f

distributions sur X par transport de structure en transposant la restriction aux polyn6mes cylindriques de l'aoplication

->

O!.

"transposee

j

de derivee partielle¥ sur 1es series formelles. Alors pour tout entier

V.e f

R! ":'lj J.

->

j V f

Pour la preuve, on se ramene aux series formelles par transport de structure et 1'on utilise (11.5). Les mon6mes

zn

sont relies tres simplement entre eux par les annihi-

lateurs et les crea teurs , on en dedui t par transport de structure ceci en appelant Cn(z)

=

n

I(z), le polyn6me chao t vque assoc i e

((11.8) Proposition

a zn pour zeH' .

(identites verifiees par les

C (h) n

=

de structure, les relations dans le Fock ecr i tes ci-dessous

1 (h'1l). Par transport

a gauche IfheH

190

entrainent les relations eerites ei-dessous

a

droite pour les polynomes

chaotiques k h 1

a*(h)k 1

a) de meme b) Donc

P

Ck(h) P

/'(h)k+l

hk+1

a*(h)k+l P

Ck+1(h) P

k+1 a(h) h

k (k+l) h

a(h) Ck+1(h)

(k+ 1) Ck(h)

hk+1

a*(h) h k

Ck+1 (h)

a*(h) Ck(h)

c)

a( h) h k

a*(h)

d)

a*(h)k

k h

k

k C (h) k

a*(h) a(h) Ck(h)

La relation a) signifie que la derivee directionnelle absolue - a*(h)

de

P a pour deris i te

Ck(h)

par rapport

a

P. Mais il y a

mi eux

(11.9) Proposition (quasi-invariance de

Soit

X'c H' , P)

a

k, la der i vee absolue

t rans formee chaotique

z

de norme image dense

"Z""h Pest la mesure

j

(_I)k

J

k

Pest la distribution

.r g)

00

xp h

c:

P

de

f

K, toute injection

V

k P a une dens i te E L2(X, K)

est une distribl:tion

a

valeurs dans

V et de l' ope re teur de produit par

f

Dans le cas Gaussien le second membre se re dui t le cas 00

M: X' = Vect i

base sur 1 'espace Euclidien

a

/lH

=

r -')

->

g(d f) - Vg. f . Examinons

dim H' = 1 .

(11.12) Proposition - Soit 2

a

une dimension engendree par

>0 • 1dentifions l e dual

X de X'

a

IR

par

tel que

b, i > = 1 . Alors l a variable a Iea to i re

tout reel

1\

M(

a M( i)

i)

un processus l i nea i re

LP(!:l.)

n< M( i),

i > = -c M( i),

M(i)

i

et posons

bt

00 decompose

BEX

est

M car pour

ni ">

Comme le processus lineaire M(i)

est

-In!

dans

a. .fri!

M centre est isometrique, la variance de Qn a pour norme 1ntroduisant le vecteur unitaire e = i ,e X'n

et de f i n i t l e mon6me suivant sur X.:

t -teQn(t) = < en , (bt)&n >

n

=

a- n/2

tn.

Alors

1 'application du procede de Gram Schmidt aux polyn6mes

M est chaotique ssi e&n(t)

donne un systeme

orthogonal total de polyn6mes

e t side plus

.fr0. .

UI ( en) II

On dit alors que

pest une probabilite chaoti-

que sur la droite.

(11.13) L'exemple 00

p

On a alors pour tout

n

t1

est la Gaussienne reduite de IR

(11.14)

Donc

Hn(t)

n!

ce qui entralne

est bien connu.

192

Vu (II. 8-b), ceci entraine de

a(e) = d/dt

est

f

a(e) = d/dt . Il en res ul te que 1e transpose

-

00 () est 1a de r i vee abso1ue

On notera d'ai11eurs que 1a formu1e (11.14) definissant 1es

ce qui est conforme

a

Cons i de rons dont de variance

a

H n

peut s'ecrire

(II. 8.a).

present 1e cas 00

II .

des distributions.

Pour

M(i)

a une 10i de Poisson de moyenne

}'l ,

decr i vant 1 'ensemble des entiers re1atifs, intro-

x

duisons 1es 2 operations suivantes de difference avancee ou retardee f(x)

f(x+1) - f(x)

f(x) - f(x-1)

f(x)

Uti1isant 1a dua l i t e nature11e = 1e transpose de dHinis pour

Donc

x

Cn(i\,.)

est

-

entier

A'

L

entre suites bi l a ter'a l es ,

f(x) g(x)

Les po l ynomes de Char1 ier de [T. HI 70J

sont

par

est 1a restriction aux entiers >0

d'un pol ynome sur 1a droite

n dont 1e terme de plus haut degre est 8- n/ 2 t . Des sommations par parties montrent que 1es

CnUl..)

forment un sys t.erne orthogonal et que

n!

Comme 1es Cn(a) forment un sys teme tota 1 dans L2 (IR), est chaotique et Pi! n) Cn( , . ) = I(e Ecrivant A' = Id - t' 00 e-' designe 1a translation d'un cran

a

gauche dans

"7, puis deve l oppan t

b'i nome, les coefficients de

Cn(R,.)

A,n = (Id _t,)n

peuvent et.re ca l cul es et l'on trouve

ainsi e

avec l a formule du

-s./F

s x

(1 +-)

Fr.

193

ce qui entraine

{? A

11.16

.)

Vu (II. 8.b) ceci entraine

are)

=

n Cn- 1

= fa

=-I'J

6. . 0'00 a'l((e)

A'

ce qui est

compatible avec (11.9).

a1 / 2

Ces formules se simplifient d'ailleurs en introduisant

C (i) n

C (e)

=

(11.17) Proposition - Soit 2·(V) L

processus l inea i re p

de moyenne

V un espace mesure reduit

est ence ndre par 1 'indicatrice 2(V) L

M base sur

C (i) n

tel que

M(i)

un point

fyJ

de masse

de fyJ . Alors le

soit une mesure de Poisson

(e

Si

01>

) (x)

nn( /1' (n) p ) p-l i\

(_l)n

L

s

n

0

( I I. 20)

a

es t chaoti que

(11.18)

(11.19)

ca r

Resumons les resultats obtenus

n

. L'espace

e

a(i)

= nfJ.

L

Utilisant la dua l i te

C i) n(

----;:;,-=e n.

: f(x) - ?

-s i\ (l+s)x

a f(x+1)

- i'I f( x)

f(x) g(x)

induite par

---7

a g(x)

2 L , le transpose de

a(i)

s ' ecri t g(x)

x g(x-l) -

11 est probable que les mesures de Poisson et les mesures de Gauss sont les seules probab i l i tes chaotiques sur la droite. Ce re sul tat pourrait alors servir de point de depart pour 1 'etude des espaces chaotiques en general.

III - Theorie des projections entre espaces chaotiques (111.1) - Introduction: cas des coordonnees Browniennes. Soient deux espaces p rob ab i l i ses Gaussiens

(X - X'cH' , P)

et

(V - V'cK'

, Q). On peut dire que

194

le couple

(s,k)

forme par une application Ti nea i re continue

: V' ---?> X'

1i nea ire

qui transpose

Q = Loi s = St(P), si

si

Alors trivialement

I(X)

H'

.t

I(V). De plus

X vers

1\

k: K'

Y

et si

induit une application l i nea i re

induit

et par k

definit une projection de

Sf

k se prolonge en une t sone tr i e

la projection orthogonale

s : X

J.:

X'

V'

etant l'inverse a gauche de

k,

ni) est l'inverse a gauche de r(k). 0'00 en transposant ces applications un ascenseur reliant par

s

Pol(X') a Pol(V')

est 1 'inverse a droite

series formelles sur

1 'operation r(k)T

e.....-.......;

1r

r( k)

(V' )

Fock H'

i

r(k)T

r(k)

> Fock H'

c

X aux distributions sur

Gaussien

X modelisant

mode l i s an t

dB/dt

r(£)T V'

des

>

c

sur JO,t)

(St' st)

V. Par exemple

dB/dt

Pol (X' )

>

c

0'00 par transport de structure un ascenseur sur

de restriction a

st

X'

(X' )

n.e )

c'est-a-dire que le re l evemerit

1T

r(l) T

Pol (V' )

reliant les distributions

\It >0 , l'espace probab i l i se

des espaces

est isomorphe au produit

et sur Tt , +00[. Ce qui precede s'applique alors

aux deux projections canoniques. 11 apparait ainsi que le conditionnement par ou par la tribu sur

tj:t

du passe a l'instant

s'etend a to utes les distributions

X en une application lineaire continue d'image directe

(St)t : P(X)'*

qui admet le re l evemerit

On a donc des injections canoniques et sur

+

Xt

sur 1 'espace

(St)

dans les distributions sur X modelisant

dB/dt

= fa = a o < a 1 < ... < an

t

et

(St)'* +

t

comme inverse a droite.

des distributions sur

X ... On peut donc travailler sur

X t X ou w

exactement comme L. Schwartz nous a appris a

travailler avec les distributions sur a-

t

St

rn 2

. De meme pour toute subdivision

on travaille sur

X comme sur

rn n+1

La suite de ce paragraphe peut etre omise en premiere lecture car elle etend tout ceci aux espaces chaotiques. Le fait surprenant est que si 1 'on definis-

195

sait une projection entre espaces chaotiques comme ci-dessus, eh bien n'induirait pas forcement

1 'etude par le point de vue dual

I(Y).

(111.1) Injections lineaires dans

I(X)

X'

compatibles avec la decomposition de

Gram-Schmidt de M un processus l ineat re

Soit

injection l tnea i re l i nea i re

X'

() LP(S1.)

k d'un espace vectoriel

M' = M 0 k

est le complete

d'intensite

Y' dans X' , l'intensite du processus

K' de Y'

pour le produit scalaire A

H' .0'00 un prolongement i s orne t r'i que Pour tout

n, Mn et M'n de s i qnent les restrictions

k: K' ---?H'

a

fePn(M')

k de Y' dans X' de

si

)

et

_ n

P n(M') des projecteurs orthogonaux sur les sous espaces 2 de L (Sl.). 0'00 pour tout

H' • Pour toute

et

de

k

a P< n(M')

la figure ci-contre. Avec ces notations, l' i njecti on

est dite compatible avec 1a decomposition par Gram-Schmidt \In

\lfEPn(M')

Figure (III.2)

1T"n f = 11"'n f P",(M') c

P",(M)

o autrement dit si le diagramme ci-dessous commute 00 relative

a

M' .

R'n designe l'application

Rn

196

.;;;

Rn

(H')- .:>(X')-

n

r

n

n k

( II1.3)

.fni.

(K'

fn

Idn -"lfn

f

R' n

::> (Y')- ------;> n

Pl!: n(M' )

Id'n -11'n

Pc::: , n

r » P

Comme les deux fleches verticales extremes sont isometriques, cela entraine que la ligne i nfe r i eure est i s orne t r i que si la l i qne s uper i eure 1'est. Par ailleurs pour to ute injection l i nea i re

IhT'n f - f II

IIIn f - f II

(I I I.4) Donc si

k de Y' dans X'

Met M'

sont chaotiques, ces inegalites sont des egalites et

compatible (cette derniere remarque tres utile est due

k est

a H. Dermoune).

(III.5) Theoreme sur les decompositions chaotiques induites - Avec les notations qui precedent supposons

P(M) et P(M') denses dans

a) Si l'application l i neai re injective la decomposition par Gram Schmidt de

et

k:

2

resp.

est compatible avec

et si

M est chaotique, alors

est chaotique b) Reciproquement si

Met M'

sont chaotiques, alors

k est compatible

avec la decomposition par Gram-Schmidt de c) Dans chacun de ces deux cas, la decomposition en chaos de M' induite par celle de M i.e. le diagramme ci-dessous commute Fock H' (I 1. 6)

r(k)

I (M)

n

ffi k

Jj

J Fock K' En particulier pour tout

v'6 K'

)

I (M' )

(5l. )

est

M'

197

exp

A

A

, k'

V'

exp -c kv'

Z>

, Z>

I (M)

A

>E.Xp kv'

r exp

(111.8) Corollaire I vers un autre

Tn k) exp
s xp v'

telle que

c s x , v > = pour tout

V

et presque tout

induise une application faite si

.i:

x

et telle que la projection orthogonale

H'

K'

Y'---7X' . Cette derrri ere condition est toujours satis-

dim Y est fini. 11 en res ul te alors comme dans le cas Gaussien un

ascenseur

(s*, s*)

reliant les distributions sur

X aux distributions sur

Y .

La composee de deux projections entre espaces chaotiques est une projection entre espaces chaotiques. 0'00 une notion d'isomorphisme d'espaces chaotiques.

(111.9) Corollaire 2 = decomposition en produit d'un espace chaotique X

= (.: -

avec

j = 1 ou 2 , une decomposition

(Sj,k j)

X. = (X.-X'.CH'.,P.) J -J J J J est definie par deux projections

X' c H' , P). Etant donnes deux espaces chaotiques

de

X vers

l'espace chaotique tout temps

Xj

telles que

X fV Y x Y 1 2 s

= (sl,s2)

X vers l'espace chaotique

t , 1 'espace chaotique

definisse un isomorphisme de

Xl x X . Exemple typique : pour 2

X modelisant

dB/dt

est isomorphe

a

1 'espace chaotique

La theorie des phenomenes cylindriques est un autre corollaire mais ceci necessaite une definition:

198

(111.10) Systemes finiment orthogonaux et filtrants de generateurs d'un espace

X'

prehilbertien

J

a) Pour toute fami 11e

... , op (J)

Pour p = 1, 2 nables Ii par d.

p

j

On dit que

teurs de

X'

X'

si

designe 1 'ensemble des parties j'

elements de

X'0( = Vect

Soit

= (a i

le sous espace de

0(

orthogo-

=1

X'

engendre

et si

X' + X'

O(J)

E.

0(

b) Pour tout processus l i nea i re introduisons les restrictions de X'

0(

est un systeme finiment filtrant et orthogonal de generaest union des

'fd..

X' , posons :)' = :l \ [OJ

de parties finies de

Mol.

et les variables a l ee to i re s

M: X'

=

Vect ::J 1'\ LP(St)

pc(' Pr

H' ,

d'intensite

Xt

M aux so us espaces

de

' M

c X' 't

qui decomposent ces restrictions.

0 '00 le diagramme commutatif suivant 00 les petits points symbolisent des dualites Fig u re (I I I . 11)

Sl X...

7"j

&,

X-(

..

>(

M

n L"(n) p>"

Un argumen t de pol ar i te mont re ceci c) Supposons que i.e.

E(expcllpcln)

P(M)

es t dense dans

'fd.

E;

O(

j) ,

mol = Loi

fini pour C = f.(o0

6-

est

a

dec ro i ss ance exponentie11e

assez petit. Alors pour

l

z p e ee

,

St)

d) Avec les notations b) on dit que

a/ = (i 1 , ... , iN)

Pc(

0(:])

les variables

M est M(i

1)

J -j ndependant

, ...

et

M(i

N)

si pour tout sont i ndependan tes .

199

e) Par exemple si

M est un processus l i nea i re Gaussien (ou espace Gaussien

au sens de [J. NE 68] , alors

J -j ndependa nt

M est

si

J

des i qne 1 'ensemble de

Xl •

toutes les parties finies non vides de

(111.12) Corollaire 3 et theorie des phenomenes cylindriques. Soit

J

un systeme

finiment orthogonal et filtrant de generateurs de 1 'espace prehilbertien complete separable

IJcI.. o(:l),

a

p: n

On suppose que

il existe une classe d'applicatiJns mesurables

qui decompose la restriction

M.,(

de

un processus l i nea i re d'inten-

M: X'

decompose par une variable a l ea to i re

H'

site

H' • Soit

X'

Ma

de

, te11e que

So(

Pq

= So( 0

r

ait une l o i

decroissance exponentielle et te 11e que

P(M)

M"" soit chaotique. Alors vu (IIL15.c), 2 L (st ) . De p1 us pour tout couple (0( ,0) tel que x: t» X'

est dense dans

co(

et vu (IIL5.b) la decomposition chaotique de est chaotique. De pl us l' appl ication , Loi Pel)

-

XJ

sur

So(

de

induit celle de - X'c H'

la base orthonormee

normant les elements de 0< • Alors pour toute distribution

composantes de

Zl ... zn

sur

X.

a

(111.13) Remarque - Rapportons

L

P)

(e.) J

obtenue en

9 sur X(J,.' et pour

d'une operation vectoriel1e fondamenta1e, on peut calcu1er les Lg

a

l'aide des i t e rees des operations scalaires. Par ex. natant

les coordonnees par rapport

k de la serie forme11e O

G sur

a

une base orthonormee de

est 1a serie formel1e suivante

la derivee

a

valeurs dans

)k k!

(111.14)

a vee

p.1 -- rr; i

l'annihilateur

M

def i n i t une projection. I\insi la theo r i e des distributions

est induite par celle de

toute iteree

, Loi

Me( . Alors

I

: ...

Pn'"I

a(e.). J

:L

f->. 00

symbo1yse en fait

200

0'00 par transformation chaotique inverse 1 'expression suivante de k

(111.15)

9 -7 V

(111.16)

Application

SoH

a

I F'I = k

tout espace probabilise Gaussien

1 'ensemble de toutes les parties finies de

de decomposition chaotique induite montre que Pour tout

v -I 0

de H' v v vu II

Vect v

H

(111.17)

dans

H'

0'00 pour tout (111.18)

soit

Pv

u6H'

X' . Vu le II , le theo reme

X est chaotique.

la variable a Iea to i re

2

(s-7 exp(s UvU - UvD /2))

=

(X - X'cH' , Pl.

a

valeurs dans le dual

M . Par decomposition chaotique induite on a v

qui decompose

(z xp v)

k

'L

k!

9

V

Pv

0

la formule (op( u+v)) . exp < u, v >H'

(z xp u) (EXp v)

Passons maintenant au cas Poisson. (111.19) Systemes finiment orthogonaux et filtrants d'indicatrices. Soit

nais muni d'une mesure positive a) Lorsque parties de

Ib

m o--finie, H = H' =

est fini, Y est muni d'une bonne algebre de Boole

Y, i.e. Bales deux proprietes suivantes

Bor-e l i enne de trices

m(Y)

Y et m(b) = 0

(bsB)

Y . Vu i) 1 'espace

entraine

b =

ii)

contient une partie denornbrabl e Vect

j

Y polo-

B de

i) B engendre la tribu 1 'ensemble

j

des indica-

D qui s epa re les points de

des fonctions e ta qee s snqendre par

J

s'injecte dans

H separable et l ' injection a une image dense. De pl us l' integration des fonctions en escalier par rapport aux mesures donne une dua l i te entre MSp(Y)

des mesures sur

Vect:J et l'espace

Y (dites signees ponctuelles) qui sont combinaisons

lineaires finies de masses de Dirac.

201

b) Lorsque m(V) croissante

(V j)

est infinie, on suppose que Vest union d'une suite

de parties polonaises telles que m(V)

On suppose que chaque V. J

i nduise

j

pour tout

Bj

est muni d'une bonne algebre de Boole B. et que Bj + 1 J Posant B = U B. et notant j 1'ensemble des indicaJ

(be B), on a encore une injection

trices De plus

j

soit fini pour tout

Vect Vest en dualite avec 1 'espace

l'ensemble des mesures e Msp (V j) pour tout est muni de sa topologie polonaire usuelle.

a image

dense de Vect ::J dans H

Msp (V) defini maintenant comme j . On definit de meme Mp(V) qui

Application aux processus lineaires de Poisson' (non prolonges).

(111.20)

L2(Sl.. )

Avec ces notations, soit M un processus l i neai.re de Poisson: Vect d'intensite

m decompose par une variable aleatoire

a valeurs

dans la partie

Mp(V) de Msp(V). La T.F de M est

(111.21)

a) Vu II le theoreme de decomposition chaotique induite entralne que chaotique et induit I a decomposition chaotique de Me( pour tout

est

«ECj'(j).O'oO

en oubliant Sl la structure d'espace chaotique de Poisson (Mp(Y) - Vect :J c

, l.oi

b) Expl icitons les e.= J

So(

P) ;

pour tout

c>l

J

J

fication de X",

b.

a JIRn

. Comme

M(e j) =

*

M(bj)' la variable X..

-'"

i =1

J

(111.22) 00

qj

a.J = m( b.)J

Xo( . 0' 00 une i denti-

forment une base orthonormee de

MJ prend ses valeurs dans le sous espace ....

n] . Posant

= L]b. ' j = 1

des i qne le nombre de points wR. s i tue s dans b. J

IN

pc(

decornposan t

de X... 11 vient ..

202

q e Vect j

Vu II et le theoreme de decomposition induite on a pour toute

(c 0'00

Cf

SC/'dm

-

)( c..:»

= e

1fi e n (w) (1 +

la formule analogue de [J. NE 68J exp Cf

(, Lorsque

K le lemme des Scub o'

a

f

est l' esperance

P(Xo()

K est le dual d'un espace de

applique

a

B

=

LP (X , K o)

ne donne

qu'une partie de ce que donne la theorie des martingales vectorielles. Etudions les Sobolev dans le cas general en suivant la methode de

rH. KR 74 + 77J relative au cas Gaussien. (IV.7) Espaces de Sobolev

WP,s(X,K)

pour

s

reel, l"'p
Comme

9 G f = 1(G) G 1(F) = (I G I) (G G F), ceci montre que le noyau de

Berezin est la

TC

du noyau de L. Schwartz et le theoreme est demontre. Bien

noter qu'en dimension finie car

a He remplace par

Ls tP(X)

n'est pas le noyau de la theor ie de L. Schwartz et car

dx

a ete remplace par

souvenant des ecritures symboliques suivantes utilisees pour continue t;l:)

et

(A

ou meme

Ix A(x,x')

Cf(x') dx'

211

nous pouvons uti1iser en dim. que1conque l'ecriture symbolique suivante rappe1ant que

L s

travaille comme 1es noyaux de la theorie de L. Schwartz

(L f) (x) =< Ls(x,x') , f(x'»

f

ou meme

Ls(x,x') f(x') P(dx')

X

(V.l) On definit l'app1ication symbole par

Qp

;:}.!:.

N

---+ L(z .z ') e

-zz'

Pol ( X' )( X' )

E

C'est un homeomorpht sme car c'est l e compose de 1 'application noyau "operation consistant forme11e fixee

(exp -

a multip1 ier une ser i e zz') a terme constant

(V.8) Ecriture matricielle de est 1a some de 1a ser ie

L = L Lk,t ,00

Lke e Op

Pour tout couple

1:

L

(k ;t)

par

par 1a ser i e

non nul.

'"L

de tout

L60p

qui converge dans les series formelles. est un horneomo rph i sme, on en dedu i t que

a pour noyau Lk,R.

0'00 la formule suivante pour

X'l( X'

(V.6). D'apres (11.18) le noyau

Lk,.t(Z'z')

Puisque l'application noyau

formelle sur

L -+ L

.

d'entiers et tout couple

z et Z'

(u; v)

X',.. X'

X'

Donc (V.6) peut etre ecrit sous la forme equiva1ente CP

(V.IO) mais rant

a

(LF) k (z) =

.t

present 1e deux i eme membre

F0

j

=1

On se donne un processus simple etage adapte

a

qui s'ecrit pour une certaine subdivision

(a o = 0O. 4. CASE OF ORNSTEIN­UHLENBECK PRJCESSES

The W­valued Ornstein­Uhlenbeck process is cylindrically characterized on H by (4.1)

= e­ t

[X (h) + 12 It e S dB (h)] o o s

h H

243 where X is a W--valued randan variable whose distribution is o

)J

and B is a W-valued

£;-Bro.vnian rrot.Lon Independent; of X X (h) satisfies the equat.i.on o' t

with initial value X (h). It is a centered Gaussian random variable wi th variance o 2 II h[[H' Therefore, as a W-valued random variable, the probability distribution of X coincides with t

)J.

For this process X we have the follcwing result.

4.1 PROPOSITION Let X be the W-valued Omstein-Uhlenbeck process, then for FE [)2,2 we have

(4.2)

F(X

t)

= F(Xo ) +

Jt LF(X ) ds + /2 ft (OF(X , dEs) 0 s 0 s)

proof : Let again F be in [) and put F (X n t)

= E)J (F IVn ) (Xt)

as in the proof

of 3.2. Then Formula (3.5) beCaTES

For m,nEN we can write E[sup [ft (OF (X ) - OF (X), d3 )]2] toms ns s 2 ( 4E Jl [[OF (X ) - OF (X )II ds oms n s H

Similarly, E [sup [Jt(LF (X ) - LF (X )dsj] 2 n s) tom s

By prooeeding in the same way as for 3.2 Pr'opos i t.Lon and 3.3 Theorem, we get the

proof. _

244

In case B is the n-dimensional Brownian sheet on

Formula (4.2) takes

a rrore explicit form. In this case, W is the space C ([0,1]; lIP) of IFP-valued conn tinuous functions on [0,1] and H is the space of absolutely continuous IR -valued functions f vanishing at 0 with density in L that of its density

f

2([0,1],

n dt ; IR ) . The norm of f is

E:L . The procesa X characterized by (4.1) is then the two2

pararreter Ornstein-Uhlenbeck process defined by

(4.3)

where {X ; o,s

SEC

[0,1]} is an n-dimensional Brownian rrotion independent of B. X has

a continuous version on [0,1]2. We put X = (Xt,s ; sdO,l]}and consider it as a t n). n C([O,l]; IR ) - v alued randan-variable. It induces the Wiener measure ]J on C([O,l];lR We then consider X

=

{X ; t E: [0,1]} as an infinite dimensional Ornstein-Uhlenbeck t

process with values in this space. 4.2 POOPOSITIGl n Let X be the C ( [0, 1], IR ) -valued Ornstein-Uhlenbeck process defined by (4.3).

F(X

(4.4)

t)

- F(X ) + f t J 1 0 s= u=

DF(x)su

X(s,6u) ds

where ----DF(X is the density in L n of DF(X s) 2([0,1J,dt;lR) s)

; the first integral is a

Skorohod integral, whereas the second is a two-pararreter one-sided integral in the sense of Cairoli and Walsh [1]. Proof: The above formula is an explicit form of (4.2). The first integral replaces f

• 1 ,.----..

r

0

t

L(X) ds , As L=6D, we only have to show that 6DF(X ) = s s

o DF(Xs) (u)X(s,6u). The operator 6 coincides with the Skorohod integral on the

Wiener space, [12]. Since X induces the Wiener measure 6DF(X ) can be evaluated as s s a Skorohod integral. Let un o

=

[0,1] such that lim max Fl-e-cc k

1 -

0


1, k

is defined coordinatewise. D (M) is defined as the projective limit of E

ll} and its continuous dual is denoted by D'(M). Let us note that if

M =R. we shall simply write Dp,k' D, D' instead of Dp,k (R), D (R), D'(R). For the needs of the filtering, we shall be studying simultaneously with two independent Brownian Motions

wl,w 2

to which will correspond the respective Sobolev and distribution

spaces and we shall distinguish them by indicating in the parenthesis the corresponding Brownian motion, e.g.: the space Dp,k (M) defined with respect to the trajectories of W' will be denoted as

Dp,k (M, W'), D p ,k (W 1 ) if M =IR, the Ornstein-Uhlenbeck generator on the paths of W 1 will be written

asA w" etc.

The space of H -valued, adapted distributions is denoted by an upper index added to the above notations, e.q.: D;,k(H, W') means the Sobolev space Dp,k(H) defined on the trajectories of W' , whose elements are adapted to the filtration of W 1 (c.t, [11],[12],[14]). II. Conditional Laws and Filtering We shall treat the classical filtering problem of a diffusion process (x t ) where

Xo

where band R

m

o are bounded,

=x ,

- vector fields on R d with values respectively in R d and

® IRd , having polynomially bounded derivatives. The observation Iyt) is defined with t

Yt

= fh(xs)d s +Bt o

n

where h is an IR -valued, bounded n

field on R d , with polynomially bounded derivatives

and B is an IR -valued Brownian motion independent of W. We want to study the regularity of

250 the mapping

{Ys; s

f

H

E if (Xt) I

is the o-alqebra generated with the process

where

eventually completed with the negligeable sets.

As usual we shall attack the problem with the reference probability method: Let

Zt

be the

process defined by

1t

t Zt

=exp{Ih(xs)dys -2flh(xs)12ds}

°

°

then, under the probability P" defined by

where P denotes the probability on the product space Q(w )xQ(B), (Yt) and (Wt) are two independent Brownian motions and E

if (xt ) I

is equal to

f (Xt) I EO[Zt I

EO [Zt .

(c.t. [7],[9]).

We can now give our first result: Theorem 1:

Let F be an element of D (w) (D (w ) represents intersection of all Sobolev spaces defined on the trajectories of w ) and suppose that the mapping continuous extension from S'(R

d

)

f

H

f F 0

from S (JRd) into D (w ) has a

(Le., the tempered distributions on lAd) into D '(w) ( i.e., the

distributions defined on the Wiener space corresponding to the W). Then the conditional law of

F, given Yt has a density which belongs to S(R d ) almost surely. Proof:

As we have explained above, it is sufficient to prove that the conditional measure

f

H

EO [f (Xt)Zt I

has a density in S(R d ) under the above hypothesis.

From the assumptions on h, it is easy to see that, for any

z, ED (w

,y) where

t

the exponential martingale

D (w ,y) denotes the space of the test tunctionals defined on the product of

251

the Wiener spaces corresponding to Wand y. Hence, if TeS'(Rd ) , then from the hypothesis

T (F) is in D '(w) which can be injected continuously into D '(w ,y), and the rnultlpllcation with an element of D (w ,y) the elements of D '(w ,y) induces a linear, continuous mapping of D '(w ,y) into itself (c.f. [19)). Consequently, the mapping T H T(F)'Zt is a continuous mapping from

S'(R d ) into D '(w ,y). Furthermore, the Ornstein-Uhlenbeck operator defined on the trajectories of y commutes with the conditional expectations with respect to 'Yt (c.f. [12},[14)), hence the mapping T H EO [T(F)Zt I'Yt] is a linear, continuous mapping rom S'(Rd ) into D'(y) since it is the composition of the following continuous mappings:

(T H T (F)) 0 (T (F) y

T (F)) 0 (T (F) H

where the second mapping corresponds to the injection D '(w) y

z, T (F)

0

D '(w ,y).

Since D '(w ,y) can be written as U p>1

U kEN

D p -k(w,y) , '

for given TeS'(R d ) , there exists some (P,k) such that T(F)eDp,_k(W,y)-Zt being in

D (w ,y), we have also (1_Ay)k Zt in D (w ,y) where A y is the Ornstein-Uhlenbeck operator defined on n(y). If we multiply T (F) by (1_Ay)k z., the result is then again in the same Sobolev space Dp ,-k (w ,y). Consequently, the conditional expectation

is in the Sobolev space D p ,-k (y) since A y and EO[. I 'Ytl commute. If we apply now to this conditional expectation the nice bounded and smoothing (in the sense of the Malliavin calculus) operator (1_Ay)-k , then the result should be in D p

,0

(y )=LP (n(y )), but, since A y commutes

with EO[- I 'it] and since wand yare independent, we obtain

As a result we see that, for any T eS'(Rd ) , in fact the conditional expectation EO [T (F)Zt

I 'Yt]

is

252 a

(class

of)

random

T HEO[T(F)Zt

Iytl

variable

in

some

LP (n(y»), p > 1; consequently,

is a linear mapping from S'(lRd ) into u

p>1

the

mapping

LP(n(y). fly). A classical

closed graph argument shows in fact that it is also continuous, therefore the bilinear form on

S'(Rd ) x n LP (n(y). fly), defined by p>1

is separately continuous, the second space being a Frechet space, a classical category argument (c.f. [8]) implies that it is uniformly continuous. Since S'(lRd ) is a nuclear space, the Kernel Theorem of A. Grothendieck (c.f. [8]) implies that the above bilinear form can be represented as

L.'A;,fi i=1

where

('A;, )el 1, (fd

C

0 Gi

S(Rd ) and (Gd c u LP (n(y» are equicontinuous sets. It is now p>l

trivial to see that the map

can be represented as

L.A.; «r.r, >(5',5) .

i=l

o; Q.E.D.

Let us remark that the hypothesis of Theorem 1 is satisfied, for instance, when F is a nondegenerate Wiener functional in the sense of Malliavin-Watanabe (c.1. [19]). In fact, in this case, one can say much more as the following theorem shows:

Theorem 2: Suppose that every1hing is as in Theorem 1 and that F is nondegenerate, i.e., the matrix

{('i7wFi , 'i7wFJ); i jSd} (where 'i7 w means the Sobolev derivative defined on the trajectories of W) has an inverse which belongs to

253

n LP(llw

;

p>1

R d 0 Rd )

,

then the mapping T H EO [T (x t )Zt I Yt] is continuous from 5'( R d) into the space of the test functions D (y ) on the trajectories of y and the conditional measure can be written as

where

(A; )E [1. (fj) c

5 (R d ) and (CJ'j) cD (y) are equicontinuous sets.

Proof: It is well known that any TE5'(R d ) can be represented as T=Dag where (XE N d • g is a continuous function of polynomial growth. We have, then, using this representation:

now we use the integration by parts formula and the selfadjointness of A w to reduce the order of the derivation of g, we study under the conclitional probabilities, but this does not make a big difference with respect to the ordinary case (c.f. [10],[14]) and the last equality can be written as

where

P a(t) =

L

a" ... , a.. t

t

where H u, are the polynomials in terms of fn "h. (xs )dy s' fD ah (xs)D "h. (::s)ds • Dj and r jj o

°

denotes the inverse of the matrix (VF j • VFj )jj' By the hypothesis, P a(t) belongs to D (w ,y) and since g is of polynomial growth the resulting conditional expectation belongs to all

LP-

spaces. Let us now look at

A;EO[T(F)Zt I Ytl :=EO[T(F)' A;' z; I Ytl := EO fg (F)zJj a k (t) I Ytl where

Pa k

is similar to P a but takes into account the effect of A k on Zt also. Nevertheless, the

result is again in all LP -spaces, i.e., T H EO [T(F)zt I Yt] is a linear mapping from 5'(R d ) into

254

D (y) (l.e., the space of the test functions). We have again its continuity by the Closed Graph Theorem. Let us denote this mapping by Tet' if we look at the bilinear form b on S'(Rd)xD'(y) associated to Tet by

b(T,G) = D,D'(Y) we have only the separate continuity of it. Since neither S' nor D' are metrizable, we cannot deduce its uniform continuity as in the preceding theorem. At this point we shall need some results of the theory of stochastic processes with values in the nuclear spaces (c.f. [11] and the references therein). We know that Tet is continuous from S'(Rd ) into D (y), hence, in particular from the Theorem of Minlos-Sazonov-Badrikian, it induces an S(R d )-valued random variable that we shall denote by itt. From a theorem of [11], we know that, there exists an absolutely convex, compact set K in S(lRd ) such that itt in fact takes almost surely its values in the subspace

u nK := S[Kj, moreover K can be chosen such that (S[Kj,PK) is a (separable) Hilbert space

n=O

where PK denotes the gauge function of K (In practice we say that any nuclear, Frechet spacevalued random variable has a Hilbert-support and in [13] this result is proved for the general semimartingales!). Hence we have

Tet (T) where iK : S[Kj y

= S.S' = since S[Kj and S[K]' are Frechet spaces and (T ,G) H (i;(T),G) is continuous on

S'(Rd)xD'(Y), we see that the composite map (T,G) H is continuous and then, we apply the Grothendieck's kernel theorem thanks to the nuclearity of S'(Rd

).

Q.E.D. III. The Case F

=x t

In this section we shall study in detail the measure defined by f

H

EO[f (Xt )Zt I 'Yt j and

its requianty with respect to the space and time variables under the non-degeneracy hypothesis

255 about x.:

Nondegeneracy hypothesis: For any 00, p >1,

!

1E [

1. . ] dt O, TES'(R d), we have the Ito formula for

T (Xt) in the following way: t

t

T(xt) = T(x e) + fLT(Xs)ds + fLO'ij(Xs)diT(xs)dWl e

e

where L is the infinitesimal generator of the diffusion process, the Lebesgue integral should be understood as a Bochner integral in some Sobolev space Dp ,-k where (p ,k) depends how bad the distribution T is, and the stochastic integral is the weak Ito integral constructed in [11] and [14] as a topological isomorphism between D; = {G-; G ED'} and the space of the adapted distributions tD" (H))'. The- same formula applies to calculate T (xt ) . Zt (since Zt ED (w ,y)) and taking the conditional expectations with respect to EO[. I Yd, by the independence of wand

y, we have t

t

1tdT) = 1t[(T) + f1ts (LT)ds + f1t s (hT)'dys , [

e

where 1tt (T) denotes the (generalized) conditional expectation

As in the preceding section, we can write T

= D "e

where g is a continuous function of

polynomial growth and then using the integration by parts formula on the trajectories of w to make disappear the derivation D

0.,

we have

256 t

t

J1ts(LT)ds = JEO[LT(x s)' Zs £

=

£

t

= JE [g (xs )zsP a(s) I o

]ds

£

where

t

t

t

where the Ha;'s are the polynomials of Jh(xs)'dys' Jlh(x s )12ds , VgJDYh(xs)·dys, and o

r ij (t), where (r ij

0

0

(t)) is the inverse of the matrix {(Vxf, Vxi) ; i j lo then SD e can be injected into S'D £' by restricting the time to the interval [lO',1]. From the calculation that we have made above, it is obvious that the mapping T

1t.(T)

defines a linear, continuous mapping from S'(R d ) into SD o ' Indeed, we have Theorem 3: Q

The biunear form (T ,11) --t is continuous on denotes the semimartingale {EO [1te(T)zt

I ytl;

)x(SDo y, where 1t.(T)

tE]O,1]} in the sense of Laurent Schwartz (c.t.

[10]). Proof: Let I

be any open interval containing zero, we should show that Q-1(I) is open in

S'(R d )x(SD o )' whenever (SD o )' is equipped with its inductive limit topology. To do so, it is sufficient to prove that Q-1(I)

is open when Q

S' (R d )xSD ; In' "i n > 1. However,

1t.

is restricted to the product space

restricted to the interval

[J..., 1] is a semimartingale with n

values in S(R d ) , hence from a general theorem about the nuclear, Frechet space-valued semimartingales (c.t. [13]), it is concentrated in some Hilbert space S[K], K c S(R d ) being compact, absolutely convex and the same argument as in the proof of Theorem 2 can be applied. Q.E.D.

258

We have the following Corollary: The filter

1t

can be written as

1tt (x ,y)

= Llld;(x )TJf(y) i;1

where (/li)E [1, (f;) C S(R d ) and (TJi) c SD o are equicontinuous sequences. Proof: It follows from the Kernel theorem, as before. Q.E.D. IV. Finite Dimensional Approximations of the Zakal Equation Suppose that we have a sequence (h n

)

in the Cameron-Martin space H and denote by 1

8h n the Wiener integral (under the reference probability measure PO) V n the a-algebra generated by the Gaussian random variables {8hi imate 1tt, i.e., the filter

f

H

EO [f (Xt )Zt I ytl with respect to V n

EO [1tdf) I V n ], for any n:2:1 , t :2:0, (we suppose that

f

Jh n (8 )'dys'

Denote by

o

i 0l}. We want to approx-

;

,

Le., we want to calculate

is a smooth function). If (h n ) is a com-

ptete basis for H, then, because of the martingale convergence theorem, this would be a good approximation. However, in general, the conditional expectation with respect to V n troubles the causality of Zakai's equation because of the Itointegral and to circumvent this difficulty we shall be obliged to use the divergence operator or the so-called Skorohod integral. For this we need the following result, for its importance, we shall give it in its full generality and using the general notations: Theorem 4: For any TJE D '(H), we have the following relation:

259

E[8Tl JVnl =

L

JVnl

i,kSn

denotes the inverse of the matrix {(hi,h j ) ; iJ5n}. In the particular case where

where

(h n ) is a complete, orthonormal basis of H, we have

where P« is the orthogonal projection from H into the space spanned by {h 1, . . . ,h n }·

Remark: Note that

is nothing but the inverse of the Gram-Malliavin matrix corresponding to

the random variable (8h 1, .•. , M n). Proof: It is sufficient to prove the above relation by evaluating each side on the test functionals of the type

where

f

E

S (Rn ). We then have

=

i,kSn

L

i,kSn

,8hn), (Tl,hd>

••• ,

8h n), 8hk (Tl,h i) - Vh.(Tl,hd>

where the last equality follows trorn the integration by parts formula with respect to the Wiener measure. QED.

260 For the calculations we shall need the following:

Corollary: Suppose that integral of

(Le., an adapted distribution). If

denotes the extended Ito

in the sense of [12], we have also

L

i,k$n

!:,.iI:tV)h k

Proof: For the adapted distributions, the divergence operator coincides with the Ito integral (c.t. [14]).

Q.E.D. Suppose now we have the following diffusion

dx, = b (XI )dt + cr(XI )dW I with the observation I

YI = JH(xs)ds +B I o

where b .e.H ,B ,W satisfy the hypothesis of the preceding sections. If we apply the conditional expectation operator EO[. I V n 1to the Zakai equation corresponding to the above filtering problem, we have:

EO [ltl (f) IVn 1=

I

lto

(f) + JE o[lt s (L f) o

I

+EO[Jlts(Hf)dYs

IVnl

o

We can now calculate the last term using the Corollary:

IVn lds +

261 t

EO [J1ts(Hf)dys IVnJ =

°

t

1

=

I. t.lk){Jhk(s)dYs-Y'hJJEO[1ts(Hf)lVnlhi(s)ds i,k91 ° °

If we denote EO[1t t (f) I V n J by 1tt(f), we see that

satisfies the following partial differential

equation:

Since we are taking the conditional expectation with respect to a finite number of random variables, a combination of Doob's Lemma and the kernel theorem implies that there exists a mapping Qt such that, for any f

E

S (R d), we have

where Qtn is linear with respect to preceding section, f

Qf(f ;

f.

Moreover, under the nondegeneracy hypothesis of the

oh 1, ..• , oh n ) extends as a linear, continuous mapping from

S'(Rd ) into D (y). Moreover, using the Sobolev injection theorem (finite dimensional case) we see that the map (y 1, ... , Yn) H Qf(T ; Y 1> write the above equation for

••• , Yn)

is

for any T E S'(R d ) . Hence, if we

Qf using the coordinates of R d xRn , we have

(1)

where Qtn(x ; Y 1, ..• , Yn) denotes also the kernel defined by

JQt(x; Y1,···,Yn)f{x)dx=Qt(f; Y1>···,Yn) We know by the results of the last section that Qt(x ; Y1""

,Yn) is in

s;

hence a "good" approximation for the filter would be solving the deterrninistlc equation (1) and then replacing the free variables Y l'

...

,Yn by the white noise observations

oY 1, . . . , OYn .

262

For the sake of simplicity, let us suppose that the sequence (h k ) is orthonormal in IH. Then, the Gram-Malliavin matrix becomes the identity matrix of IAn and if we define Stn(x;y) by 2

S?(x;y)=(exp- IY 1 ) . Q;'(x;y) 2 where Y denotes the vector in IAn, (y 1, ..• , Yn), the equation (1) can be rewritten as

(2)

as;,(x;y). = L;rS;'(x;y) -

a

t

L

a

-:;;-Sr(x;y)(H (x),

iSn 0Yi

.

hdt» .

If we take the partial Fourier transform of Sr(x ;y ) with respect to the variable Y , the equation (2) becomes

(3)

and this equation can be solved explicitly using the functional integration and the Feynman-Kac formula. V. On the Solutions of a Simple Anticipative S.D.E. In this section we want to illustrate with a simple example that the method we have used to calculate the solution of the approximation of the Zakai equation can be used in the reverse order, i.e., to show the existence of the solutions of the original equation. In fact, we can even work with the anticipative stochastic differential equations thanks to Theorem 4 of Section IV. We shall use the classical notations with a one-dimensional Brownian motion W: Theorem 5: 1

Let X; in D 2,0 = L

be such that JE [Xo2exp-2wt ]dt

E:

and by the

O. We will say that a family of continuous

269

processes Zt converges in Lq to Z if the Lq norm of Zt - Zt converges to 0 uniformly in t; we will say that the convergence holds in sq if the Lq norm of SUPt IZt - Zt I converges to O. Similarly the convergence in LO will mean lim sup lP[IZt - Ztl

>

OJ = 0

(1.7)

0, and the convergence in SO will mean the convergence of SUPt IZt - Ztl to 0

in probability. The constant numbers involved in the calculations will be denoted by the same letter J.t, though it can vary from line to line.

2. A simple example On (0, J', ft, lP), let Yt be ad-dimensional

Ji

Brownian motion and let Yt be

Ji

adapted

absolutely continuous approximations. In this section, we will study the behaviour of the integrals (2.1) These processes can be viewed as solutions of bilinear equations, so that it is a particular case of (1.1); moreover we will prove in next section that, when dealing with more general equations (1.5), the form of the limit (1.6) will be determined by the limit of (2.1). To begin the study, write the integration by parts formula (2.2) We deduce

Suppose that Y converges to Y in L 4 ; then the second and third terms of the right side of (2.3) converge to 0 in L 2 , so defining

(2.4) if

r

converges in L 2 to some process "I, we immediately get the convergence of X, to

(2.5) Note that

so if the convergence of It =

1t/2, the limit

r

holds, the limit must necessarily satisfy "It

+ "I; =

It; in particular, if

It is the double Stratonovich integral of y and the approximation

Y will be

said to be symmetric; in the other case, the limit contains an additional skewsymmetric term.

270

Now our aim is to explain how the convergence of I' can be checked. We will first consider the case of the approximations

. Yt

=

A -(Yt e

- Yt),

where A is a fixed stable d x d matrix. Then Y t - Yt is of order f

Al

= --

t

c

(2.7)

Yo = 0,

V

and

t

0

(2.8)

(Ys - Ys)(Ys - Ys)*ds.

By writing the equation satisfied by (Yt - Yt)(Yt - yd*, one verifies the following mixing condition: lJ > 0 and all s -S i;

for some J.L and

(2.9) One easily deduces from (2.9) that the variance of I' t is of order e , so we only have to estimate its mean value. But for any t > 0,

(2.10) which is solution of AK + K A *

+ I = 0,

so

r

converges to "It

=

-AKt. In particular, note again

that It +'1 = It. If A is symmetric, then K = A-I /2 so "It = It /2; conversely, if It = It/2, then

K = A-I /2, which implies that A is symmetric (since K is symmetric). Thus the approximation

is symmetric if and only if A is symmetric. As another example, consider the perturbed delayed polygonal interpolation of Y defined by

Y t = 0 for t -S e and Y t = Y(k-l).

for ke -S t < (k

t-kc

+ --(Yk. c

+ l)c,

- Y(k-l).)

+A

((k+1)c-t)(t-kc) c

2

(Yk< - Y(k-l).)

(2.11)

k :::: 1 and where A is again a d x d matrix. Then Y converges to Y in

sq

(Yt , Yt - Y t ) and (Y., Ys - Y s) are independent as soon as It - sl > 2c, so the mixing condition (2.9) is again satisfied. By computing the mean of f t , one can prove that f t converges

for every q; to

It =

I

A* - A

2t + - - 6 - t .

(2.12)

Thus the approximation is symmetric if and only if A is symmetric.

3. Convergence for adapted approximations In this section, we will prove a convergence theorem for the solution of (1.5), when Y t is a family of 1t adapted approximations. This result can be compared with those of [6]; note that the regularity assumptions on the coefficients are slightly weaker; some conditions are more stringent but this is because we study both convergences in L 2 and

S2.

We restrict ourselves to

a framework in which the list of assumptions is not too cumbersome; the more general case will be studied extensively elsewhere.

271

Let us first describe the processes and coefficients involved in the equations. On the space

(11,1, It, IP), consider two independent 1 t Brownian motions Wt and Yt with values in ffi.P and rn. d, a 10 measurable rn. n valued variable Xo, and for each e > 0, a It adapted process Y t with absolutely continuous paths and a 10 measurable variable Xr: Consider also locally bounded It 1

adapted processes t and at with values in IRpf ® IRF and IR'p ® IR d and put

1 t

mt =

Let b, in

I

and

g

0,

-

+

(3.10)

and a similar inequality can be obtained with IX t - xtl and IAtl replaced by suP.::;! IX. IA.I. From Gronwall's inequality, the theorem will be proved as soon as lim

II

sup IAtll1 = 0 2

x.1

and

(3.11)

will be checked. Actually, since SUPt IAtl is uniformly square integrable, we only need to prove its convergence to 0 in probability. The different positive constant numbers which will be involved in the subsequent calculations will all be denoted by J1. Let (ti, 1 SiS n) be a subdivision with step p = max Iti+l - til; for every time t , denote ott) the greatest point of the subdivision which is less or equal to t. Then

(3.12) so, from our assumptions,

(3.13) On the other hand, one has

273

f

so Ihp!At -

-1: j ]1111 < JL(1111Xs -

(3.15)

1/2.

Moreover, one can prove from (3.7) that

IIXs -

X 6 (s) 112

JL SUpl1 Y t t

Yt 112

+ JLVP

(3.16)

so, by combining (3.13), (3.15) and (3.16), we obtain lim limsup IlsUPIAt t

10t'" Gk9j(X6(B))d[r:

j

-1: j ] III

1

= 0,

(3.17)

and therefore, to prove the convergence of SUPt IAt I to 0 in probability, it is sufficient to check that for every fixed subdivision, the variable \It = supl

r

-1:i ] I

. 10

(3.18)

converges to 0 in probability; but

(3.19) so it is sufficient to use the convergence of

rt

to It for t fixed to complete the proof. D

Remark 1. Suppose that the approximation is symmetric (in the sense of §2) and that at = 0, so that the joint quadratic variation of m and Y is zero; then the limit equation (3.3) can be written in the form dx; = b(t, xddt

If f is of class

+ f(t, xddmt + g(Xt) a dYt.

(3.20)

one concludes that the family of equations

dX t = b(t, Xt)dt

+ f(t, Xd a drri, + g(XdYtdt

(3.21)

converges to

(3.22) We can say that the Stratonovich equations are stable with respect to symmetric approximations. However, in the general case (at not necessarily zero), it appears that the limit of (3.21) is

dXt = b(t, xt)dt

+

2

(FkgJ - GJik)(t, Xt)o;j dt

+ f(t, xd a dm, + g(xd

a

dYt.

(3.23)

Roughly speaking, the property "Y is a symmetric approximation of y" does not imply "(Y, y) is a symmetric approximation of (y, y)".

Remark 2. When It is deterministic, in order to prove the convergence of

rt

to It, one can

proceed as in the examples of §2; first one proves that the variance of I' t tends to 0; this holds if for every s

1= t,

the covariance of Y s (Y. - Ys)* and Y t (Y t - Yt) * tends to 0 as e

sufficient to prove that lEft

---> It.

lim {tlE[YsYs*]ds=-Yt.

10

Note that unlike f of

t,

---> O.

It is then

From (2.3) and (2.4), one is reduced to check

(3.24)

this condition does not involve Y and can be related to the condition (6.7.5)

[2]. Now we are going to prove that, by a localization of our arguments, the convergence in

probability is preserved with weaker assumptions.

274

Proposition 3.2. Assume that band

f satisfy (3.4) and

Ib(t,x)1 ::; K(1 + Ixl)'

If(t,x)l::; K, 1

(3.25)

.

that g is C;, that 0, such

uniformly in e and secondly, the processes X t , Yt ,

I; h./ ds stopped at time 2

> oj. By applying

are bounded by J1. on

theorem 3.1, one can prove the convergence in L 2 of the stopped processes X, and deduce the proposition.

0

Proposition 3.3. Assume that band

f are locally bounded, locally Lipschitz with respect to

11

are locally bounded, that 0 Irt l 2 dt and SUPt IYtl are bounded in O L and that the solution Xt of (3.3) is defined up to time 1. If X o and f t converge in probability

x, that g is C

2

,

that

IEr, the probability

t

of {r > M} is of order s, so from IE[r;

r > M] ::; IErlP[r > M]

+ (varflP[r >

M]f /

2

(4.9)

we deduce (4.1). Let us now see how the mixing conditions (4.6), (4.8) can be checked with some tools taken from the stochastic calculus of variations; this calculus was used a great deal in the past years in order to study the density of diffusion processes (Malliavin's calculus), but it is also useful for approximation problems; one can refer to [7] and [8] for details concerning the stochastic differen-

°

tiation. We will use the following framework; we suppose that the space 0 can be decomposed as 0=

X

c» where (0,1, JP)

is some probability space and

c-

is the space of continuous functions

from [0,1] into IRd endowed with the uniform convergence topology, the Borel o-algebra C and the Wiener measure W; we let 1 be the completion of canonical process of

o-

1 ® C and lP = JP ® W; the process Yt

is the

and it will also be considered as a process on 0; its filtration is denoted

it, variables Xo and X o which are 10 measurable, a it Brownian motion Wt and it adapted processes rPt, Ut; the filtration on 0 will be ft = it ® Yt; the by Yt. On

0, we will

consider a filtration

approximations Y of Y will be Yt adapted. If Z is a C measurable variable, if Z is in the space

Dq , 1 (see [7]), we will denote by DZ its derivative considered as a line vector: it is an element of Lq([O, 1] X c», IRd ) ; roughly speaking, DtZ is the derivative of Z with respect to the infinitesimal increment of y at time t. If Z is a vector-valued variable, then we will consider DZ as a matrix tt» Zk) (k,j)" We recall a result which will be often used: if Z is a C measurable variable of D2 , 1 , then Z admits the integral representation (4.10) Theorem 4.2. Assume that band

f

that rPt, Ut are Let 1/J be a positive function defined on

are bounded, satisfy (3.4), that g is

uniformly bounded and that the variables

Yt

are in DB,I.

[0,00) such that

(4.11)

278

and assume that, for 0 S «

°and its rate of convergence; but . lEft

so, if we assume that as x

->

- (t-S) = ;1 Jot' 1313* - g - ds =

I

tl e

0

f3f3*(z)dz

(4.24)

oo

00, I x f3;1*(z)dz is of order 1/.,fii, and if we put It = t

then we will have

I

IlEft ­ It =

loo (3;1*(z)dz,

­ dz ds IIo /.00 (3(3*(z) t

ale

l ::::

(4.25)

it ds

(4.26)

10'

0

voS

so we will be able to apply theorem 4.2. In the particular case (2.7), (3 is the exponential function

f3(x)

= Ae- A x

and all the conditions are satisfied.

Up to now, we have obtained a class of approximations Yt such that the solutions of the differential equations (1.5) converge for 'regular' coefficients b, f and g, and we have estimated the rate of convergence. Our next goal is to prove that, if we restrict ourselves to a particular family of coefficients, we can make the rate of convergence more accurate; the assumptions which will be involved are commutativity conditions on the vector fields Gj and Fj; when

f

=

0, these are just

the Doss­Sussmann condition and our results can be compared with those about the discretization of equations (112]). One can note that if the Gj's commute, using the relation It

+ ,t =

It, the

limit equation (3.3) does not depend any more on It so that all the approximations work like symmetric ones: the limit of equation (3.21) is (3.23); if moreover Gj and Fk commute (and this will be assumed in next proposition), the limit (3.23) becomes

(4.27) so the Stratonovich equations are stable. We are going to use the spaces Dq ,2 consisting of twice differentiable variables.

280

Proposition 4.3. Suppose that b is uniformly

tives bounded in (t,x)), that f is uniformly

(differentiable with respect to x with derivathat g is C:, that ¢;t, at are uniformly bounded

and that the vector fields Gj and Fk satisfy and

(4.28)

for all possible indices (j, k). Let Yt be a family of approximations of Yt such that we assume that for some bounded positive function l/; such that

10

00

l/;(z)dz
0 such that for every (s, t),

. II J.L vlt - sl II D.Yt 8 :S -E: exp - - -E:- , and for every K and

1]

> 0, there exists IEexpK

(5.1)

J.L such that for s

I

t

•

.

IY"ldu:S

:S t, 1](t - s)

(5.2)

J.Lexp - - - . E:

Then the solution X, of (1.5) is, for each t, in O2 , 1 and there exist J.L and v

> 0 such that for

s :S t,

(5.3) Moreover,

(5.4) Sketch of the proof. For any

rn.n

valued function h(t, z) defined on [0,1) x

with respect to z ; denote Jh its Jacobian matrix and consider the

rn.

n

0

rn.

n

rn.n ,

differentiable

valued process Zt

solution of the equation

(5.5) with the initial condition Zo = I. This process has the following interpretation: the solution of

(1.5) can be written as X; =

xfo, where

X] is the solution with the initial condition

xg =

x,

284

the map x

>->

X] is a stochastic flow of diffeomorphisms of lR n, and Zt is its Jacobian matrix

taken at x = X«. Put also for s :::: t

(5.6) which can be viewed as the derivative of X at time t with respect to its value at time s. From (5.2), one can deduce that for any

'Y)

> 0, there exists Ji, such that s)

IIZ"dl s :::: Ji,exp - . E: 'Y)(t -

(5.7)

Now, by a classical calculus of variations on (1.5), one can prove that X, is differentiable and for all sand t

D.Xt Thus, after choosing the

'Y)

=

it

(5.8)

Zu,tg(Xu)D;Vudu.

of (5.7) strictly less than the v of (5.1), for s :::: t,

The result (5.3) is now immediate. To prove (5.4), we note that from (5.1), the £s norm of

Yt

is of order 1/y/ , so one can deduce on (1.5) and (5.5) that for every TJ, there exists f.J. such that for u :::: t,

Ilg(Xtl- g(Xu )114 < f.J.(vt=U + t {

IIZu,t - 1114:::: Ji,(vt=U +

t-u

ftu),

y/ ) exp

(5.10)

TJ(t-u) 0:

The estimate (5.4) is then easily proved by applying (5.10) on (5.8).

•

0

We now state and prove the analogue of theorem 4.2; the process

rt

of (2.4) has to be

corrected into

(5.11) Since D.Y. = 0 in the non-anticipating case, this definition is compatible with the previous one. With this modification and some additional assumptions, we prove that, as in §4, the processes X, converge with a rate y/ to the limit

Xt

solution of (5.12)

Theorem 5.2. Assume that the coefficients band f are uniformly

ct

with respect to x, that 9

is that the variables Yt and DtYt are in Ds ,1 and that for some f.J. and v> 0, the following estimates hold for every (s,t):

IID.Y.t I 8 :::: -f.J. e x pvlt- -- -sl, vlt - sl { II D.DtYt I s:::: f.J. yO: 1

0:

0:

0:

IID.(Yt -

vlt - s] ytl I 8:::: f.J.exp---, E:

(5.13)

285

Suppose also (5.2) and (4.13) for the process r t defined in (5.11) and some absolutely continuous function It. If IIXoof order

ye.

xol12

is of order

ye and Xt is solution of (5.12), then SUPt IIXt -

Xtll2 is also

Sketch of the proof. The main difference with respect to previous proofs appears at equation (3.7); to develop g(Xt)Yt, one has to use the integration by parts formula of Skorohod's calculus rather than that of Ito's calculus. One gets g(Xt)Yt = { d 9(X8)Y.

+

dg(X 8)y.

+

=

it

it it

g(X.)dY8

+{

g(X.)dy.

+{

+{ where

f

J g;

(5.14)

-

g(X8)dy. is a Skorohod integral; let us explain briefly how this formula is derived: if

= 0, it is a particular case of Ito's formula for Skorohod calculus; for the general case, consider a

Wr of absolutely continuous approximations of W; this sequence can be constructed for instance with a scheme of the type (2.7); for each fixed, one can deduce a sequence of processes Xf which converges, as well as their derivatives to X; and using the independence of Wr

sequence

E

and Y, one can write, for each n, a formula of the type (5.14) for g(Xf)Yt; then one lets n to conclude. Then, with the new definition of

r t, one

-+ 00

can write the equation (3.8) modified as in

theorem 4.1, provided that the integral with respect to Y is understood as a Skorohod integral and that the last term of (5.14) is added; note however that from lemma 5.1, this term is of order

ye so will not be

awkward in our estimates. When one computes the £2 norm, one has to use

(5.15) One can check from (5.3) that the double integral is of order similar to §4.

g2.

The remaining of the proof is

0

Remark. As in the adapted case, one has It

+ ,;

= It and the approximation is said to be

symmetric if It = It/2. Example. A first example is the polygonal interpolation (1.3), which is a symmetric approximation. As a second example, let ;3(z) be a JRd ® JRd valued function defined on JR, which is dominated by some /lexp

-vlzl

ljJ(z)dz = I; consider the approximate noise

and such that

defined by

fa

Yt =

1

!=!!:

(5.16)

(3(z)dz dyu.

If the support of ;3 is in [0, +00), then this enters the example of §4. Put

/3(x) =

1 x>0

1+00 x

;3(z)dz -

JX

-00

(3(z)dz.

(5.17)

286

r

Then .

Yt =

1

Jo f3

(t-U) -£-

t - U) Jt'o f3 (-cdyu

Yt - Y t =

dyu,

(5.18)

and I' t converges to

f3iJ*(z)dz -

'tt =

J(O)).

(5.19)

One can verify all the assumptions of theorem 5.2; for (5.2), one decomposes with

u, = { U; =

L L

e i

into Ut

r(4i+3).,,1< f3C - U)dyu, e

e i 1 -

Yt

J(4t-11."l< 1{(4i+21."lt is related to the limits of the iterated integrals of Y up to order 3. This procedure can be repeated an arbitrary number of times: the more singular the approximation is, the greater the order of iterated integrals of Y to be studied is. This remark can also be stated in another form: in [13], it is proved that if the Lie algebra associated to the differential system is nilpotent, then the solution can be expressed as a function of a finite number of iterated integrals; here it appears that actually, using smooth enough approximations, one has only to consider integrals up to some order. References [1] H. Doss, Liens entre equations differenticlles stochastiques et ordinaires, Ann. Inst. Henri Poincare, Section B, 13 (1977), 99-125. [2] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, NorthHolland, 1981. [3] H. Kunita, Convergence of stochastic flows connected with stochastic ordinary differential equations, Stochastics 17 (1986),215-251. [4] V. Mackevicius, On the support of the solution of the stochastic differential equation (in Russian), Lietuvos Mat. Rinkinys 26 (1986),91-98. [5] V. Mackevicius, SP stabilite des solutions d'equations differentielles stochastiques syrnetriques avec semi-martingales directrices discontinues, C. R. Acad. Sc. Paris, Serie I, 19 (1986), 689-692. [6) V. Mackevicius, SP stability of solutions of symmetric stochastic differential equations with discontinuous driving semimartingales, to appear. [7] D. Nualart, Non causal stochastic integrals and calculus, this volume. [8] D. Ocone, A guide to the stochastic calculus of variations, this volume. [9] M. Pontier and J. Szpirglas, Convergence des approximations de Mc Shane d'une diffusion sur une variete compacte, Serninaire de ProbabiIites, to appear. [10] A.V. Skorohod, On a generalization of a stochastic integral, Theory Proba. Applications 20 (1975),219-233.

[111 H.J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Annals Proba. 6 (1978), 19-41. [12] D. Talay, Analyse nurnerique des equations differentielles stochastiques, These de 3eme cycle, Universite de Provence, 1982. [13] Y. Yamato, Stochastic differential equations and nilpotent Lie algebras, Z. Wahrschein. verw. G. 47 (1979), 213-229.

BROWNIAN MOTION AND HARMONIC FffiMS K.D. Elworthy Mathematics Institute, University of Warwick, Coventry CV4 7AL O.

INTRODUCT ION In his seminal paper [7] at the 19E2 International Congress of Mathematicians

in Stockholm,

suggested how parallel translation along the sample paths of

Brownian motion on a Rienannian manifold might be defined and used in the analysis of the behaviour of the solution of the heat equation for differential forms.

In

the 1970's this became a standard tool, used in particular to obtain information about the non-existence of non-trivial harmonic i-forms

[1], [19], [23], [3], and

more recently in simpl ified proofs of the Gauss-Bonnet-Chern theorem with modifications for other index theorems, and often considered with the Malliavin Calculus [4], [16], [18]; see also [13].

In this

expository article we will give a

description of this probabilistic approach to the heat equation for forms, with an extension of [10] to give an 'elementary formula' for the fundamental solution to this heat equation.

This is used to present a proof of the Gauss-Bonnet-Chern

theorem, closely modelled on Patodi 's proof as given by Simon in [8], following Getzl er' s work, see [14], wi th frequent references to [8]. the aSYmptotics of the heat equation as t

O.

Thi s proof depends on

In §6 we use the long time behaviour

to give an extension of Bochner's theorem on the vanishing of harmonic i-forms which can also be used to estimate the spectral radius of the Laplacian on functions in certain cases; this is one of a group of results in [12], and it was these aspects rather than the short time asymptotics which were the subject of the talk given at the,most enjoyabl e,workshop at Sil ivri. 1.

THE LAPLACIAN ON FORMS A.

It would be difficult to improve on the exposition in [8], so it will be

followed very closely.

Suppose that Mis a connected Riemannian manifold of dimension

n, so that for each x in M there is an inner product < , >x on the tangent space TxM to M at x.

The distance d(x,y) between points x and y of M is given by

d(x,y)

= inf

{t{o) :

where the length t(o) of a path b

t(a) for jUlx

= fa

= Ix

0

l is a piecewise C path from a to b}

0: [a,b]

-+

M is

1;(t)lo(t) dt

when u

E

TxM.

We shall assume M to be complete in this metric.

289

B.

A I-f orm ¢ is an assf qrment of a linear map 0

(14 )

where 4>t is a p-form for t 0, with 4>0 given. For a probabilistic solution take any Xo in M and let {x t : t z O} be Brownian motion on M starting from X i.e. a o 0 sample cont i ruous Markov process with generator 6 , using 6 0 for the Laplacian on functions. Let 1\Pr M be the p-th exterior power of T Mfor each x so 4> gives x x * a map 4>x : 1\PTxM -+ R for each x if 4> is a p-f orm and the dual acts on 1\PrxM as a 1 inear map. Given Vo = v01" ... vop E: 1\ PrxoM define vt E: 1\p\tM for t > 0 by Dv t 1 P * = - - (101 ) v

at

xt

2

(15)

t

with initial point vo' Such covariant equations along the paths of a diffusion for stochastic processes are described in [9J, and given in local coordinates in [15J. The easiest way to describe them is in tenns of paral.l-el: translation along the paths of x t: if II : 1\ PT X M-+ 1\ Prx Mis parallel translation (defined for almost all o

sample paths) so

t

where {V t k : t O} is parallel translation of the tangent vector v ok for k = 1 to p, d ef i ned in local coordinates by the Stratonovich equation i

_

dV t• k - where

r1R. (x)

(I

E:

.r} j t

J.",

j

x t v odxt

i=lton

is the Christoffel sjmbo l at x in these coordinates, then

1\ PT

xoM

and our equation can be written as the linear ordinary differential

equation:

wt =

(We

t

t

(wt )

(16 )

where {wt : t O} is the fhxoM-ValUed process The basic result is that the solution 4>t of the heat equation is given by (17)

293

provided q>o is bounded and both wP and the Ricci curvature are bounded below on M, (the latter condition ensures Brownian motion is defined on M for all time. These acnditrione on M win be asswned fran now on. For proofs see [15], [9] and the referencesof !W: the basic point is that would solve

given by ¢t,x (vo) 0

= I: q>o,x 0

and our covariant equation for vt generalizes the Feynman-Kac f orrnrla to include p. th e zero order term W 3.

THE ELEMENTARY FORMULA FOR THE HEAT KERNEL WHEN M HAS A POLE

A.

The heat equation has a fundamental solution kP on p-forms:

so that the solution ¢t x

, 0

is given by

= JM

'

y)dy 2

where dy refers to the Riemannian volume, and now q>o is assumed to be in L • We will find a probabilistic expression for when Yo is a pole for M t ,e, when each point of M can be joined to y by a unique geodesic, so that there o are global normal coordinates for M. This is true for all Yo if M is simply connected with non-positive sectional curvatures, by the Cartan-Hadamard theorem. Assuming this take w APT Mand choose a p-form ¢ whose support is compact and o Yo with qI (-) = -

(!lPV)* its 8upertrace is given by

(-I)P trace BP.

p=1

There is the Berez in-Patod-i. [OL'muZa: for such B str(B)

= (_I)n

E{l, ... ,n}' {l, ... ,n}'

For the proof see [8J.

c.

Using the Berezin-Patodi formula it is a simple matter to compute the short

time limit of str k

t

(x,x) acting on (liTxM)*: From

It

H t = I - 1. H 2 0 1:,S 1:,

0

R(s)d s

it follows by substitution of the corresponding expression for H1: ,r into the integral, and then iterating, that

H1: ,1

=I

+ ZI +

where

R, =

2,3, ...

Now suppose that M is even dimensional, n then yields Str with

H t,1

= Str

L; + 0(/+1) N

2t say.

The Berezin-Patodi furnu la

298

L 5gnhr )5gn(0) RTI(1)0(1)7r(2)0(2)(s£) ..•

l\r (n -1 )a (n -1 )n (n )0 (n ) (sl ) (still a random variable) where the sum is over all permutations TI and

0

of {l, ... ,n}.

Here we have used the

anti commutation relations

and

Now we restrict to x

= y,

although the more general case follows similarly.

From the elementary formu la (21) lim Str k (x ,x ) 1 1+0

As 1

+

=

lim (2TI1) 1tO

-z

I{exp[1/2

Jl fJ'(Zs}lI6 k

1

0

-k

r

'(zs}dsJ Str ¢

1,

I}·

0 the semi-classical bridge converges to the process which stays constant at

x, and parallel translation along it, for each PI see [13J. Since

Iii,

converges to the identity map of i\PTxM

and all quantities are bounded lim 1+0

Str k (x,x) 1

= E(x)

(25)

where

E(x)

= (21T) -t (-

1 t 1 2) £!

l: sgn(TI) sgn (O}R TI {I ) 0(1 )1T(2) 0(2} (O)

(26) RTI (n- O o(n- l )TI (n ) a (n }( O)

In particular the right hand side is independent of the choice of frame e 1 , ... ,e n

at x .

299

5.

THE GAUSS-BONNET -CHERN THEOREM The G-B-C tneor en states that for M an even dimensional compact manifold,

A.

dim M = 2R"

the Euler characteristic x(M) is given by X(M)

=

1 M

E(x)dx

for E given by (28).

= 1.

theoran when

Ll

where

= exp

0,

tii, t

and maps

,;,P.

For our point of view

is the Laplacian on p-forms.

Let Pt of

It is easy to see this reduces to the classical Gauss-Bonnet

R let

For A

(For example see the Appendix to [9

be the heat semigroup on forms.

J).

It is trace class

np(A) be the multiplicity of A as an eigenvalue

Using the fact that d + d* maps eigenspaces to eigenspaces and even forms to

odd forms it is easy to see that the non-zero eigenvalues of ii cancel out in pairs in the supertrace of P t: Z(-I)P n (A)

P

P

=0

(A ., 0)

so that Str P t

=Z

(-I)P Z n (A)epAP

Z A

tA

e- t A Z (-I)P n (A) P

P

Z (-I)P np(O) p

X(M) independently of t ,

This remarkable fact, due to McKean and Singer is the basis of

Patodi's approach to the G-B-C theorem: to cOl1lpute x(M) it suffices to calculate lim Str Pt' t+O

Using standard analytical results to show that

lim t+O

Str Pt

=J

lim Str Pt(x,x)dx M t+O

the theorem follows fr-on equation (25). For the corresponding scheme of proof (without the use of an 'elementary formula') which will give the Atiyah-Singer Index theorem see [14J.

300

6.

A GENERALIZED BOCHNER THEOREM AND THE SPECHJM OF

l'l O.

Now suppose that M is complete with Ricci curvature bounded below. but not necessarily compact. The operator L 0 acting on L2 functions is non-positive. so that if Ao is the suprenum of its spectrum then A $ O. When M has finite volume then Ao = 0 since the constants lie in L2 but fo/simply connected manifolds of constant sectional curvatures-1 it is known that A = (n-l)2. Let o C = inf{inf Ricx(u.u) : lul x = I}. Then there is Cheng's [6 J estimate when C < 0: x A.

-!

Ao

1

4" (n-l)C.

We will prove the following result fram Elworthy and Rosenberg [12J: Theorem. Suppose there is a non zero one-form 11 which 1 ies in LZ and satisfies Ll¢ = O. Then

Remarks (t ) In fact it is shown in [lzJ that if C A and Ric > A at same point x then x o o there are no non-zero harmonic one-forms. This extends the corresponding result for C = 0 which is the classical form of Bochner's theorem when M is compact and is inunediate from the Weitzenbock fonnula:

(ii) There are analogous results in [12J for p-forms. proved the same way. and also some results for the first cohomology group with compact support: for example if M covers a compact manifold then 1 0 implies Ao C. Whereas it is not difficult to give an analytical proof of the theorem as stated here. analytical proofs of same of these other results look harder to obtain. B.

The proof depends on a simple lemma:

Lemma. of M

Let xt : t 1im

t--

t

0

log

be Brownian motion on M.

P {xt

E

U} s

Then for any bounded Borel set U

Ao'

Proof. There is a C2 function h : M-.. R(> 0) with Llh = Aoh. [7J: it can be obtained hk where hk is the positive Dirichlet eigenfunction for the ball. about as the some fixed point. of radius k, From formula

301

- (x ) Eh t

= eJ,"At() 0 h X o

so that if m = inf {h(x) : x

TIm t--

t = log

F (x ,

Tiiil

C.

1

Proof of Theorem. 2

U}

t log

2" 1. 0 •

s

U}

[ h(x t)

II

First it is necessary to observe that on the intersection of

their domains the L

semigroup {exp : t O} defined by the functional 2 calculus for operators on L agrees with the semigroup P defined probabilistically t as in 52. For example see [12J. From this it follows that if f : M 0) is smooth with compact support and such that w = fep converges in LL as t

00

is not identically zero then Ptw

to the harmon ic component Hw of w.

Moreover Hw

. no t

1s

zero since

From Fatou's lemma it follows that there exists that

I[

x t (vt)1

W

[{ IWx I t

where Dvt

-at = - -21

by equation

and

(15),

1

*

(Wx ) vt t

so that by

(13)

IV tl

}

X

o

EM with sane

V

o

T M such X

o

302

Tll..ts

°

s lim

t+ro

:£ -

t

10g\ (Ptw)

1

Xo

-1

2" C + 1im t log E

by the prev i ous 1 erma,

II

AC KNOWLEDGEMENTS I benefitted greatly from a series of talks given by B. Simon at Swansea in the summer of 1984 and a series by S. Watanabe at Warwick in August 1985.

This research

was helped by SERC grants GR/C46659 and GR/D23404. REFERENCES [IJ

Airault, H. (1976) harmon i cu es ,

LL]

Subordination de processus dans le fibre tangent et formes

C.R. Acad. Sc. Paris, Si!r. A, 282 (14 ju i n 1976), 1311-1314.

Azencott, R. et a1. (1981). San i na ire de p robabil i

et diffusions en temps petit. Uni vers i ti! de Par i s VI 1.

Asteri qu e 84 - 85.

Soci He

matnemat i ur e de france. L3]

Berthier, A.M. and Gaveau, B. (1978).

Cr i t er e de convergence des fonctionnelles

de Kac et app lications en mecanique quanti que et en geomHrie.

J. Funct. Ana1.

L9 41b-424. L4]

Bismut, J.-M. (1984).

The Atiyah-Singer theorems: a probabilistic approach.

C5]

Chavel, 1. (1984)

L6]

Cheng, S.-Y. (l975)

[7]

Cheng, S.-Y. & Yau, S.T. (1975).

I & II J. Funct. Anal. §I, 56-99 & 329-348.

app 1 i cat i on s.

Eigenvalues in Riemannian Geometry.

Math. Z.,

ill,

289 -297. Differential Equations on Riemannian Manifolds

and their Geometric Applications. [li]

Academic Press.

Eigenvalue comparison theorems and its geometric

Camm. Pure Appl. Maths., XXVIII,

Cycon, H., Froese, R., Kirsch, W. & Simon, B. (1987).

Schrbd inqer Operators

with applications to quantum mechanics and global geometry. in Physics. L9]

Stochastic differential equations on manifolds.

ZQ.

Elworthy, K.D. & Truman, A. (1982). mechanics: an elementary formula.

London

Cambridge: Cambridge University Press.

The diffusion equation and classical In

ed. S. Albeverio et al. pp. 136-146. Verl ag.

Texts and Monographs

Springer-Verlag.

Elworthy, K.D. (1982).

Mathematical Society Lecture Notes LI0]

333-354.

I

Stochastic Processes in Quantum Physics I

Lecture Notes in

Springer-

303

[l1J

[12J [13J

[14J [15J

[16J

[17J

[18J [19J [20J [21J

[22J [23J

[Z4J

Elworthy, K.D., Ndumu, M. & Truman, A. (1986). An elementary inequality for the heat kernel on a Riemannian manifold and the classical limit of the quantum partition function. 'From local times to global geometry, control and physics' ed. K.D. Elworthy, pp. 84-99. Pitman Research Notes in Maths Series 150, Longman Scientific and Technical. Elworthy, K.D. & Rosenberg, S. (1986). Generalized Bochner theorems and the spectrum of complete manifolds. Preprint: Boston University, M.A., U.S.A. Elworthy, K.D. (1987). Geometric aspects of diffusions on manifolds . .!!:!.- Ecole d'Ete de Probabilites de Saint-Flour XVII - 1987, ed , P.L. Hennequin. To appear in Lecture Notes in Maths, Springer-Verlag. Getzler, E. (1986). A short proof of the local Atiyah-Singer index theorem. Topology, 25, no. 1., 111-117. Ikeda, N. & Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. Tokyo: Kodansha. Amsterdam, New York, Oxford: NorthHo11 and. Ikeda, N. & Watanabe, S. (1986). Malliavin calculus of Wiener functionals am its applications. Fran local times to global geometry, control, and physics, ed • K.D. Elwort hy, pp. 132-178. Pitman Research Notes in Maths. Series 150. and J. Wiley. Ito, K. (1963). The Brownian motion and tensor fields on a Riemannian manifold. Proc. Internat. Congr. Math. (Stockholm, 1962), pp. 536-539. Djursholm : Inst. Mittag-Leffler. Leandre, R. (1986) Sur le theoreme d 'Atiyah-Singer. Preprint: Dept. de Mathematiques, Faculte des SCiences 25030 Besancon, France. Malliavin, P. (1974). Formule de la moyenne pour les formes harmoniques. J. Funct. Anal., .!L, 274-29l. Molchanov, S.A., (1975). Diffusion processes and Riemannian geometry. Usp. Math. Nauk, 30 3-59. Engl ish translation: Russian Math. Surveys, 30, 1-63. Ndumu, M. (1986). An elementary formula for the Dirichlet heat kernel on Riemannian manifolds . .!D.. 'Fran local times to global geometry, control and physics' ed. K.D. Elworthy. Simon, B. (1982). Schrlldinger Semigroups. Bull. Amer. Math. Soc., L, no. 3, 447-526. Vauthier, J. (Ell:!). Theoremes d'annulation et de finitude d'espaces de I-formes harmoniques sur une variet de Riemann ouverte. Bull. Sc. Math., IL:J-l77. Witten, E. Supersymmetry and Morse theory, J. Diff. Gean. 17 (1982),661-692.

304

Additional references (a) Azencott, R. (1986). Une Approche Probabiliste du d'Atiyah-Singer 7-18. d'Apres J.M. Bismut, Seminaire Bourbaki 1984-85, (b) Pei Hsu (1987). Brownian Motion and the Atiyah-Singer index theorems. Preprint: Courant Institute (author's oresent address: University of Illinois at Chicago).

AN EXTENSION OF VENTSEL-FREIDLIN ESTIHATES

P. BALDI - Dipartimento di Hatematica Citta Universitaria Viale A. Doria, 6

95125

CATANIA (Italia)

H. CHALEYAT-HAUREL - Laboratoire de Probabilites. Universite Paris IV 4 Place Jussieu

75252

Paris Cede x 05

(France)

This paper was originaly motivated by the study of the modulus of continuity of diffusion processes. In

[3J

we showed that it can be

reduced to prove a large deviations type estimate a little more general than the classical ones of A. Ventsel and H. Freidlin Hore precisely, if

y

E

is a diffusion process associated to the

following stochastic differential equation:

we need, under suitable hypotheses, an estimate for the quantity

Log P [ IlyE- g

(1)

where

g

II> ph(

II EB--f II

E)

h(

E)J

is the solution of the controlled system

b(g )+o(g )f t t t band

o

being the limits of

b

E

and

o

s

and

h

a function submit-

ted to growth conditions. In this article, we begin by estimating the expression (1) for tubes depending on

E

then

we derive a large deviations theorem for

306

Log P[yEE A]. This extends the well known results of

P.

and

Priouret

ding on

[6J

R. Azencott [1]

to the case of diffusion coefficients also depen-

E. Large deviations problems of this type have already arisen

in questions connected with the iterated logarithm law (P.Baldi

[2J).

In the third paragraph we give an application of (1) to the study of the modulus of continuity of diffusion processes which was proved in

[3J

by a different method.

1. -

THE r1AIN ESTIMATE. Let us consider on

b

E

mm

for

c>O

a family of vector fields

: m m -->-m m and a family of matrix

fields

mx k

o . Let E

be a function such that lim ,0

T>O

cm=

us paths on

[O,T]

If

C([O,TJ,mm)

f

m

H

starting at =

H([O,T] ,mm)

m m, endowed with the topology

will denote the closed hyperplane of all

x

paths

will denote the space of allco'1tinuo-

taking values in

of uniform convergence. em

o

h (c)

x. will be the subspace of

of all absolutely

continuous paths such that

f: f I

(s)

shall also denote for

II f

II

t

=

B(f,p)

I2 ds
O

sup s s t;

= {h, Ilh-fll

o}

If I

=

1:

I

f

(s) I 2 d s

307

em

Obviously

Hm

is

II II

is a Banach space w.r. to the norm

an Hilbert space w. r. to the norm

1

1

whereas

.

We shall make the following assumptions (A. 1)

There exist On field

JR m

a vector field.

1

lim h (E) E:-"O

(1. 1b)

lim h(E:) E-"O

1

uniformly for

(A.3)

b

y

mx k

matrix

E:

(y) -b (yl I

10 E (y) -(] (y)

0

I

0

are locally Lipschitz continuous for every

E

E>O

the equation

has a solution on

a

Ib

in compact sets.

For every

Let B be

and a

locally Lipschitz continuous and such that

a

( lola)

(A.2)

b

[O,T]. We shall note

g

S

x

(f).

k-dimensional Brownian motion on some probability spa-

ce

Y

E

be the solution of the

SDE

(1.2) = x

Theorem 1.1. - Under the assumptions R>O, p>O, a>O, c>O

there exist

(A.l),

E:O>O, a>O

(A.2),

(A.3), for every

such that, if

E p h t rl

for every function for every

x

, II E

II

ah (

such that

E)

and

in a compact set. Horeover if

and the convergence in (1.1) is uniform in in

B-f

band

(}

}

R

-

and

S (f)

g =

x

are bounded

y, then (1.3) is uniform

x.

Before the proof of Theorem

. 1 we shall make a few remarks and

state some auxiliary Lemmas.

Remarks.

1.

[lJ

Theorem 1.1 is an extension of Theorem III 2.4 of Azencott and of Theorem 4 of Priouret h (E)

::

and that

(}

E

[6J

where it was supposed that

= o. Section 2 and 3 will show that the

extension provided here have real applications; however. our proof is just an improvement of Priouret's paper.

2.

In assumption

(A.2)

we demand that the coefficients

b

E

,

0

E

are locally Lipschitz continuous, but we don't need any uniformity in

E.

A closer look to the proof would show that actually the

only assumption needed on

o , b, E

E

besides (1.1), is that equa-

tion (1.2) has strong solutions.

3.

Since (1.3) depends only on the joint law of

(yE ,B)

suppose in the theorem that the Brownian motion Proposition 1.2. -

Let

c(s,x), c

E

(s,x)

B

one could depends on

be vector fields such that

E.

309 (1.4)

[c

(1.5)

Ic(s,y)-c(s,z) I

E

1 lim E-+O h t s l

(1.6)

Let

a

E

,

0

q,E L

l/!(s) ly-zl

l/!

a

be matrix fields such that

M) and such that

2

([O,TJ)

1 L ([0, TJ )

E

o

sup Ic (s,y)-c(s,y) Ids c y

ous, bounded (with bound in

ots )

(s,x) [+lc(s,x) I

is Lipschitz continu

(1.1-b)

holds uniforDly

y. Let

x

E

, y

be respectively the solutions of

c (s, y s) ds .

Then, for every tha t,

for every

R>O

x

E JR

,

p>O

a >0 , a > 0 , there ex i st.s 2 1

,

m , q" l/!

sa.tisfyi..'1g

E

Ill/! II

sO>O such

1

a 2'

E< E0

L

we have p[ IlxE-y[1 »ph t »)

,

IIEBII.$ah(El]

exp(-

R 2)

E

Proof. - For every

n

IN

let us set T

n

kT n r

:>>

and define the discretization x

s,n t

if

Then the two following results hold.

.$tO , 6>0 , a>O

there exist

and

n

such that if

E

E 6} )

) ds I>

By Schwarz inequality

(s)1

so that for

n>n

2

ds )

large enough, independently from

1/2

...

1h

'-, the events

t

t

sup k

k+1

ift

, k=O, ... ,n-1} k

are empty. Moreover from the exponential inequality of martingales

(t

p{

sup

lEI 1

J-,

Thus for every

2

E

n>n , E>O

E

E

log p{ llx -x '

n

II> B)

s

r;

2

log(2mn)-

and the last quantity, for a larger value of

n

eventually, is less

311

than - R

Lemma '.4. ry

£O ,

(Assumptions and notations of Proposition' .2). For eve-

p>O

there exist

such that if

Proof.

II £ f . 0 ( X E) dB II o E S S

> ph (

II fB II

,

ah ( E) }c A u B u C

where

A

II E ,('

B

(0

) 0 (

II E I

C

(x E) - 0 (x E, n) ) dB ESE S s

,

)0

0

E

(x E, n)

s

dB

s

II >

II

> 1:' h ( e l , 2

II x E-

1:' h ( e), 2

In order to give an upper bound for BCB,UB B = ,

B = 2

{E

P(B)

E, n

X

E,n

J('0 (0 E (xE)-o(xE))dB II> S S S

'

E n

C

II o (o I x s )-o(x s ' f

r

)0

(0

(x E , n) -

s

Since for every

0 ,E

n>O

))dB

s

we write

Q h (c) } 6

pEE n h(E),11 x -x '

II> -6

(x E , n) ) dB S

S

II

>

there exists

Q

6

h

II

"S}

II s. II cB II

where

2UB 3

{ell

X

II x E-

II-$S}

(c ) } such that for

«h ( c) }

312

sup 10 (y) - o(y) I nh ( cl . the exponential inequality gives for small n y E

P (E

2 p

exp [ -

1)

72 T n p

P(B ) czm exp [3

Moreover since

h ( E) 2 2 E

2

2

2

]

< exp [- R E

2

72 Tn

2

exp

2 E

[-

R

h ( El 2 2 E

2

]

1

is Lipschitz continuous (with constant

0

again by the

inequality, for small

2 p

72

S

2 2 2 h

T K

K)

2 ] $ exp [_ R h

]

S E E

Thus

P(B)

(1 .7)

3 exp [- R

h (E) 2

2

]

E

for

En

exp (E

and finally on the set (t

n

E

lEI o(x ' )0

)dB

which gives

s

I =IE

c =!/!

{ II EB II

n.

2

ch ( E) }

n-l

E [ o(x N=O tk

if

a
O

Ic(s,y) - c(s,z) I

such that

EO

then, denoting

A

dP

j

-c dB s

315

-

E [exp (-

2)

E

2

exp(

Thus, for every

f

with

f

Jo

(1.11) x

(T

1 2

If I

If

dB

S

4 -

s

1 2

exp(

)

fao If. Ids) ]

1 2

2

2

x

S

If I

)

a h( )

2

P(A ) {. exp[- h( ;

})J

(R'-

2

We conclude by a localisation argument in the following way. Let us remark first that the set of paths

with

f

varying in

{If I

the ball

and

x

which solve

in

a

is relatively compact in of radius

Kc JR m

compact set

m C . In particular J the set

relatively compact in for some

a}

1

g

K = {y E JR

m

1

]Rm. Let

H>O

is

, gt = y

be such that

and centered at the origin contains

H

and let us define

'\,

b

and in

it

if

lv l

if

lyl

H

(y)

similar way

'\,

b

'\,

a

'\,

a

>

H

The new coefficients

are obviously bounded and Lipschitz continuous and

'\,

b

'\,

,0

'\,

,

'\,

b, a

316

lim

1 ----10

'C

e-+ 0 h (e)

uniformly in

y. Moreover, if

=

'CE

the joint laws of

from

B , which means that if

(y ,B)

H

{Il/-

(y)- o(y) I

is the solution of the

o

SDE

x

then

p

'C

E

gil> ph(E)

and If11

E

coincide until the exit

(y, B) a , XEK

IIEB-fll

IIEB-fll

which concludes the proof of Theorem 1.1.

2. -

A LARGE DEVIATIONS

In this section we state a large deviations result for the family of diffusion processes satisfying (1.3). It

will be derived from the

main theorem of section 1 exactly as it is done in Azencott [1J, chapter

3.

We shall reproduce here the main points of the proof only for

reasons of completeness. We consider a family of diffusion processes (1.2) and assume that assumptions (A1),

(A2),

(A3)

E

(y )E>O

satisfying

are satisfied

317

with

h:= 1 .

For

x E lR

m

and

gEe

m x

let us define

A (g)

(2.1)

f E

k

H

S (f) x

g}

It can be proved (Azencott [1J, proposition I I I 2.10) that if A(g)

is finite, then the above lower bound is attained. If

Ac em

is a Borel subset let us set

A(A)

Theorem 2.1. -

Suppose that

Then for every

Ace m

lim E

(2.2)

2

inf A(g) gEA

(A1), (A2), (A3)

(

are satisfied with

log p{y E A}

-

A(A)

E log p{y E A}

-

A(A)

h=1.

(-TO

(2.3)

lim E-TO

(

2

Moreover these estimates are uniform for

x

in compact sets.

Proof. Let

gEA

A(g)

such that

A(A)

+ (

and

f E Hk

such that

S (f) =g x

and Thus if at

g

is such that the tube of radius

0>0

is contained in

p{y ( E A}

IIy

E

-gil

Now for every

A

and for every

IIEB-fll

a>O

0

and centered

a>O

IlyE -gil

>0, II(B-fll

a}

(Schilder [6J; see also Azencott [1J, Vent-

318

sel-Freidlin [4])

II EB-f II

Lf.ra E2 log P {

a}

21 Ifl 21

-

A(A)

-

- E

whereas, by Theorem 1. 1

lim E

so that, if

R

2

log p{

II yE_

is large enough

lim 2 E E->O E log p{ YEA}

which,

E

If

A(A) - E

m

respectively)

{gEC

a

Then

K nA = I/; a

B(g,p)nA = (/)

The sets

Let

there exist

R>O

B(f,a) g

f

1,

... ,f

m x

I :\(g)

and for every

Theorem 1 . 1 for every

g

r

gEK

a}

there exist

a

f = f E C be such that g a there exists

, gE K

a

B (f. ,a,)

such that ,

1

a=a

g

S

x

P= Pg

(f)

such

g. From

such that for small E

form on open covering of C so that a, 1

sue covering. Let us set F =U B{f, ,a,)

(2.5)

A(A)- E)= -

a < A(A), let us consider the compact sets (in C x

K

that

min (-R, -

being arbitrary gives (2.2).

UppCfl bOl:l.l1d:

and

> p

gil

1

1

1

i=1, ... , r

is a finite

g.= S (f,). Then 1 x 1

319

and since (Schilder

[6J) for small

P {EB E F

C}

E

a 2

exp

E

r

P {y EE A ,

L

EB E F}

i=l from (2.5),

(2.4), for small E

P {y E A}

p{ Ily-g.11 >p" ].].

E

r exp (

R

2 ) + exp(-

if

a E

E

which

IIEB-f,11 ].].

2

R>a gives (2.3).

3. - APPLICATION TO THE HODULUS OF CONTINUITY OF DIFFUSIONS

Let

y

be the m-dimensional diffusion

associated to the stocha-

stic differential equation

=

x

whose generator is given by 1

L

with

2

a (x)

o(x)

L t

a i j (x)

8x, ax. ]. J

+

L

b , (x ) ].

a ax,

.i,

o(x). Throughout this section we make the following

assumption

(H)

band

0

are locally Lipschitz countinuous and a is strictly el-

liptic. Thus w.e may consider the intrinsic metric of the process, that is

320

the riemannian metric defined by

a

-1

we denote by

d

the associated

riemannian distance. For every closed set from

F

and set for

F

in

F

[3J

m

denote the exit time

let

E>O

W (E)

In

JR

sup

d(ys' Y t)

It-sl

we proved the following theorem, which extended a clas-

sical result of

P.Levy for the modulus of continuity of Brownian mo-

tion.

Theorem

3.1. -

(3.1 )

lim t-+O

log

If for all

t

x

p

x

E

the estimate

x log P { d(x,y ) >n t

1

holds uniformly for

for all

n>O

x{

in a compact set

1 2t log t

K

of

n

JR

m

2

then

K

lim E-+O

-----.Y'!..- (E) 2E log

1 } E

K

Thus in order to prove Levy's law for diffusion processes i t is sufficient to prove (3.1) . This can be se e n as a large deviations estimate but for moving (with

t) sets. In

[3]

we showed that i t is easy

to get (3.1) when there are scaling properties available. This is the case of the principal invariant diffusions of nilpotent Lie groups. We also proved (3.1) in the elliptic case using S.Molchonov's work

321

on the equivalent of transition densities

[5J

Here we deduce (3.1), under ellipticity hypotheses, from theorem 1.1. For this purpose we prove a slightly more general result. Let

be a function from

lim t->-O

3.2. -

For all

(3.2)

It (t)

o

+

to

and

(Assumption

lR

lim t+O

+

such that

(t)

o

(n)).

n>O

2t

lira t->-O

uniformly for

Proof. -

lR

x

- n

in a compact set.

We can suppose that

nuous and that

2

a

0

and

b

are bounded Lipschitz conti-

is uniformly elliptic; the general case will follow

from localization arguments. It is classical to transpose this problem in small time into a problem of small perturbations. We set y

(t) = y ; then s st

is a solution of the stochastic

differential equation

x

the hypotheses of theorem 1.1 are satisfied with b =tb s

and

T=l

, s=1t" ,

0

s

:::0

,

So, from theorem 1.1, we deduce that for eve-

322 ry

R>O , p>O , a>O

, there exists

to>O,

such that if

t (1+0)

2 2 n

in order to obtain

(3.7)

for all

0>0.

The uniformity in independent of

(ii)

x

comes from (3.6), the estimate (3.5) being

x.

Proof of the upper bound. - For

0>0, let us set:

324

A

t

Clearly

f

t

n A

0.

=

t

Moreover, if

with center disjoint from

f

t

and radius

oA n

0'

=--

2

O'\)!(t)

the open ball

in

it follows from

'

d(x,f )O, if we denote by

from the ball (for the metric

then, uniformly for

x

(ljJ (t) ) 2 2t

lim

(ljJ(t) )

the exit time of

x,P

d) with center

x

and radius

p,

in a compact set, we have

2t

lim

1

2

Log P [d(x'Yt)

> nljJ(t) ]

Log P [d (x,y t)

> nljJ(t)

1

> tJ x, P

Proof. - It is enough to notice that:

, 1 x,

p

>t

]

+ P

[1 x,

o

J

t

, T

Classical results

on large deviations

x,p

>

t]

(see R. Azencott

OJ

p.164)

imply lim t Log P

[1

2

x,p

_2-

or: J

t >

o

is choosed such that L

_

tAL

(4.9) n}

is an uniformly integran

ble strictly positive martingale; Z can be proved to increase up to n infinity. Then, equation (4.2) is developped with the Stratonovitch

* and Y and transformed in integral written with local coordinates Ito integral with respect to [,-Brownian motion B(cf. 1.2. or 1.3). After tedious computations, we get Il t f(.,U t) - J

t

o

so

IlsA'f(.,Us)ds

t

J o

i

.. 11

1

s

(dB s

-

Il

s

)ds),

(4.10)

Qn is a solution whatever n.

Step 4 : At last, it remains to prove that hypotheses H1 to H3 imply H1 and H2 for (A', Q(A')). The construction Q(A') and H1 prove that Q(A') is a dense subalgebra of ColE x O(M)), by applying the Stone Weierstrass theorem to the one-point compactification of E x O(M). Then, Q(A') is the set of finite linear combinations of product f.g, f in !I(A) , g in so the image by A' of l1(A') is in ColE x OeM)) indeed, let us remark that in (4.7) the last term can be written (c f . [13] p. 513) : f(x)(u-

1

ljJ

* (x,y))lL.g(U) . 1

i

=

f(x)ljJI(x) (Adh-1)ILig(U)

( 4 • 11 )

-1

; Adh is an unbounded function on G, o but Lig(U) is with compact support, so is the product and (4.11) belongs to ColE x M), and hypothesis H1 is verified by (A', l1(A').

where h is such that U

=

h.U

Now we are concerned with hypothesis H2. We adapt the proof of KURTZ and OCONE' lemma 4.4 whel! the signal function is depending on the observation. First, we prove that uniqueness holds for the martingale problem deduced from the filtering model under the reference probability. Indeed, the martingale problem associated to has a unique solution: the Brownian motion on OeM) (6]. So, hypotheses H1 and H2 and lemma 4.3 of [7] show:

337

uniqueness holds for the martingale problem associated to (B, with B defined by linear extension of 1 B(f.g)(x,V) = Af(x)g(V) + Zf(X)6g(V), f.g E Besides, let (rtX', f:J, D, (X t, martingale problem associated to (A', -1 :'Ii Vt = gt ,u t such that:

(4.12)

be any solution of the and let process

-1 '" -1 dg -1 t = gt (- 1/I(J\))dt ,go = e

(4 . 1 3)

-1

(that is to say gt (- 1/1) is a left invariant vector field on We want to prove for each f.b in

-

f(Xs)b(V s)

t +

0)I ). u

and s lesser than t (4.14 )

E(Js

Xlt Xlt "So, (rt ' , , , (X Vt)' will be a solution of martingale problem t' (B, as the initial system under reference probability (rt, (Xt,Vt),IP)Jo) is. Then, (4.12) proves that X and V are independent and V is a brownian motion on O(M) as (X,V) are under P )J ; furthermore, laws of X and are the same: so, ltt is the of brownian motion Vt by gt solution of dg t = 1/I(Xt)gtdt (consequence of 4.13, see [15 J 2.16) and uniqueness holds for the martingale problem associated to (A', (A' )) • Now let us prove (4.14) : thanks to derivation rules on a manifold (4.13) and (4.11), i f we define R(s,t) as begs-1 Vt), we get: R(s,t) - R(O,t)

s

= J

o

aR av

(v,t)dv

=

(4.15)

For instance, if s equals t, we get : f(Xt)t(V

t)

=

f(Xt)b(V) - /f(X tot

v

J

'Yt)(L.b)(g-1 Ut)dv(4.16) 1 v

Expression - as a function of (Xt,V - is a bounded t) function of peA') thanks to (HS) and expression (4.11), when v is fixed. Thus, to compute mean of (4.16); we cut the integration at v = s, and after s, we apply mean before G -one : =s s t a R

J f(X o s +E(J

o

t

I

t

svv

v

+

E(I

0

A'(faaR(v,.)(X,V)dudv/G). v u u =s

(4.17)

338

This last term, by exchanging the order of u and v - integration becomes : t. u , aR E(f f A (f'ay(v,.)(Xu,Uu)dv (4.18) o 0 Let u be fixed and consider LR as a function on E x OeM) x [O,T] : f.R : (x,U,v)

+

-1

f(x)b(gv

.U) =(f.boL -1)(x,U). gv

(4.19)

Since belongs to Q(A') and A'(LR(v,.)) is time-differentiable, operators A' and aa are commuting, so we get v u

f

o

«v

1I

,U )dv

(A

U

- A' (f. R (0, • ))) (Xu' Uu) (4 . 20)

(f . R (u , . ))

I

Definition 4.19 shows that the first term is A' (f . R (u , . )) (X

,U )

u

= A'

u

(f. b 0 L

gu

1)(X

:

,U )

1I

(4.21)

U

We use the fact L. (boL ) (U) I g

= (L ). (L.) (b) (gU) = (L. b) (g.U) g I I

because L.

i = i

I

= (i .)(A.), L I

g

0

same is true for /', which equals

0

Land A. g

1:L. (L)

I

the

. so we z e t

ill'

A'(LboL g) ( x,U) = B(f.b)(x,g.U) + (u -1 Thus, we can write (4.20)

is left invariant 0

w• (x,y)) i (L

ib)(g.U)f(x)(4.22)

:

(A'(f.R(v,.))-A'(LR(O,.))(x ,U )=B(f.b)(X ,V )-B(Lb)(Xv'U v) v v '" v ':' (4.23) 1 • + [(uw (X,Y ))(B.b)(V )_(u- 1 w (X ,Y ))1(L.b)(U ))]f(X v VV I V V VV I V v) Hence, applying t i ori (4.15) of

mean to

and using (4.17), defini-

and (4.18) to (4.23) conclude

the proof of

(4. 14) . So, assumption H2 is satisfied by A' F.M.P.

(A', Q(A')). Whatever n ,

n

and the uniqueness holds for . 'd e Qn almost Z (f) COInCI

(f) and TI

t"

everywhere and we get the results as n grows to igfinity.

REFERENCES [1]

J.M. BISMUT, "Me c an i qu e a l e a t o i r e '! , L.N.

in Mathematics n° 866,

Springer-Verlag, 0981).

[21

I.E. DUNCAN, "Stochastic Filtering in Manifolds", Proc. IFAC World Congress Pergamon (1981).

339

[ 3] [ 4

1

[ 5 ]

[ 6] [ 7]

[8I [ 9 ]

[ 10]

[ 11 ] [ 12 ]

[ 13 ]

[ 14 ]

K.D. EL WORTHY, "Stochastic Differential Equations on Manifolds", Cambridge University Press, (1982). P. GANGOLLI, "On the construction of certain diffusions on a Riemannian Manifold", Z.f. lV 0 tel que E( exp a:IIX W) < 00. On trouvera une demonstration de ce lemme dans Ferniquej l]. Nous utiliserons plutot le resultat intermediaire suivant. Si s est un nombre tel que a = P(IIXII s) > 1/2 alors :

Vu

s

C a))

P(IIXII> u) < a exp

Nous allons appliquer ce resultat avec: E

muni de la norme et si on pose :

II/II =

={/: R +

--->

1/(t)l.

R , continue et bornee }

Soit X

(3t

un mouvement brownien unidimensionnel

(3t

t

= ¢(t)"

Il est clair que X est un vecteur gaussien a valeurs dans (E, precedent a ce vecteur gaussien, on obtient :

11.11). En appliquant le resultat

LEMME 2.7

posons:

Soit uo verinant P(N(w) era

On a, si u

uo :

uo)

= uo Vrv: t;g3

348

On en deduit :

COROLLAIRE 2.8

, si p E N*

Demonstration: On a, en appliquant le theorerne de Fubini :

2p Done:

l" u 2p - 1 p

Juo

(N 2 u) du

E (N 2P)
t}. De plus ,comme E(IX tI 2 ) "'0 on obtient :


E p

ou

j3 =

-

( su {(3P'p- : (3p'-' }) 1