CRC Research Notes in Mathematics Series)

stochastic functional differential equations

3t

PITMAN PUBLISHING LIMITED 128 Long Acre, London WC2E 9AN PITMAN PUBLISHING INC 1020 Plain Street, Marshfield, Massachusetts 02050 Associated Companies Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto

© S-E A Mohammed 1984 First published 1984 AMS Subject Classifications: (main) 60H05, 60H10, 60H99, 60J25, 60J60 (subsidiary) 35K15, 93E15, 93E20

Library of Congress Cataloging in Publication Data Mohammed, S. E. A. Stochastic functional differential equations. (Research notes in mathematics; 99) Bibliography: p. Includes index. 1. Stochastic differential equations. 2. Functional differential equations. I. Title. II. Series. QA274.23.M64 1984 519.2 83-24973 ISBN 0-273-08593-X British Library Cataloguing in Publication Data Mohammed, S. E. A. Stochastic functional differential equations.(Research notes in mathematics; 99) 1. Stochastic differential equations 1. Title II. Series 519.2

QA274.23

ISBN 0-273-08593-X

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise, without the prior written permission of the publishers. This book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, without the prior consent of the publishers.

Reproduced and printed by photolithography in Great Britain by Biddles Ltd, Guildford

To Salwa and Yasar with 1 ove

Contents Preface I.

PRELIMINARY BACKGROUND §1.

Introduction

§2.

Measure and Probability

1

§3.

Vector Measures and the Dunford-Schwartz Integral

8

§4.

Some Linear Analysis

11

§5.

Stochastic Processes and Random Fields

14

§6.

Martingales

16

V. Markov Processes

18

§8.

1

Examples:

21

(A) Gaussian Fields

21

(B) Brownian Motion

22

(C) The Stochastic Integral

24

II. EXISTENCE OF SOLUTIONS AND DEPENDENCE ON THE INITIAL PROCESS

30

§1.

Basic Setting and Assumptions

30

§2.

Existence and Uniqueness of Solutions

33

§3.

Dependence on the Initial Process

41

III.MARKOV TRAJECTORIES

46

§1.

The Markov Property

46

§2.

Time-Homogeneity: Autonomous Stochastic FDE's

58

§3.

The Semigroup

66

IV. THE INFINITESIMAL GENERATOR

70

§1.

Notation

70

§2.

Continuity of the Semigroup

71

§3.

The Weak Infinitesimal Generator

76

§4.

Action of the Generator on Quasi-tame Functions

97

V.

VI.

REGULARITY OF THE TRAJECTORY FIELD

113

§1.

Introduction

113

§2.

Stochastic FDE's with Ordinary Diffusion Coefficients

114

§3.

Delayed Diffusion: An Example of Erratic Behaviour

144

§4.

Regularity in Probability for Autonomous Systems

149

EXAMPLES

165

§1.

Introduction

165

§2.

Stochastic ODE's

165

§3.

Stochastic Delay Equations

167

§4.

Linear FDE's Forced by White Noise

191

VII. FURTHER DEVELOPMENTS, PROBLEMS AND CONJECTURES

223

§1.

Introduction

223

§2.

A Model for Physical Brownian Motion

223

§3.

Stochastic FDE's with Discontinuous Initial Data

226

§4.

Stochastic Integro-Differential Equations

228

§5.

Infinite Delays

230

REFERENCES

234

INDEX

240

Preface

Many physical phenomena can be modelled by stochastic dynamical systems whose evolution in time is governed by random forces as well as intrinsic dependence of the state on a finite part of its past history.

Such models may be ident-

ified as stochastic (retarded) functional differential equations (stochastic FOE's).

Our main concern in this book is to elucidate a general theory of stochastic FDE's on Euclidean space.

In order to have an idea about what is act-

ually going on in such differential systems, let us consider the simplest stochastic delay differential equation.

For a non-negative number r > 0,

this looks like dx(t) = x(t-r) dw(t)

(SDDE)

where w is a one-dimensional Brownian motion on a probability space (R,F,P) and the solution x is a real-valued stochastic process.

It is interesting to

compare (SDDE) with the corresponding deterministic delay equation dy(t) = y(t-r)dt.

(DDE)

One can then immediately draw the following analogies between (SDDE) and (DDE): (a)

If both equations are to be integrated forward in time and starting from

zero, then it is necessary to specify a priori an initial process {A(s): -r < s < 0} for (SDDE) and a deterministic fucntion n:[-r,0] -* R for (DDE).

In the ordinary case (r = 0), a simple application of the (Ito) calculus gives the following particular solutions of (SDDE) and (DDE) in closed form: ew(t)-its

x(t) =

y(t) = et, t E R.

For positive delays (r > 0), no simple closed-form solution of (SDDE) is known to me.

On the other hand, (DDE) admits exponential solutions y(t) = eAt, t ER,

where A E L solves the characteristic equation

(i)

a (b)

e-ar = 0.

When r > 0, both (SDDE) and (DDE) can be solved uniquely through e, n,

respectively just by integrating forward over steps of size r; e.g. e(0) + x(t)

f t e(u-r)dw(u) JO

0 < t < r

-r -r} of (SDDE) in R when r > 0.

Heuristically speak-

ing, a positive delay upsets Markov behaviour in a stochastic delay equation. (d)

To overcome the difficulty in (c), we let C = C([-r,O],R) denote the

Banach space of all continuous real functions on [-r,0] given the supremum For each t > 0, pick the slice of the solution paths over the

norm 11 11 c.

interval [t-r,t] and so obtain trajectories {xt: t > 01, {yt:t > 01 of (SDDE) and (DDE) traced out in C.

It now follows from our results in Chapter III

(III.2.1), (111.3.1)) that trajectories {xt:t > 01 des-

(Theorems

cribe a time-homogeneous continuous Feller process on C. (e)

As functions of the initial path n E C, trajectories {nxt:t > 01,

{nyt: t > 01 of (SDDE) and (DDE) through n define a trajectory field

Tt:C

- t2(St,C),

t > 0,

nNnxt and a semi-flow

Tt:C -rC , n -r. nyt (ii)

t>0

respectively.

Both Tt and Td are continuous linear, where L2(c,C) is the complete space of all F-measurable 0:c2-'C such that IcIle(w)IIc dP(w)
0, does not admit 'good' sample function behaviour.

Thus,

despite the fact that Borel measurable versions always exist, no such version of the trajectory field has almost all sample functions locally bounded (or even linear) on C (cf. Corollary (V.4.7.1), V §3, VI §3).

It is intriguing

to observe here that this type of erratic behaviour is peculiar to delayed diffusions (SDDE) with r > 0.

Indeed for the ordinary case r = 0 it is well-

known that the trajectory field has sufficiently smooth versions with almost all sample functions diffeomorphisms of Euclidean space onto itself (Kunita [45], Ikeda and Watanabe [35], Malliavin [51], Elworthy [19], Bismut [5]). At times t > r, the deterministic semi-flow Tt maps continuous paths

(f)

into C1 paths, while the corresponding trajectory field T

takes continuous t

paths into a-Hdlder continuous ones with 0 < a < } (cf. Theorem (V. 4.4)). Now our discussion has so far been with reference to the rather special examples of stochastic and deterministic delay equations (SDDE) and (DDE) above.

However, this is indeed no serious restriction; it is one of our main

contentions in this-book that the observations (a) - (f) are essentially valid for a much wider class of stochastic FDE's than just (SDDE).

Thus in Chapter

II we establish existence, uniqueness and continuous dependence on the initial process for solutions to general stochastic FDE's of the form dx(t) = g(t,xt)dz(t),

x0 = 0 E

t > 0

C2(S2,C)

where the coefficient process g: R>0 x L2(S2,C) - L2(S2,Rn) and the initial process 6 E L2(Q,C) are given, with z a McShane-type noise on a filtered probability space (S2,F,(Ft)t>0,P)

(Refer to Conditions (E)(i) of Chapter

II).

Chapter III essentially says that for systems of type (iii)

dx(t) = H(t,xt)dt + G(t,xt)dw(t),t > 0, x0 = n E C e trajectory field {nxt:t > 0} describes a Feller process on the state Here the drift coefficient H:R>0 x C - Rn takes values in Rn and

ace C.

e diffusion coefficient

G:e

x C - L(Rm,Rn) has values in the space

Rm,Rn) of all linear maps Rm + Rn. ownian motion.

The noise w is a (standard) m-dimensional

If the stochastic FDE is autonomous, the trajectory field

a time-homogeneous diffusion on C.

In Chapter IV we look at autonomous stochastic FDE's dx(t) = H(xt)dt + G(xt)d w(t)

d investigate the structure of the associated one-parameter semi-group t}t>0 given by the time-homogeneous diffusion on C.

A novel feature of

ch diffusions when r > 0 is that the semi-group {Pt}t>0 is never strongly ,ntinuous on the Banach space Cb(C,R) = Cb of all bounded uniformly con-

nuous real-valued functions on C endowed with the supremum norm (Theorem V. 2.2)).

Hence a weak generator A of {Pt}t>0 can be defined on the

.tter's domain of strong continuity Cb + Cb and a general formula for A is ;tablished in Theorem (IV. 3.2).

Due to the absence of non-trivial differ-

itiable functions on C having bounded supports, we are only able to define weakly dense class of smooth functions on C which is rich enough to generate ie Borel c-algebra of C.

These are what we call quasi-tame functions (IV §4).

i such functions the weak generator assumes a particularly simple and con-

ete form (Theorem (IV. 4.3)). Distributional and sample regularity properties for trajectory fields of itonomous stochastic FDE's are explored in Chapter V.

We look at two extreme

samples: the highly erratic delayed diffusions mentioned above, and the case stochastic FDE's with ordinary diffusion coefficients viz. dx(t) = H(xt)dt + g(x(t))dw(t), t > 0. F g satisfies a Frobenius condition, the trajectory field of the latter lass admits sufficiently smooth and locally compactifying versions for t > r Theorem (V. 2.1), Corollaries V (2.1.1) - V (2.1.4)).

In gereral, the

)mpactifying nature of the trajectory field for t > r is shown to persist in distributional sense for autonomous stochastic FDE's with arbitrary i.pschitz coefficients (Theorems (V. 4.6), (V. 4.7)). iv)

There are many examples of stochastic FDE's. highlight only a few.

In Chapters VI and VII we

Among these are stochastic ODE's (r = 0, VI §2),

stochastic delay equations VI §3), linear FDE's forced by white noise (VI §4), a model for physical Brownian motion (VII §2), stochastic FDE's with discontinuous initial data (VII §3), stochastic integro-differential equations (VII

§ 4 ) ,

and stochastic FDE's with an infinite memory ( r= -, VII §5). Chapter

VII contains also some open problems and conjectures with a view to future developments.

From a historical point of view, equations with zero diffusions (RFDE's) or zero retardation (stochastic ODE's) have been the scene of intensive study during the past few decades.

There is indeed a vast amount of literature on

RFDE's e.g. Hale [26], [27], [28], Krasovskii [43], El'sgol'tz [18], Mishkis [56], Jones [42], Banks [3], Bellman and Cooke [4], Halanay [25], Nussbaum [62], [63], Mallet-Paret [49], [50], Oliva [64], [65], Mohammed [57], and others.

On stochastic ODE's, one could point out the outstanding works of

Ito [36], [37], [38], It6 and McKean [40], McKean [52], Malliavin [51],

McShane [53], Gihman and Skorohod

[24], Friedman [22], Stroock and Varadhan

[73], Kunita [45], Ikeda and Watanabe [35], and Elworthy [19].

However,

general stochastic FDE's have so far received very little attention from stochastic analysts and probabilists.

In fact a surprisingly small amount

of literature is available to us at present on the theory of stochastic FDE's. The first work that we are aware of goes back to an extended article of Ito and Nisio [41] in 1964 on stationary solutions of stochastic FDE's with infinite memory (r = co).

The existence of invariant measures for non-linear

FDE's with white noise and a finite memory was considered by M. Scheutzow in [69], [70].

Apart from Section VII §5 and except when otherwise stated, all the results in Chapters II-VII are new.

Certain parts of Chapters II, III and IV were

included in preprints [58], [59], [60], by the author during the period 19781980.

Section VI §4 is joint work of S.E.A. Mohammed, M. Scheutzow and H.v.

Weizsacker.

The author wishes to express his deep gratitude to K.D. Elworthy, K.R. Parthasarathy, P. Baxendale, R.J. Elliott, H.v. Weizsgcker, M. Scheutzow and S.A. Elsanousi for many inspiring conversations and helpful suggestions. For financial support during the writing of this book I am indebted to the British Science and Engineering Research Council (SERC), the University (v)

of Khartoum and the British Council. Finally, many thanks go to Terri Moss who managed to turn often illegible scribble into beautiful typescript.

Salah Mohammed Khartoum 1983.

(vi)

I Preliminary background

U.

Introduction

In this chapter we give an assortment of basic ideas and results from Probability Theory and Linear Analysis which make necessary prerequisites Due to limitations of space, almost

for reading the subsequent chapters. all proofs have been omitted.

However, we hope that the referencing is

adequate.

§2.

Measure and Probability

A measurable space (S2,F) is a pair consisting of a non-empty set c and a

If E is a real Banach space, an E-valued

a-algebra F of subsets of Q.

measure u on (c,F) is a map u:F -> E such that (1) p(O) = 0, (ii) p is a-additive i.e. for any disjoint countable family of sets {Ak}k=1 in F the series M E

k=1

00

p(A ) converges in E and P( U

CO

A ) =

k=1

k

k

E p(A ). k=1 k

When E = R, u is called

a signed measure and if u(F)=J O it is called a positive measure. sup {!p(A)J: A E F} < -, p is a finite measure. P on (Q,F) such that P(Q) =

1

If

A positive finite measure

is a probability measure on 0; the triple

(S2,F,P) is then a probability space.

The set of all finite real-valued

measures on 0 is denoted by M(S2) and the subset of all probability measures

by Mp(S2). A probability space (S2,F,P) is complete if every subset of a set of P-

measure zero belongs to F i.e. whenever B E F, P(B) = 0, A c B, then A E F. In general any probability space can be completed with respect to its underlying probability measure.

Indeed let (0,F,P) be an arbitrary probability

space and take

Fp= {AUA: AEF, Ac A0EF, P(A0) =0} to be the completion of F under P. A c A0S P(A0) =0.

Extend P to Fp by setting P(A U A) = P(A),

Then (Q,Fp,P) is the smallest P-complete probability space

with F c TP

Because of this property, we often find it technically simpler to assume from the outset that our underlying probability space is complete. 1

When 0 is a Hausdorff topological space and F is its Borel c-algebra, Bore! 0, generated by all open (or closed) sets, a measure u on 12 is regular if u(B) = sup {p(C) = inf {u(U)

C c B, C closed)

:

B c U, U open).

:

If 12 is metrizable, every u E M(S2) is regular and hence completely determined

by its values on the open (or closed) sets in Q.

(Parthasarathy [66], Chap-

ter II, Theorem 1.2, p. 27).

An E-valued measure

Let 52 be a Hausdorff space and E a real Banach space.

u on (0, Borel S2) is tight if (i) sup { Iu(B) I

:

B E Borel S2} < -, where

I.

I

denotes the norm on E; and (ii) for each e > 0, there is a compact set K. in

0 such that Iu(a2,KE) I < E Theorem (2.1):

Let S2 be a Polish space i.e. a complete separable metrizable

space.

Then every finite real Borel measure on 12 is tight.

Proof:

Parthasarathy ([66], Chapter II, Theorem 3.2, p. 29); Stroock and

Varadhan ([73], Chapter 1, Theorem (1.1.3), pp. 9-10).

o

Let 0 be a separable metric space and F = Borel Q. Denote by Cb(S2,R) the Banach space of all bounded uniformly continuous functions 0:0 - R given the supremum norm

110 II Cb = sup { Iq(n) I : n E st}. The natural bilinear pairing

Cb(S2R) x M(S2) - R

(n) du(n), 0 E Cb(12,R), V E MM,

= J TIER

induces an embedding

Cb(S2,R)

M(12)

u

>

where Cb(c,R)* is the strong dual of Cb(12,R).

sponds to the continuous linear functional

Cb(S2,R) 0

2

;-

I ----->

R

q'(n) du(n)

Indeed each u E M(12) corre-

because for every m E Cb(52,R)

if

O(n) du(n)I < II0IICb v(u)(0) nE52

p where v(u)(0) = sup { E

p lp(Ak)I: Ak E F, k = 1,...,p disjoint, 0 =

k=1

U Ak, k=1

p < -} is the total variation of u on 1 (Dunford and Schwartz [15], Chapter III, pp. 95-155). ology.

As a subset of Cb(c,R)* give M(S2) the induced weak * top-

Now this turns out to be the same as the weak toplogy or vague top-

ology on measures because of the following characterizations.

Theorem (2.2): Let S2 be a metric space and u, uk E M02) for k = 1,2,3,...

Then the following statements are all equivalent:

(i)

uk - u as k - - in the weak * topology of M(s2) ;

(ii)

lim

k-

4(n)duk() = fnER

4(n)du(n), for every $ E Cb(S2,R); JnEQ

(iii) lim sup uk(C) < u(C) for every closed set C in Sl; k

(iv)

lim inf uk(U) > U(U) for every open set U in Q; k

lim uk(B) = u(B) for every B E Borel S2 such that u(aB) = 0.

(v)

k-),w

For proofs of the above theorem, see Parthasarathy ([66] Chapter II, Theorem 6.1, pp. 40-42) or Stroock and Varadhan ([73], Theorem 1.1.1, pp. 7-9).

The weak topology on M(c), when S2 is a separable metric space, can be

alternatively described in the following two equivalent ways: (a)

Define a base of open neighbourhoods of any u E M(2) by

U11 (019...90p's E12...sep) _ {v:v E M(2), IJok dv -J 4)k

dpi < Eks

k = 1,2,...,p}

where 41s...94p E Cb(S2sR)s EVE: 2,...,Ep > 0.

3

Furnish M(S2) with a metric p in the following manner.

(b)

Compactify the

ti

separable metric space 9 to obtain Q.

Then Cb(S2,R) is Banach space-isomorphic

to the separable Banach space C(S2,R) of all continuous real functions on S2,

given the supremum norm.

Pick a countable dense sequence {Ok}k=1 in Cb(c,R)

1

and define the metric p on M(S2) by

E

p(u,v) =

k=1 p,v

2

Ok du - J

k

11

1 I1

J

Cb C

0k dvj st

sz

E M(Q) (Stroock and Varadhan [73], Theorem 1.1.2, p. 9; Parthasarathy

[66], pp. 39-52).

$ote that M(c) is complete if and only if S2 is so.

ilarly,Mp(S2) is compact if and only if 0 is compact.

Sim-

More generally compact

subsets of M(S2) are characterized by the well-known theorem of Prohorov given

in Chapter V (Theorem (V.4.5)). There is a theory of (Bochner) integration for maps X:S2- E where E is a real Banach space and (S2,F,u) a real measure space (Dunford and Schwartz [15], Chapter III §1-6). On a probability space (S2,F,P) an (F, Borel E)-measurable map X:f -> E is

Such a map is P-integrable if there is a

called an E-valued random variable.

sequence Xn:S2+ E, n = 1,2,..., of simple (F, Borel E)-measurable maps so that

Xn(w) -> X(w) as n -

lim J

for a.a. w E S2 and

m,n-'

IXn(w)-Xm(w)IEdP(w) = 0. wES2

Define the expectation of an integrable random variable X:52- E by EX = J S2

X(w)dP(w) = lim n

Xn(w)dP(w) E E.

J

0

This definition is independent of the choice of sequence {Xnf

.00n=1

converging

a.s. to X (Rao [67], Chapter I, §1.4; Yosida [78], Chapter V §5, pp. 132-136).

For a separable Banach space E, X: Q- E is a random variable if and only if one of the following conditions holds: (i)

There is a sequence Xn:52 -* E, n = 1,2,..., of simple (F, Borel E)-measur-

able maps such that Xn(w) -> X(w) as n

- for a.a. w E S2;

(ii) X is weakly measurable i.e. for each A E E*, AoX:c -* R is (F, Borel R)measurable.

(Elworthy [19], Chapter I, §1(C) pp. 2-4; Rao [67], Chapter I §1.4). Denote by L°(s2,E;F) the real vector space of all E-valued random variables X:S2-> E on the probability space (0,F,P).

The space L°(S2,E;F) is a complete

TVS under the complete pseudo-metric d(X1,X2) = inf[e+P{w:w E S2,IXI(w)-X2(w)IE >E} : E > 0]

for X1, X2 E ro(S2,E;F).

The norm in our real Banach space is always denoted

by I.IE or sometimes just I.I.

A sequence {Xn}n=1 of random variables 00

Xn:S2 -* E converges in probability to X E JCO(1,E;F) if for every e > 0 lim P{w:w E S2, IXn(w) X(w)IE > E} = 0. A random variable X:1+ E is n(Bochner) integrable if and only if the function Sa

le0

W i---;. 1X(W)1E is P-integrable, in which case 1IJQ X(w)dP(w)IE
R, k = 1,2,..., converges p-almost everywhere

to f:Q - R if fk - f as k

- a-a.e. i.e. if there is a A-null set A E FA

such that fk(w) -> f(w) as k - - for all wE a2,A. A function f:SZ - R is (D-S) p-integrable (over f2) if there is a sequence {fk}k=1 of simple functions

such that fk -* f as k +

u-a.e. and the sequence

{fA fk du}00 converges k=1 9

The limit

in E for every A E F.

f du = l im

f

fA

A

du E E 00

is independent of the choice of sequence (fk}k=1 and is called the (D-S) integral of f over A with respect to the E-valued measure U.

Theorem (3.1)

( Properties of the Integral): Let E be a real Banach space and

p an E-valued measure on a measurable space (0,F). (1)

Let

f,g:S2- R be u-integrable over A E F and a,B E R.

Then of + gg is

u-integrable over A and

(af + sg)du = JAJA g du.

JA

(ii) If f:9 -> R is u-measurable and u-essentially bounded, then f is u-inte-

grable over any A E F and

J f du1E < IIu 11 (A) u-essup If (w) I. w EA

A

(iii)

The map >

F

E

AIA f du is an E-valued measure for any u-integrable f:S2 + R. (iv)

(Dominated Convergence): Fix A E F and let fk:S'i - R, k = 1,2,... be a

sequence of u-integrable functions converging V-a.e. to a function f:c -> R. Suppose there is an integrable g:Sl -* R>' such that Ifk(w)l < g(w) for u-almost

all w and all k > 1.

f du = lim fA

Proof:

(A fk du

J

Dunford and Schwartz [15], Theorem IV.10.8 (pp. 323-325); Theorem IV.

10.10, p. 328.

10

Then f is p-integrable over A and

a

Some Linear Analysis

§4.

If E is a real Banach space, its strong dual is E* = L(E,R), the space of all

continuous linear functionals A:E -.R with 11x II = sup {1),(v) :v E E, ME =1}, The weak topology on E is obtained by taking as base of open neighbourhoods of points x E E the sets N(x;r,c) = {y:y E E, r c E* is finite and c > 0.

Jx(y) - a(x)l < c, A E r} where

In particular a generalized sequence {xa} in E

converges weakly to x E E if lim a(xa) = A(x) for each a E E* (Dunford and Schwartz [15], §V.3).

If E, F are real Banach spaces, say a map T E L(E,F) is weakly compact if the weak closure of the image T(B) of the closed unit ball B = {x:x E E, IxIE < 1} in E is weakly compact in F.

A subset of a Banach space is weakly

sequentially compact if every sequence in it has a weakly convergent subsequence.

Thus T E L(E,F) is weakly compact if and only if it maps bounded

sets in E into weakly sequentially compact subsets of F; e.g. a compact linear map is weakly compact.

Let S be a compact Hausdorff space and C(S,R) the Banach space of all continuous functions n:S - R given the supremum norm IIn1IC = sup ln(s)l. Then SES for any real Banach space E, a weakly compact linear map T:C(S,R) - E corres-

to an E-valued measure on (S, Borel S).

Indeed we have the following

generalization of the Riesz Representation Theorem for continuous linear functionals due to Bartle, Dunford and Schwartz ([15], Theorems VI (7.2)

-

(7.3), pp. 492-496). Theorem (4.1):

Let S be compact Hausdorff and T:C(S,R) -. E a weakly compact

linear map into a real Banach space E.

Then there is a measure u:Borel S - E

such that (i)

T(n) = JS n(s) du(s), for all n E C(S,R);

(ii)

IITh

=

Ilull(S);

(iii)

for each a E E*, )op is a regular finite Borel measure on S;

(iv)

T*(a) = aau for all a E E*.

Conversely, every E-valued measure p on (S, Borel S) which satisfies (iii) above defines a weakly compact linear map T:C(S,R) - E which fulfills (i), (ii) and (iv).

11

Remark

The integral in (i) is the D-S integral introduced in the previous section (§3).

A class of weakly compact linear maps C(SR) -* E is provided by

If the Banach space E is weakly complete, then every contin-

Theorem (4.2):

uous linear map C(S,R) -* E is weakly compact.

Proof:

Dunford and Schwartz ([15], Theorem VI 7.6, pp. 494-496).

a

If E, F are real Banach spaces, their projective tensor product E ®

F is

a real Banach space with respect to the completion of the norm p

IIzIl = inf

Ixi IE Iyi IF:

p

Z=

E

xi 0 yi}.

i=1

i=1

Here the infimum is taken over all possible representations

z=

p

E xi®yi EE®F, i=1

the algebraic tensor product of E and F. continuous bilinear maps S:E x F

Let L(E,F,;R) be the space of all

R viz. all those bilinear a such that

there is a constant K > 0 with IB(x,y)I < KIxIE IYIF for all x E E and all y E F.

Then L(E,F;R) is a Banach space when furnished with the norm

II s II

= sup {IB(x,Y)I

:

x E E, Y E F, IXIE < 1, IYIF < 1}.

The projective tensor product allows us to identify each continuous bilinear map E x F -* R with a continuous linear functional on E ® F viz.

Theorem (4.3):

Proof:

L(E,F;R) is norm isomorphic to [E 87r F]

Treves [75], Proposition 43.12, pp. 443-444.

As an.example consider the following situation. dorff space and E = F = C(S,R).

.

a

Let S be a compact Haus-

Then every continuous bilinear form

g:C(S,R) x C(S,R) -* R corresponds to a signed bimeasure on S x S viz. a set function v:Borel S x Borel S -* R such that for any A,B E Borel S, are measures on S. 12

and

An integration theory for signed bimeasures has

been developed by Ylinen [77].

Note that a continuous bilinear form on

C(S,R) does not necessarily correspond to a measure on S x S because of the strict embedding C(S x S, R)

C(S,R) ®,m C(S,R)

n 0 E a

>

{(s,s')

F-3'

n(s)C(s')}

(Treves [75], cf. Exercise 44.2, p. 450).

The following theorem of Mercer will prove to be useful.

For a proof see

Ri.esz and Sz-Nagy ([68], Chapter VI, §97-98, pp. 242-246) or Courant and Hilbert ([10], pp. 122-140).

Theorem (4.4) (Mercer's Theorem): Let I c R be a closed bounded interval and K:I x I - R a continuous symmetric kernel which is positive in the sense that

l I ft

K(s,s')n(s)n(s')ds ds' > 0 00k=1

for all n E C(I,R).

Suppose

c

C(I,R) and

{Ak}k=1

c R>O

are solut-

ions of the eigenvalue problem K(s,s')Ck(s')ds' = Ak Ys), for all s E I. JI

Then K can be expanded as a uniformly convergent series K(s,s') =

E k=1

AkCk(s)Ck(s'), s,s' E I,

on I x I. Remark

The validity of JI

K(s,s')n(s)n(s')ds ds' > 0 for all n E C(I,R) is JI

equivalent to Ak > 0 for all k > 1.

Theorem (4.5)

(Douady):. For real Banach spaces E, F, every Borel-measurable

linear map T:E - F is continuous.

13

Schwartz. ([71.], Part I.I, Chapter I., §2 pp. 1.55-1.60).

Proof: §5.

a

Stochastic Processes and Random Fields

Let (S2,F,P) be a probability space, T a complete separable metric space and E a real Banach space.

An E-valued stochastic process x on (0,F,P) para-

metrized by T is a map x:T x S2 - E such that x(t,.) E1t52,E;F) for all t E T. If 0 < a

-, a filtration (Ft)0 1 is an integer such that x(t,.) E 9k(c,in) for all t E [0,a].

Let Rn carry the norm. n E

Ivl1 =

Ivil

,

v = (vt,...,an) E

n.

i-l

Then the real process

[0,a] x 12 -> R (t,W)

>

I

Ix(t,W) Ik

is an (Ft)0 0

Ix(t,W)I1 > E} < -1

and

E{ sup

I

I

is any other norm on Rn, then there are positive constants K1, K2

independent of c, a and x such that K1

P{W:w E S2,

sup Ix(t,w)Ik 0 0, and E{

Proof:

sup O 0 and n E E, Ps,n {w.w E St, x(s,w) = n} = 1.

(v)

NO If B E Borel E, 0 < s < t < u and n E E, then the Markov Property E B)

E BIFt) = holds a.s.-Ps,n on S2.

An E-valued Markov process defines a family of transition probabilities E E, 0 < s < t < co} on E given by

p(s,n,t,B) =

E B), B E Borel E.

These have the properties: (i)

for fixed 0 < s < t, B E Borel E, p(s,.,t,B) is Borel-measurable;

(ii)

for fixed 0 < s < t, n E E,

is a Borel probability measure

on E; (iii) the Chapman-Kolmogorov identity

19

p(s,n,u,B) =

JSEE p(s,n,t,dE) p(t,C,u,B)

holds for 0 < s < t < u, n E E, B E Borel E (Dynkin [16], p. 85).

A Markov

process is time-homogeneous if its associated family of transition probabilE E, 0 < s < t < -1 satisfies

ities

p(s,n,t,B) = p(O,n,t-s,B)

whenever 0 < s< t, n E E, B E Borel E. Let b(E,R) denote the Banach space of all real-valued bounded Borel-

measurable functions 0 on the Banach space E with the supremum norm 11011b = sup {I4(n)I:n E E}.

Then a time-homogeneous Markov process on E

with transition probabilities

n E E, t > 01 yields a contraction

one-parameter semi-group {Pt}y of continuous linear operators defined by

Pt:b(E,R)

Pt( )(n) = JSEE 4,(F) p(O,n,t,dC), n E E,

for each 0 E b(E,R) (Dynkin [16], Chapter LI., §1-2, pp. 47-61; Chung [8], pp. 1-12).

A time-homogeneous Markov process on a Banach space E with semi-

group Pt:b(E,R) -, b(E,R), t > 0, is called a Fetter process if

{Pt}t>o leaves invariant the closed linear subspace Cb(E,R) c b(ER) of

(i)

all bounded uniformly continuous functions E -. R; (ii)

lim

{Pt}t>O is weakly continuous on Cb(E,R) at t = 0 i.e. for every n E E Pt(4)(n) = 4)(n) for all 0 E Cb*

t-+0* As a consequence of our results in Chapter IN (Theorem (IV. 2.2)), for infinite dimensional E the last condition (ii) in the above definition of Feller process does not imply strong continuity at t = 0 of the semi-group

{Pt}y on Cb(E,R).

Compare this with the locally compact case in Chung ([81,

pp. 48-56) where strong continuity is implied by weak continuity at t = 0. For a further study of (time-homogeneous) Markov processes, their transition semi-groups and infinitesimal generators, the reader may consult Dynkin ([16]).

20

Examples

§8.

Let us wind up by looking at some useful examples of stochastic processes which are indispensible for our forthcoming discussion of stochastic FDE's. Gaussian Fields

(A)

If T is a metric space, a real stochastic process x:T x S2 -).]R on a probability space (52,F,P) is a Gaussian process if for every t E T the random

variable x(t,.) E r2(2,R;F) and has a Gaussian distribution (Hida [31], pp.

31-43). For a Gaussian process x:T x S2 -. R define the mean m:T -). R and covariance V.:T x T --+ R by setting t E T

m(t) = E x(t,.), V(s,t) =

E T.

A Gaussian process x:T x Q a R is called a Gaussian system if every finite linear combination from. the family (Hida [31] §1.6).

t E T} has a Gaussian distribution

The following results concerning Gaussian systems are well-

known.

Theorem (8.1):. (i)

Any subsystem of a Gaussian system is also a Gaussian system.

(ii)

For any Gaussian system x:T x c + R the closed linear hull of t E T} in r2(S2,R;F) is a Gaussian system.

(iii) If x:.T x 0 - R is a Gaussian system, then a necessary and sufficient

condition for the family {x(t,.):t E T} to be independent under P is V(s,t) = 0 whenever s # t. (iv)

If x:Z+ x Q-). R is a (discrete-time) Gaussian system, then the sequence converges in probability i.f and only if i.t converges in C2(c2,R;F).

In this case lim x(n,.) is a Gaussian random variable. n->oo

Hida [31], pp. 34-35.

o

If T is a real Banach space, a Gaussian system x:.T x 0 + R on a probability

space (c,F,P) will be called a Gaussian (random) field or just a Gaussian field.

Gaussian fields

parametrized. by i.nfini.te-dimensional Hilbert or

Banach spacesdo not in general admit continuous or locally bounded versions 21

which are defined on the whole of the parameter space (Dudley [13] and Chapter V §3 of this book).

For a deeper study of sample regularity properties of

Gaussian processes, the interested reader could refer to the works of Dudley [13], [14], Feldman [20] and Fernique [21].

Brownian Motion

(B)

Let (c ,F, (Ft)i>0,P) be a filtered probability space. A process w:Rex S2 -R is a one-dimensional Brownian motion (or a Wiener process) on (Q,F,(Ft)i>0,P) if

it Is (Ft)t>O-adapted.

(i)

(ii) it Is a Gaussian system on (52,F,P) with mean and covariance given by

E w(t,.) = 0, E w(t,.)w(s,.) = min (t,s), for all t,s > 0.

A Brownian motion w can always be normalized so that w(0,.) = 0 a.s. process w

A

x 0 - Rm is an m-dimensional Brownian motion if it is of the

form w = (w1,w2,...9wm) where the coordinate processes {wi}m=1 are independent one-dimensional Brownian motions on (c,F,(Ft)i

,P).

For easy reference, we

list here some of the basic properties of Brownian motion in Rm.

These are

well-understood and the reader may look at Hida [31], Chung [8], Friedman [22] and McKean [52] for proofs.

Theorem (8.2):

Let w 1

Rm be an m-dimensional Brownian motion on the

x S

filtered probability space (S2,F,(Ft)t>O,P).

Then the following is true:

(i)

w is an

(ii)

The sample paths of w are almost all nowhere differentiable.

t)t>O -martingale.

ti

(iii) w has a version w such that for any a > 0 and any 0 < a < }, almost all ti

sample paths of wl[0,a] x S2 are a-Holder continuous.

(iv) For a.a. w E S2, )-i

lim sup (2t log log lim sup (2t log log

t->00 22

t

jw(t,w)j = m1/2

t)-l

jw(t,w)j = m1/2

m

where

I

denotes the Euclidean norm IvI =

I

E

v?, v = (V49 ...,v ) E R.

m

i

i=1.

This property is called the Law of the Iterated Logarithm. (v)

t > 0} is an

If t0 > 0, then the process

m-dimensional Brownian motion on (f,F,(Ft)t>t ,P). 0

(vi)

Let

S2

=

C(R>O,Rm) be the space of all continuous paths R>O

-

Rm and f

be the a-algebra generated by all cylinder sets in SZ of the form {f:f E (f(t1),...,f(tk)) E Bk}' tl,...,tk E Rte, Bk E Borel (Rm)k

Then there is a ti

unique probability measure uw (called Wiener measure) on (S2,F) giving the

finite-dimensional distributions of w viz. uw{f:f E 2, (f(tl,...,f(tk)) E Bk}

= P {w:w E c2,(w(tl,w),...,w(tk,w)) E Bk}

=

JBk

where the kernel g:(R>O)k x (Rm)k , R>O is given by k

g(u1,...,uk;y1,...,yk) =

(2i u.)-m/2

IT

j=1

for ul,...,uk E R0, '11 ...,yk E R. (vii) If 0 < s < t, then 2k

IFs) _ (2k-2+m)(2k-4+m)...mjt-sI k

a.s. for every integer k > 1. (viii) For 0 < t1 < t2 < t3, the increments

w(t3,.)-w(t2,.)

are conditionally independent under P given the c-algebra (xi)

u < t1}.

w is a Feller process with stationary transition probabilities t > 0, x E Rm}

given by

-Ix- 12

p(0,x,t,B) =

m/2 (2,,t) m/2

e t dy, B E Borel R m. J

Furthermore, the associated contraction semi-group {Pt}t>O is strongly 23

continuous on b(Rm,R).

Its infinitesimal generator A:D(A) c b(Rm,R) -b(Rm,R)

is given by (A4)(x) = }AO(x), x E Rm,

where g:Rm

R is a C2 function with compact support and A is the Laplacian a

A0(x) =

(x).

E

i=1 8x (C)

The Stochastic Integral

We adopt the viewpoint of McShane ([53], Chapters II, III, IV) which contains adequate information for all our purposes.

However, it is interesting to

note that Elworthy's recent treatment ([19], Chapters III, IV, V) contains far-reaching generalizations of McShane's stochastic integral to infinite dimensions.

Other references for stochastic integration include Ito [39],

McKean [52], Meyer [55], Friedman [22], Gihman and Skorohod

[24], Ikeda and

Watanabe [35], Metivier and Pellaumai.l [54], and Arnold [2].

In this section we shall only content ourselves by giving a brief account of the stochastic integral including some of its most important properties. ,P) be a filtered probability space.

Let (c,F,(Ft)

For a closed inter-

val [a,b] c R a partition 11 = (tl,...,tk+1;T1,...,Tk),

a = t1 < t2 < ... < tk = b, Ti E R, i = 1,2,...,k, is belated if Ti < ti, i = 1,...,k and is Cauchy

if Ti = ti, i = 1,...,k. If f:[a,b] x 52 - L(Rm,Rn) and z:[a,b] x S2 -Rm are (Ft)a 0 such that K(t-s)

IFs) < K(t-s) t

a.s. whenever a < s < t < b.

Then the stochastic integral I(t) =

f(u)dz(u)

f

a

exists and belongs to L2(S2,Rn;Ft) for every t E [a,b]. EIJb f(t)dz(t)I2

< {2K(b-a)1/2

+ K1/2}

Indeed

E IIf(t,)IIdt]

Ja where

I.

I

denotes Euclidean norms and

II

II

the standard operator norm on

L(Rm,Rn) (§2).

If, furthermore, z is a continuous

t)a 0 such that Ia(w)(t2) - a(W)(ti)I < 11t t2-t11

a.a. W E 9,

and all t1, t2 E [0,a]. (2)

The stochastic FDE(I.) includes both the ones with random coefficients

and the (ordi.nary) stochastic differential equations (without retardation, ti

[22], [53]); for suppose g:[0,a] x C(J,Rn) x S2 - L(Rm,Rn) is a coefficient process corresponding to a stochastic FDE with random coefficients. Assume

that g is L2 with respect to the third variable. Then we can define the coefficient process g:[O,a] x t2(S2,C(J,Rn)) -> C2(S2,L(Rm, Rn)) by setting g(t,W)((,)) = g(t,W(w),w)

a.a. W E 0

for all t E [0,a], and all W E C2(S2,C(J,Rn)).

§2.

Existence and Uniqueness of Solutions

The following lemmas will be needed in order to establish the main theorem for existence and uniqueness of solutions to the stochastic FDE(I).

Lemma (2.1):

Suppose x:[-r,a] x S2

paths continuous.

-*

Rn is a process with almost all sample

Assume that xj[0,a] is adapted to (Ft)tE[O,a] and x(s,.)

is F0-measurable for all s E J.

Then the process

33

y C(J, Rn)

[O,a] x 2

(t,w) 0----

xt(w)

>

is adapted to (Ft)tE[O,a]' with almost all sample paths continuous.

Proof.

Since xI[-r,t] has continuous sample paths, it induces

Fix t E [O,a].

an F-measurable map xI[-.r,t]:0 - C([-r,t],Rn).

This map is in fact Ft-measur-

able; to see this observe that Borel C([-r,t],Rn) is generated by cylinder sets of the form

ptt(Bi) where Bi E Borel Rn, ti E. [-r,t],

n i.=1

pt

i

> Rn is evaluation at ti for 1 < i < n. Thus it is suff-

C([-r,t],Rn)

i.cient to check Ft-measurability of xI[-r,t] on such sets.

With the above

notation, n [ n

(xj[-r,t])-1

1

pt (Bi)] =

i=1

By hypotheses, if ti E J, {w:x(ti,w) E Bi} E Ft

is

n {w:x(ti,w) E Bi}. n i=1

{w:x(ti,w) E Bi}

E FO; and if ti E [O,t],

.

Hence {W:x(ti,w) E Bi}

E Ft c Ft if 1 < i < n. i

So n

(xl[-r,t])-1[ n i=1

Pt(Bi)] E Ft* i

and the deterministic memory

Now xt =

Rn)

> C(J, Rn)

is continuous, so xt must be Ft-measurable. Lemma (2.2): tion E(i).

Suppose that z:92

o

C([0,a], Rm) is a process satisfying condi-

Let f:[0,a] - C2(st,L(Rm, Rn)) be an t2-continuous process adapted

to (Ft)tE[O,a]' Define the process F:[-r,a] t E [0,a]

(O

0

34

,

tEJ

L2(S2,Rn) by

a.s., where the integral is McShane's belated integral of f with respect to z..

Then F corresponds to a process belonging to r2(s1,C([-r,a), Rn)) and

Indeed there is an M > 0 such that

adapted to (Ft)tE[O,a].

E(

sup tE[0,a]

IJt f(u)dz(-)(u)I2) < M Ja E( IIf(u)II2 )du 0

(1)

0

The process [O,a] 3 t t--> Ft E r2(S2,C(J,Rn)) is adapted to (Ft)tE[O,a] with almost all sample paths continuous (i.e. it belongs to CA(0,a], 92(52,C(J,Rn))). M is independent of f.

Proof:

With the notation of Condition E(i), t

rt

+ (w)

F(t)(w) = f 0 o

f(u)dzm(-)(u), t E [O,a] I

0

(2)

a.a.w E Q.

The first integral on the right-hand side of (2) is a Riemann-Stieltjes integral for a.a. w and is therefore continuous in t for a.a. w E S2; it thus

defines an F-measurable map S2+ C([0,a],Rn).

As f(u) is Fu-measurable for

all 0 < u < t, then f(u)(-)da(u)

It

0

is Ft-measurable (being a.s. a limit of Ft measurable Riemann-Stieltjes sums).

Since zm is a martingale adapted to (Ft)tE[O,a] with a.a. sample

paths continuous, then so is the McShane integral on the right-hand side of

(2) 0I.8(C)). E(

Hence by the Martingale inequality we have

sup tE[O,a]

< 4C

4E(IJa f(u)dz m

IJt f(u)dzm 0

Ja

0

)du (Theorem (1.6.1))

E(

(3)

0

where C = 2Ka1/2 + K1/2 (Theorem (1.8.4)).

If R > 0 is the Lipschitz constant for a, then it is easy to see that for

a.a. w E S2 II

f(u)(w)da(u)I2


1.

Then by Condition (E)(ii),

(iii) and the continuity of the stochastic memory, it follows from P(k) (ii) that the process [O,a] 3 u

g(u, kxu) E JC2(Q,L(Rm, Rn))

is continuous and adapted to (Ft)tE[O,a]'

We can therefore apply Lemma (2.2)

to the right-hand side of (6) obtaining P(k+1)(i) and P(k+1)(ii).

To check

P(k+1)(iii), consider 37

k+1xli22

-

IIk+2x

t

< E(

a

IIg(u,k+lxu)

< M J

sup

t

tE[O,a]

(S2,C)

J0

- g(u,kxu)IIL2 du,(P(k+1)(ii) and Lerma 2.2)

o

a




(S2,C) .

Thus viewing the right-hand side of (6) as a process in C2(St,C([-r,a], Rn)) and letting k -) co

,

it follows from the above that x must satisfy the

stochastic FDE(L) a.s. for all t E [-r,a]. ti

To prove uniqueness, let x E 9A(2,([-r,a], Rn)) be also a solution of (I) Then it is easy to see by the Lipschitz condition

with initial process 6.

(E(ii)) and Lemma (2.2) that

lixt - xtil2

(9.0

t < ML2 fo llxu

-

112

ti for all t E [O,a].

Therefore we must have xt-xt = 0 for all t E [O,a]; so

x = x in r2(s2,C([-r,a], Rn)) a.s.

The last assertion of the theorem follows immediately from Lemma (2.1). o

Remarks (2.1)

Let 0 < t1 < t < a.

process W E

Then one can solve the following stochastic FDE for any

r2(St,C(J,Rn); Ft ) at time t1:. 1

x(w)(t) = f

t

t1 < t < a

W(w)(0) + (w) Jt1

(II)

ip(w)(t-t1)

t1-r < t < t1

where the (unique) solution x E r2(I,C([t1-r,t1], Rn)).

This gives a family

of maps

W

I

. xt

When t1 = 0, we define Tt, t > 0, to be

Tt = Tt:r2(c,C(J,Rn); FO) ->

c2(Q,C(J,Rn), Ft) 39

The following theorem. on continuation of trajectories of a stochastic FDE is

a consequence of Theorem (2.1).

If 0 < t1 < t2 < a, then

Assume Conditions (E) of §1..

Theorem (2.2):

t1

-T

T

t2 =

t2

oT tt Then for S E J

Let 0 E E2(S2,C(J,Rn); F0) and t1 < t < a.

Proof:

t+s

g(u,Tu(0))dz(-)(u)

t+s > 0

f0

t+st1

re(-)(0) +

t t+s

0 < t+s < t

J 0

t+s < 0

t+s t+s > t1 J =

t1

t

Tt (e)(t+s-t1)

t1-r < t+s < tt

(7)

I

Putting s = 0 in (7), we can compare with (II) for W = Tt1 (e); thus by unique-

ness of solutions of (II) one gets a.s. Tt(e) =

(0)),

Ttt l(Tt

t1

t < a.

o

1

Remark (2.2)

In the case A = 0 (when there is "no drift"), z = zm is an (Ft)0

-martin-

gale, and hence for each e E C2(S2,C(J,Rn);FO) the solution ex E L2(SZ,C([-r,a], Rn)) through 0 is also a martingale on [0,a].

the trajectory t

- ext is not in general a C(J,Rn)-valued (Ft)0Ft), t E [O,a],

is Lipschi.tz; indeed for all t E [O,a], 61,92 E C2(S2,C(J,Rn);F0),

IITt(01) - Tt(02) II

l

2

(52,C)

< -71161 - 6211 2 (2, X

eML2t

(1)

C)

where M is the constant of Lemma (2.2) and L the Lipschitz constant of g. N.B.

Proof:

M does not depend on the coefficient process g but only on the noise z.

The result follows immediately from Gronwall`s Lemma and the obvio',s

inequalities:

sup ITt(e1)(w)(s) - Tt(e2)(w)(s)12 dP(w)

I

S2 sEJ sup le1(w)(s) - 02(w)(s)I2 dP(w)

< 2 fR

+ 2

fQ

sEJ

M

t+s

dP(w)

ssupt,O]I() fo

< 2 Ile1-021122(S2,C) +

t 2ML2

10

IITu(01)-Tu(02)1122(Q.C)du.

o

With suitable Frechet differentiability hypotheses on the coefficient process g, each Tt becomes C1 as in Theorem (3.2) below.

Condition (D):

The coefficient process g has continuous partial derivatives with respect to the second variable i.e. the map

[O,a] x L2(S2,C(J,Rn) ) > L(r2(R,C(J,Rn) ), L2(S2,L(Rm, Rn)) ) (t,W)

1

D(2)9(t,W) 41

is continuous, where D(2)g(t,y,) is the partial derivative of g in the second variable at (t,W).

Suppose g satisfies Conditions (E)(iii.) and (D).

Lemma (3.1):

Then for any

processes y1, y2 E CA([O,a], L2(S2,C(J,Rn))), D(2)9(t,YI(t))(Y2(t)) E L2(S2,L(Rm, Rn);Ft), for every O< t< a.

Proof:

The conclusion of the lemma follows directly from the fact that, for

each 0 < t < a, D(2)g(t,y1(t))(y2(t)) is the L2-limit as h -> 0 of the family

I {9(t,y (t) * hy2(t)) - 9(t,yI(t))} , h 1 0, which belongs to L2(S2,L(Rm, Rn);Ft) by virtue of Condition (E)(iii).

Theorem (3.2):

o

Suppose the stochastic FDE(L) satisfies Conditions (E) and

M. Then for each t E [0,a],

Tt:t2(c2,C(J,Rn);F0)

-> L2(c,C(J,Rn);Ft) +IL2(St,C(J,Rn)1Ft)

is C1.

Proof: L

Fix t E [O,a].

We first prove that T

C(J,Rn)F0) i.e. for each 6,S

t

is Gateaux-differentiable on

E 92(S2,C(J,Rn);F0),

Tt(6+hJ)-Tt(e) lim h-0

-

exists. Because of Conditions (D) and Theorem (2.1), the coefficient process u ---> D(2)g(u,Tu(6)) gives a linear stochastic FDE satisfying Conditions (E); so by Theorem (2.1), there is a unique solution a E L2(S2,C([-r,a]Rn)) of

t

a(')(t) = {

J0

0 < t < a (III)

t E J

for fixed a E L2(0,C(J,Rn);F0). 42

Define Y E CA([O,a], L2(Q,C(J,Rn))) by

Y(t) = at for all t E.[O,a]..

We claim that a Gateaux-derivative

GDTt(6):C2(52,C(J,Rn);FO)

-+

C2(St,C(J)Rn)'Ft)

of Tt at a is given by GDTt(e)(5) = y(t) for all t E [O,a] Using (1.),

(2)

(I.I.1) and Lemma (2.2), it is easy to see that if h ' 0

III [Tt(e+ h8)

t M J

0

Tt(e)] - Y(t)II2 2 L

(Q,C)

Ilht-[g(u,Tu(e+hs))-g(u,Tu(e))]-D(2)g(uTu(e))(y(u))II2du (3)

where M > 0 is independent of-g,0,0. Look at the integrand in (3); viz.

IIF [g(u,Tu(6+h8)) - g(u,Tu(e))] - D(2)g(uTu(O))(y(u))Il

< Fi 1

IIg(u,Tu(e+hB)) -. g(u,Tu(e)) - D(2)g(u,Tu(e))(Tu(0+hB)-Tu(e))II

+ F IID(2)g(u,Tu(6))(TU(O+hB)

Let c > 0 be given.

Tu(6) - hy(u))!I

(4)

Then because of the continuity of the partial derivative

D(2)g and the compactness of [0,a], one can show that there is a 6 > 0 depending on 0, a and independent of u E [O,a] such that if IhI < 6, then IIg(u,Tu(e+h S)) - g(u,Tu(0)) - D(2)g(u,Tu(e))(Tu(6+h6) - Tu(e))II

< c IITu(e+hs) - Tu(e)II

for all u E [0,a].

(5)

If IhI < 6, then (3), (4) and (5) give

[Tt(e+h$)-Tt(e)] - Y(t) 1122

t + 2M

< (s2,C)

M E2

t JO

II2 du

IID(2)g(u,Tu(e))II2IIh1-[Tu(e+h6)-Tu(e)]-Y(u)II2L2(0,C)du. (6) 10

43

Now let N > 0 be such that

IID(2)g(u,Tu(O))II< N for all u E Ma]. Then by Theorem (3.1),

III [Tt(e+h8) - Tt(e)] - Y(t) II2 2 L

< if IhI < 6

e2ML2a

2E2 118112

,

t 2MN2

(n.C )

Jt III T (e+h8)-T (e)]-Y(u)I12 F[ u u r2

(c2,C) du

(7)

t E [0,a].

Apply Gronwall's lemma to (7), obtaining

1/2 EIIBIIeM(L2a+N2t)

II- [Tt(o+h8) - Tt(e)]

Y(t)II < 2M

2N

(8)

for all t E [0,a] if IhI < S.

If we let c -> 0 in (8), lim

IIF [Tt(e*h8) - Tt(e)] - Y(011 = 0

and so y(t) = GDTt(e)(8).

By linearity of D(2)g(uTu(e)) and uniqueness of solutions of (III) it follows that GDTt(0)(8) = y(t) is linear in 8; by continuous dependence (Theorem 3.1) it is also continuous in 8.

Thus for Tt to be C1 it is suff-

icient to demonstrate the continuity of the map

2(cI.C(J.Rn);F0) 3 0 y GDTt(8) E L(r2(SI.C(J.Rn);FO).L2(Q.C(J.Rn);Ft))

So let e, $ E r2(,1,C(J,Rn);FO).

Then for any 8 E 92(Sl,C(J,Rn);FO) the

following estimate holds for all t E [O,a]: II[GDTt(e) - GDTt(0)](8)112

< M ft IID(2)9(uTu(e))(GDTu(e)(8))-D(2)9(uTu(O))(GDTu(0)(8))II2 du 0

44

< 2M I0 IID(2)g(u,Tu(e))(GDTu(e)(0))-D(2)9(u,Tu(O))(GDTu(e) W )II2 du

- GDTu(0)(0)II2 du 2M J D IlD(2)9(u1'Tu(0))112 I1GDTu(e)(s)

(9)

The map u ;-> GDTu(e)(R) = y(u) is a solution of the stochastic FDE (III) whose Lipschitz constant is N; so by Theorem (3.1), IIa1IeMN2

IIGDTu(e)(B)II 0 be given.

U

for all u E [0,a]

(10)

Then by the continuity of D(2)g, the compactness of

[0,a] and Theorem (3.1), we can find d > 0 (depending on e, c but independent of 4,u) such that

t 2 (c,C)

< 6 implies

e (11)

for all u E [0,a].

< S, then (9), (10) and (11) give

Thus if 110-011

I1[GDTt(0) - GDTto)](a)112

2 < 4Me2a 116 112 eMN a

+ 4M Jt {IID(2)g(u,Tu(e))112 +IID(2)9(u,Tu(e))-D(2)9(u;,Tu(O))112} 0

IIGDTu(6)(0) - GDTu(4)(0)II2 du


ti

(2)

J

i.e. t Tt1(W)(M ) =

t

k

Tt1(nj)(w)Xj (w) for a.a. w E S2.

E

j=1

J

Remark In particular, if £2D E Ft , then 1

Tt1(Xn)

=

Tt1(1)X

0 Proof:

+ Tt1(0)X

.

0

0

is Ft -measurable because Sl. E F Solving n J n 1 the stochastic FDE(II) at nj E C(J,R ), we get a solution ix satisfying Let 1 < j < k.

Then X.

t1.

J

n(0) +

n

Jx(')(t)

J

t > t1 (III)'

tl

n(t-t1) Since X.

ft

t1-r < t < t1, a.s.

is Ft -measurable, then the process u

adapted to (Ft)t

-a

is J

1

> t1;

and so by property of the stochastic integral (Theorem 49

(I.8.3)(i)) we get X12

J

a.s.

Jt

Jt t1

t1

J

Using the last relation with (III)'yields

nj(0)Xn nj = {

dw(u),

+ Jt

J

1

t > t1

J

t1-r < t < t1, a.s.

nj(t-t1)X,j ,

Clearly,

k

kn

n G(u,

E G(u, Jx u j=1

for all u > ti.

E Jx u j=1

J

Thus k

k

E G(u,ix n

n.(0)X + ft t E j=1

k E

j

0

j=1

J

j

u

1

t > t , 1

j=1

k E

nj(t-t1)xa ,

j=1

t1-r < t < t1, a.s.

J

rt r

J

t1

kn G(u,

E

j=1

ix

J )dw(u),

u

t1-r < t < ti,

t > t1,

a.s.

Therefore by uniqueness of solutions to stochastic FDE(II) (Theorem (11.2.1)), we obtain k E

n.

t1-r < t < a, a.s.

j=1

J

This implies that

tt

k

Tt(W) = Wxt =

k

t

Tt1(nj)xnJ,

Jxt X0j = jEt

a.s. on Q.

n

t1 < t < a,

jE1

o

We are now ready to prove the main theorem in this section.

50

(The Mzrkov Property):.

Theorem (1.1)

by the stochastic FDE(I). {nxt

:.

Suppose Hypotheses (M) are satisfied

Then its trajectories

t E [0,a], n E C(J,Rn)}

describe a Markov process on C(J,Rn) with transition probabilities given by p(t1,n,t2,B) = P{w:w E S2, Tti(n)(w) E B}

(3)

for 0 < t1 < t2 < a, n E C(J,Rn) and B E Borel C(J,Rn).

Indeed for any

8 E L2(Q,C(J,Rn);F0) the Markov property

P(Tt2 (6) E BIFt

)

= p(t1,Tt

2,B) =

P(Tt2 A)E BITt

1

1

(0))

(4)

1

holds a.s. on 9. Proof:

Observe first that for 0 < t1 < t2 and every n E C(J,Rn),

P o

{Tt2i(n)}F1

and is therefore a probability measure on C(J,Rn) because

We would like to show that, if 0 1,

a:B - R' is uniformly continuous and vanishes outside Ua, and 00

E a(n) = 1, for all n E B. a=1

fm:C(J,Rn)

Define the sequence

100

of functions

R by setting 53

m

nEB fm(n)

n¢.B

0

Clearly fm(n) = 0 for all n ¢

Ua, so each fm is uniformly continuous and

VU

a=

Also it is easy to see that

bounded on C(J,Rn).

0

E 4(n)=1 a

lim

TI EB

a=1

fm(n) _

M+W

n 4 B

0

XB(n) for all n E C(JRn).

=

is a sequence of bounded and uniformly

Therefore fm:C(J,Rn) -r R, m = 1,2,...

continuous functions converging pointwise to XB.

We can therefore replace f

by fm in (7), then pass to the limit as m - co using Lebesgue dominated con-

We obtain (6) in this way for any open set B c C(J,Rn).

vergence theorem.

To get (6) for any Borel set B in C(J,Rn) we reason as follows.

Let

v:Borel C(J,Rn) - R be the set function

v(B) = IA IS, XB{[Tt2(Tt1(e)(w')](w)}dP(w)dP(w'), B E Borel C(J,Rn),

(11) where A E Ft1 is fixed.

Then v is a finite measure on C(J,Rn); to prove this

let {Bi=1 be a countable disjoint class of Borel sets.

Then

m

v( U

Bi) = I

i=1

E

A I

XB {[Tt1(Tt (6)(w'))1(w)}dP(w)dP(w')

i=1

i

2

1

m =

(12)

E v(Bi) i=1

m Since

E XB (n)-* X U B,(n) as m i=1 i i=1 1

for all n E C(J,Rn). then by the dom-

inated convergence theorem we can let m -

in (12) to obtain

00

E

i=1

v(Bi) = lim I

mA

I

E

n i=1

XB {[Tt1(Tt (e)(w'))](w))dP(w)dP(w')

i

2

1

{[Tt1(T

X

fA

tt

U

i=1

54

Bi

t2

(e)(w'))](w)}dP(w)dP(w')

t1

= V( U

Bi).

i=1

Therefore v is a finite (Borel) measure on C(J,Rn).

Since C(J,Rn) is metric,

v is regular (See §(I.2)).

The set function v*:Borel C(J,Rn) - R

v*(B) = JA XB(Tt2(e)(w))dP(w), B E Borel C(J,Rn),

is also a regular (finite) measure on C(J,Rn); so for any B E Borel C(J,Rn), * v (B) = inf{v*(U): U c C(J,Rn) open, B c U}

= inf {v(U): U c C(J,Rn) open, B c U}

= v(B). Therefore (6) holds for all Borel sets B in C(J,Rn). To see that P(t1,Tt1(e)(.),t2,B) = P(Tt2(e) E BIT

ti

(0)) a.s.

it is now sufficient to check that the left hand side p(tl,T

ti

(e)(-),t

29

B)

is measurable with respect to the a-algebra generated by Tt (e); this is valid because the function

1

C(J,Rn) 3 n f------ p(t1,n,t2,B) = JQ x (Tti(n)(w))dP(w) is measurable: If B c C(J,Rn) is open, let {fm}m=1 be a sequence of uniformly bounded, uniformly continuous functions fm:C(J,Rn) -r R converging pointwise to XB.

Therefore t

f m (Tt1(n)(w))dP(w).

P(t1,n,t2,B) = lim J m

2

t

If nk + n in C(J,Rn), then Tt1(nk) -T t1(n) in Z2 (0,C); so f (Tt1(nk)()) m t2

2

2

t

fm(Tt1(n)(w)) as k - - in probability; so by Lebesgue dominated convergence 2

theorem, for each m,

55

f (Ttl(n)(w))dP(w) as k

f (Ttl(nk)(w))dP(w) Sm t2

J

m

J

t2

Hence n - fJ fm(Tt2(n)(w))dP(w) is continuous (for each m).

Therefore

is measurable because it is a pointwise limit of continuous functions.

If B E Borel C(J,Rn), then

p(t1,n,t2,B) = inf {p(t1,n,t2,U): U open in C(J,Rn), B c U}.

But

must be measurable.

each n - p(ti,n,t2,U) is measurable, so Therefore P(t1,Ttl

E[P(Tt2 (0) E BIFt1 )ITti (B)]

= P(Tt2 (0) E BITti (0)),

since the o-algebra generated by T

t1

(0) is contained in F

t1

Finally we check the Chapman-Kolmogorov identity P(t1,n,t2,B) = JCEC P(u,C,t2,B)

for all B E Borel C(J,Rn) 0 < t1 < u < t2 < a, and then construct a canonical

Markov process on C(J,Rn) with the same distributions as the trajectory {nxt:t E [0,a], n E C(J,Rn)}.

Note that if 0 < t1 < u < t2 < a, then

Tt1(n) = Tt (Tul(n)) a.s. and as in (6) we get 2

2

t

J 52

XB(Ttl(n)(w))O(w) = J t2

,

w ES2

XB([Tt (T u1(n)(w'))](w))dP(w)dP(w') 2 LE92 (6)'

Therefore, t

CEC

SEC

56

P(u,c,t2,B)

J

J

p(u,Tu1()('),t2,B)dP(w')

=

L.

-

Jw' Ec,wES2

X B([Tut2 (Tt1(n)(w'))](w))dP(w)dP(w') u

XB(Ttt2(n)(w))dP(w)

= P(t1,n,t2,B).

Transform the filtered probability space (S2,F,(Ft)

,P) using the family

of solutions nx:.S2+ C([-r,a],R") to the stochastic FDE(LI.) at n E C(J,Rn), n,

=

viz.. let

C([-r,a],R"), F = Borel Q. pu:C([-r,a],R") - R", U E [-r,a], be

evaluations at u and Ft = a{pu:-r < u

i

= 1,2,

are isonomous.

Proof:

Suppose B1,...,Bk E Borel F, tl,t2,...,tk E T.

measurable, then

Since L(t,.) is

E Borel E for all 1 < j < k, so that by isonomy

of x1 and x2 we have E =

E Bk)

E

E

E

E

E Bk).

E

Lemma (2.2):

Suppose x1,x2:[-r,a] x S2

0

Rn have a.a. sample paths continuous.

Then the following statements are equivalent: 59

(i)

and x2 are isonomous .

x1

(ii)

have the same

The induced random variables x , x2 :S2 -+

distribution.

(iii) The processes

[O,a] X 9 -- C(J,R") (t,w) I, xt(),

i

= 1,2,

are isonomous.

We prove the implications (i) -*

Proof:

(ii) ->

(iii) * (i).

Since x1, x2 have a.a. sample paths continuous, then

Suppose x1 - x2.

- C([-r,a],Rn) are (Borel) measurable; these will have the same

x1, x2:n

distribution if and only if the measures P o (x1)-1, P o (x2)-1 agree on cylinder sets in C([-r,a],R"). 212).

(See Parthasarathy [66], Theorem (2.1), p.

For each t E [-r,a],let Pt:C([-r,a],Rn)

-,

R"

Let t1,t2,...,tk E [-r,a], B1,B2,...,Bk E Borel R".

be evaluation at t. Then k

P

(x 1)-1[ n

j=1

pt1(Bj)] = P(x1 E Pt1(B.) for all 1 < j < k) J

j

=

E Bj for all 1 < j < k)

=

E Bj for all 1 < j < k) k

= P o (x2)-1

[ n Pt1(B)M .

j=1

Therefore P

(x1)-1

j

= P o (x2)-1.

Suppose now that P o (x1)-1 = P o (x2)-1.

m:[O,a] x C([-r,a],R') -> C(J,Rn) at 60

The memory

is continuous (and hence measurable)} so i.f B1.9 ...,Bk E Borel C(J,Rn),

tl,...,tk E [O,a], then

E Borel C([-r,a],Rn) for all

1<j xt(w), i = 1,2, are isonomous.

The implication '(iii) = (i)' follows trivially from Lemma (2.1).

a

Notation

E, F are B-spaces.

For any stochastic processes y1:T x 2 -*- E, y2:T x 0 -, F

define the process y1 x y2:(T x T) x S2+ E x F (on T x T) by (y1

x y2)(t1,t2,w) = (y1(t1,w), y2(t2,w)) for all t1,t2 E T,

Lemma (2.3):

W E 0.

C([O,a],R), i = 1,2, be processes satisfying

Let z1: S2

condition (E)(i) of existence of McShane's integral (Theorem (1.8.4)). Suppose f':[O,a]

C2(c,Rn) are t2-continuous processes adapted to (Ft)tE[O,a]'

Define the processes F1:[O,a] x S2 - Rn, i = 1,2, by

t F'(t,w) = (w) J

t E [O,a],

a.a. W E s2.

0

If f1 X z1 is isonomous to f2 x Z2, then F1 X z.1 is isonomous to F2 x

Proof: n

We approximate each FI by Riemann sums. a

z2.

For each integer k, let

2a

- (0, k, -k-,...,a) be a (Cauchy) partition of [0,a] into equal subdivisions.

For each t E [O,a], let m(t,k) be the greatest integer such that

m(t,k)a < t
L(Rm,Rn) is Lipschitz and

Lemma (2.4):

z1:St- C([O,a],Rm), i = 1,2, are processes satisfying the conditions of For each n c C(J,Rn) let {1Tt(n)}tE[O,a]' i = 1,2,

Existence E(i)

be the trajectories of the stochastic FDE's: rt

n(O) + (w) J

t E J, i= 1,2,

n(t) If z1 and z2

0 < t < a

(u)

0

r 'x(w)(t) = j

a.a. W E Q.

are isonomous, then {1Tt(n)}tE[0,a]' {2Tt(n)}tE[0,a] are also

isonomous.

Proof:

Use the method of proof of Theorem (11.2.1).

there are sequences ixk:11

(i)

i k+1

x

-

i k

2

x

< (ML

2

ak-1

2 k-1

)

!

i

x

2_i

lxk x zl .., 2xk

x

2 x II2 1

z2,

i = 1,2, 1

L 01,C)

t (11,C)

(ii)

Assume inductively that

C([-r,a],Rn), k = 1,2,..., i = 1,2, such that

where

n(0)

t E [O,a]

n(t)

t E J

ix1(w)(t) = {

a.a. w E 0, i = 1,2. Indeed we define t

n(0) + (w) j

G(

i

tE

i

E S2.

(i) holds by the proof of Theorem (11.2.1). To prove (ii), assume it is valid for some k > 1.

Therefore 1xk x

z1

_ 2xk

x

z2 implies 63

x

(1xt

z1

.

x z2

(2xk

m

1

(Lemma 2.2)

m

1

Hence (G(1xt

x

1-1

(G(2xt

)x z1 ti G(2xk,))

,,, 2xk+1 x Thus 1 xk+1 x z1

(.)))x z2, m

1

for t1,...,tm E [O,a], by Lemma (2.1); G(Ixk

-

m

1

i.e.

z2.

z2 because of Lemma (2.3).

Obviously (ii) is valid for k = 1 because (n,z1) - (n,z2).

Therefore

(i) and (ii) hold for all k. In particular (ii) implies 1xk _ 2xk for all = P o (2xk)-1 k > 1. Thus for all k > 1. Po(Ixk)-1

Now ixk

>

ix as k -)...o a.s. (or in L2) from the proof of Theorem (lxk)-1

(11.2.1); so P o

> P o (1x)-1, i = 1,2, as k - w in the weak

topology of Borel measures on C([-r,a],Rn). x)-1

we must have P o

(1

(2.2), {1Tt(n)}tE[O,a]

{t -> 2xt}.

_

Hence by uniqueness of limits,

= P o (2x)-1 on Borel C([-r,a],Rn).

{t E-->

So by Lemma

1xt} is isonomous to {2Tt(n)}tE[O,a]

_

o

The following theorem is our second main result; it says that the Markov process given by trajectories of the autonomous stochastic FDE (IV) is in fact time-homogeneous in the sense that each transition probability p(t1,n,t2,.), t1 < t2, n E C(J,Rn) depends on t2 - t1 (and n) only.

Theorem (2.1)

(Time-homogeneity):

FDE (IV) satisfies Hypotheses (A).

Suppose that the autonomous stochastic For 0 < t1 < t2 < a, n E C(J,Rn) let

p(t1,n,t2,.) be the associated transition probabilities of trajectories of (IV) (as given by Theorem (1.1)).

Then the trajectory Markov process {nxt:0 < t < a, n E C(J,Rn)} is timehomogeneous, i.e. 0 < t1 < t2 < a, n E C(J,Rn). t1x:;2

Proof: 64

Let n E C(J,Rn), t1 < t2.

Suppose

->

C([t1-r,t1],Rn) is the

solution of t

n(0) +

G(t

t > t1

j

t1

t1

X%.) _ { t1-r < t < t1

n(t-t1) t-t1

t > t1

G(

n(O) + j

ii

t-r w(u'+t1)-w(t1) is isonomous to w (Elworthy [19], Corollary 3A, p. 19). So comparing the above stochastic FDE with our original one viz: Jt,

n(0) +

X(-)(t')

0 < to < a

= {

0

to E J

n(t')

it then follows from Lemma (2.4) that the trajectories {Tt(n)}0 0,

J

A

lim

0 uniformly in A E A.

t-'0+ lim

t-'0+

I

uniformly in A E A.

66

t

4)(fX(w))dP(w) = J

JQ

4)(fA(w))dP(w)

n

Moreover,

Proof : (i)

By uniform continuity of 0, there is a 6' > 0 such

Let c > 0 be given.

that whenever E1,E2 E E and A2-

'

1-E2IE < d', then I4(

I

1)

- 0(&2)I < c

But

fx is uniformly continuous, so there is a d > 0 such that if

X,A' E A and d(A,A') < d , then t- IfX(w) - f

(w)12 dP(w) < c6'2, where d

Therefore, if d(A,A') < d, we get

is the metric on A.

If'(w)-fX1(w)IE

> d'} < _6L-2 J

P{w:w En,

If'(w)-f' (w)IE dP(w) < E S2

by the Chebyshev's inequality.

f9

Now i.f d.(A,A' < d we have

I0(f'`(w) - 4(fX1(w)) (w))IdP(w)

j1f_fA

=

IdP() CIE>d1

o of the above theorem will be studied in some detail in the next chapter.

Let M(C) be the complete topological vector space of all finite Borel measures on C

C(J,Rn) given the weak * (or vague) topology (§I.2, Partha-

sarathy [66], Schwartz [71], Stroock and Varadhan [73]). bilinear pairing 68

Then we have a

Cb X M(C) - > R, = jC 0(n)du(n), E Cb, P E M(C).

Following Dynki.n [16], define the adjoint semigroup

Pt*1(kt 0,

Observe also that the family of transition probabilities

E C(J,Rn), t > 0} for (IV) is left invariant by the semigroup

{Pt}t>0

when a = .

It would be interesting to find generic conditions on the coefficients H,

G of (IV) which guarantee the existence of a (unique) invariant probability measure.

Some partial results in this connection may be found among the

examples of Chapter VI (§VI.4).

See also Ito and Nisio ([41]) and Scheutzow

[69]).

69

IV The infinitesimal generator

§1.

Notation

For the present chapter we keep the notation, general set-up and the standIn particular, we focus our

ing assumptions of the last chapter (§III 1,2). attention on the autonomous stochastic FDE: r

f

n(0) + 10 H(xu(w))du + (w) x(w)(t)

t > 0

JO

(I)

1

t E J = [-r,0]

n(t)

Very frequently, the solution x through n will be denoted by nx; and throughout the chapter we shall assume that the coefficients H:C(J,Rn) G:C(J,Rn)

-

L(Rm,Rn) are globally bounded and Lipschitz.

>Rn,

The driving

Brownian motion w is in Rm, generating a filtration (F t)1>0 on the probability space (SZ,F,P).

For brevity, symbolize the above stochastic functional equa-

tion by the differential notation. dx(t) = H(xt)dt + G(xt)dw(t)

t > 0 1 (I)

x0 = n E C(J,Rn)

J

Similarly for any t1 > 0 we represent the equation n(0)+Jt

H(xu(w))du + (w)

XMM n(t-t1)

t1

Jt

t1-r < t < t1

by the stochastic-differential notation dx(t) = H(xt)dt + G(xt)dw(t) xt

t > t1 > 0

= n E C(J,Rn)

Recall that at the end of the previous chapter, we constructed a contraction semigroup {PtIt>O associated with the stochastic FDE (I) and defined on the Banach space Cb of all bounded uniformly continuous functions c:C(J,Rn) --.> R. 70

Indeed

Pt(m)(n) = E[ a

n E C(J,Rn),

t > 0.

Now, for an ordinary stochastic differential equation (Stochastic ODE, r = 0), it is well-known that the semigroup {Pt}t>O is strongly continuous on Cb with respect to the supremum norm, and its strong infinitesimal generator is a second order partial differential operator on the state space of the solution process ([16], [22]).

Our first objective is to show that, when r

0, the

semigroup {Pt}t>0 is never strongly continuous on the Banach space Cb. Furthermore, we shall derive an explicit formula for the (weak) infinitesimal

generator of {Pt}y.

§2.

Continuity of the Semigroup

For each n E C(J,Rn) and t > 0, define n:[-r,co) ; Rn by

ti

n(0)

t > 0

n(t)

t E J

n(t) = t

1

-r

t-r

0

t

Define the shift St:Cb s Cb, t > 0, by setting StM (N ) _ ,(nt), n E C(J,Rn), 0 E Cb.

The next result then gives a canonical characterization for the strong continuity of {Pt}

t>O

in terms of the shifts {St}t>0.

71

The shifts {St}t>0 form a contraction semigroup on Cb, such

Theorem (2.1):

t-,o+

t-'0+

t

P (4))(n) = $(n) for all

Pt(0)(n) = $(n) uniformly in Ti E C(J,Rn) if and

Furthermore lim

0 E Cb.

t

S MN N ) = lim

that, for each n E C(J,Rn), lim

St(4))(n) = ON uniformly in n E C(J,Rn).

only if lim

t-0+ Let t1,t2 > 0, n E C(J,Rn), 4) E Cb, S E J.

Proof:

Then

ti (St2 (4)))(n)

St

$C(nt1 )t2 ]

=

1

where ti

t2 + s > 0

n(0)

,..

(nt )t (s)

ti

2

1

-r< t2 + S < 0

(nt1)(t2 + s)

t2+s>0

In(0)

ti =

t1 + t2 + s > 0

nt1 (0) = n(0)

ti nt (t2+s) = n(t1+t2+s)

-r < t1 + t2 + s < 0

1

_ (nt1 +t )(s). 2

Hence I,

St1 (Si. (4)))(n) = (nt +t ) = St1 +t 2

1

2

MN

i.e.

St2 =

St1

St1+tz.

ti

n = n , it is clear that lim S ($)(n) = (n) for each t-$.0+ t-*o+ n E C(J,Rn), 0 E Cb. Also by sample paths continuity of the trajectory Since lira

t

t

{nxt:t > 0} of (I) (Theorem (11.2.1)) together with the dominated convergence theorem, one obtains

lim YON t ) = lim t-,0+ 72

J (nx S2

t

(w))dP(w) _ 4)(n)

for each 4) E Cb and n E C(J,Rn).

To prove the second part of the theorem, suppose K > 0 is such that Then for each t > 0 and

IH(n)I < K and IIG(n)II< K for all n E C(J,Rn).

almost all w E 0 we have Jt+sH(nx

r

u

0

(w))du+(w)

Jt+sG(nx

0

t+sa0

u

nxt(w)(s)-nt(s) = J

l

-r 1. 1

- W( n ')I = I

4,01 )

n(s 0)

n(s0) I

n(s0)

In(s0)1 IIn II IIn' 1

IIn' II

Illn'11

-

IInII I

+

III

On' - nil

Iln' II (iii) n E int B, n' ¢ B; i.e.

74

I

n(s0)

+

11

II n'Il I

11n II

-L n'( s )

-

lln


1.

Find n" E aB

(where 8B is the boundary of B) such that n" lies on the line segment joining n and n' i.e. find A0 E [0,1].such that n" = (1-A0)n + A0n' and IIn"II

= 1.

Define the function f:[0,1] --> R by f(A) = 11(1-A)n + an'Ii-1, A E [0,1].

Then f is clearly continuous. Also f(0) = IIn II - t < 0 and f(1) = III II -1 >0. Hence by the Intermediate-Value Theorem there exists a0 E (0,1) such that f(A0) = 0 i.e. take n" = (1-A0)n + a0n' and IIn"II

W(n) - W(n')l

= 1.

Hence

IW(n) - W(n")I t IW(n") - W(n')II

IIn_n"11 +211n"-n'II =[IIn -n"11 +

IIn"- n' II

]+ IIn"-n' II

= IIn -n'II + IIn" -n'II < 211n -n'II Therefore IW(n) - W(n')I < 211n - n'11 for all n,n' E C(JRn). Let {nn}rt=1 be a sequence in C(J,Rn) looking like this:

)

-r

b

s

s,+(1/n)

0

i.e.

s

0

nn(s) ={

+1<s 1.

{nk}k00

Then {Ck}k=

=1 are bounded sequences in

C*(s), n (s) - n*(s) as k - , for all s E J.

C(J,R ) such that C (s) -

So

ti

by (w2) for S and

one gets

S(C*)(n*) = lim

s(Ek)(nk) = S(C*)(n*)

k-

Thus W = S.

a

For each n E C(J,Rn) let nx E 92(9,C([-r,a],Rn)) be the solution of (I) ti

through n, and nt E C(J,Rn) be defined as in §2 for each t E [O,a].

Lemma (3.3):

There is a K > 0 (independent of t,n) such that

IIt E(nxt - nt)IIC < K for all t > 0 and n E C(J,R").

Also if a E C(J,Rn)*, then ti

lim

t Ea(nxt - nt) = a(H(n)X{0}) for each n E C(J,Rn).

t-O+ Proof:

84

Denote by E the expectation for Rn or C(J,Rn)-valued random variables.

Let K > 0 be such that IH (n)I < K for all n E C(J,R').

t+s

t+s H(x (w))du + (w)

J

nxt(w)(s)-nt(s) ={

Now

u

0

G(nx

J0

t+s>O u

t+s 0

0

0

t + s 0

,

using the Martingale property of the Ito integral.

Thus

t

lim

lim

ntt-O+

[E{t(nxt

0

t fo

E(H(nxu))du

s = 0

-r < s < 0

H(n)X{0}(s), for all s c J. ti

We prove next that IIt E(nxt - nt)IIc is bounded in t > 0 and Ti.

Clearly

t+s

I[t E(nxt-nt)](s)I < tl f

IE(H(nxu))Idul < t JIE(H(nxu))Idu O

< K for all t > 0 and n E C(J,Rn). ti

Therefore IIt E(nxt - nt)IIC < K for all t > 0 and n E C(JRn)

If a E C(J,Rn) *, then 1

A, I Ea(nxt - nt) = t a(E(nxt - nt)) = aCt E(nxt - nt)]

By the weak continuity property (w1), one gets

1im t Ea(nxt - nt) = a(H(n)X{0}).

o

85

For each t > 0 and a.a. W E 11, define wt(w) E C(J,Rn) by

Lemma (3.4):

[w(w)(t+s) - w(w)(0)]

-t < s < 0

wt() (s)

-r<s< -t

0

Then

Let B be a continuous bilinear form on C(J,Rn).

[t ER(nxt - nt, nxt - nt) - ER(G(n) o wt, G(n) o wt)] = 0.

lim

Proof:

We prove first that

lim

E II

(nxt - nt) - G(n) o wt II 2 = 0

(1)

Observe that

1

t+s

(n x H(x )du + -

r0

u

I0

n

t+s

G( x )dw(u)

I

u

t+s

dw(u)],

G(n)[

-t < s < 0

t to

-r<s< -t almost surely.

Therefore

I tnxt - nt)(s) - G(n) o wt(s)I2 E sup t+s

1

< 2

t

E

sup SEE-t,03

I

fo

n

2

t+s 1

+2 tE

sup sE[-t,0]

n

[G(xu)-G(n)]dw(u)I

II

0

+2 K1 It EIIG(nxu)


0, and some K1 > 0. Since H and G are Lipschitz with Lipschitz constant L, then almost surely 2IH(n)12 + 2L211 nxu - n,12,

u > 0;

and

IIG(r'xu) - G(n) 112 < L2 IInxu -nII

86

, u > 0.

2

Hence

EIH(nxu)I2 < 2IH(n)12

2L2E Ilnxu -n112

+

(3)

and

2 11C E IIG(nxu) - G(n)112 < L E I I nxu _n

(4)

Now by the main existence theorem (Theorem (11.2.1)), the map

[0,a] 3u- xuEr2(St,C(J,Fn)) is continuous; so uin

Ellnxu-nfl2=0. Therefore the

last two inequalities (3) and (4) imply that {EIH(nxu)12: U E [0,a]} is bounded and lim EII G(nxu) - G(n)II2 = 0. u-O+ Letting t - 0+ in (2) yields (1). Since B is bilinear,

8(71 xt

- nt, nxt - nt) - B(G(n) ° wt, G(n) ° wt) 11,

= B( (nxt - nt) - G(n) ° wt,

+ B(

(nxt

1(nxt-nt)-G(n)°wt,G(n)°wt)

-

nt) - G(n) ° wt)

+ B(G(n)°wt, *(nxt-nt)-G(n)°wt). rt

/t-

Thus, by continuity of B and Holder's inequality, one gets It E8(nxt - nt, nxt - nt) - EB(G(n) ° wt, G(n) ° wt)I

< 11811

Ell -L (nxt - nt) - G(n) ° w*ll2

+ 2118 11 [Ell t (nxt-nt) - G(n)°wtII2] 1/2 [E IIG(n)°wtII2] 1/2,

(5)

for all t > 0. But

E IIG(n) ° wtll2 < E

sup

I

SE[-t,0]

lw(t±s)-w(0)12 IIG(n)112
0.

t

t IIG(n)112

(6)

87

Combining (6) and (5) and letting t - 0+ gives the required result.

Lemma (3.5): Let in:Rn

o

Fn be the isomorphism in(v) = vX{0}, v E Rn, and

G(n) X G(n) denote the linear map

Rn

x

Rn

(v1,v2)

Rn X J n

--> (G(n)(v1), G(n)(v2)).

Then for any continuous bilinear form a on C(J,Rn) 1im

t

ES(nxt-ntnxt-nt) = trace

(in x in) ° (G(n) X G(n))]

for each n E C(J,Rn), where ; is the continuous bilinear extension of 0 to C(J,Rn) ® Fn (Lemma (3.2)).

Proof:

m E

ES(nxt-nt,nxt-nt) =

lim

t-0+

Indeed if

{e}M=1

is any basis for Rm, then

8(G(n)(ej)X{o},G(n)(ej)X{0}).

j=1

t

In view of Lemma (3.4) it is sufficient to prove that m lim

t-o+

ES(A ° wt, A ° wt) =

Z

R(A(ej)X{0}, A(ej)X{0})

(7)

j=1

for any A E L(Rm,Rn).

We deal first with the case m = n = 1, viz. we show that lim

(7)'

ES(wt, wt) = B(X{0},X{0})

t- O+ for one-dimensional Brownian motion w.

If C, n E C = C(J,R), let C 0 n stand

for the function J X J -; R defined by (C ® n)(s,s') = C(s)n(s')for all s, s' E J.

The projective tensor product C ®a C is the vector space of all

functions of the form

EN=1

Ci ® ni where Ci,ni E C,

i

= 1,2,...,N.

It carries the norm N

N

IIhII®r = inf {1 E1 IIEiII Ilni II :h = E1 Ci ® ni,Ci,ni EC, The infimum is taken over all possible finite representations of h E C 0 Denote by C

C the completion of C aff C under the above norm.

C.

It is well

known that C e,,C is continuously and densely embedded in C(J X J,R), the

Banach space of all continuous functions J X J - R under the supremum norm 88

(Treves [75], pp. 403-410; §1.4).

Since C is a separable Banach space, so is C an C. countable dense subset of C.

For let Y c C be a

Then the countable set

N

0 ni : Ci,ni E Y,

Y 0 Y = { E

i

= 1,...,N, N = 1,2,...}

i

i=1

is dense in C ®n C and hence in C Off C.

The continuous bilinear form B on C corresponds to a continuous linear functional a E [C &r C]* (Treves [75] pp. 434-445; Cf. Theorem 1.43).

The map

Now let W1,Vr2 E C2(S2,C).

C®C (E,n)

CO,R C

I -->

®n

is clearly continuous bilinear.

Thus

> C ®n C

st

W I--> W1(w) 0 q)2 (W) is Borel measurable.

But

II'P1(w) a w2(w) II On

< II'P1(w) II

IIW2(w) II

for almost all w E St; hence by Holder's inequality the integral

f

IIW1 (w) ®'P2(w) II

®7r

dP(w) exists and

SZ

JSI

11V1(w) ® V2 (W) II

ar dP(w) < [ Jii1 (w) 112 dP(w)]l[ JnIIV2 (w)II2dP(w)]

From the separability of C of C the Bochner integral (§I.2) W1(w) ® W2(w)dP(w)

E

JSZ

exists in Cti®,R C.

Furthermore, it commutes with the continuous linear

functional S; viz.

0

(8)

89

,

Fix 0 < t < r and consider

E [-t,0]

t s E [-r,-t) or s' E [-r,-t).

0

1

+

min (s,s')

s,s' E [-t,0]

t

s E [-r,-t) or s' E [-r,-t)

0

_ [1 + t min (s,s')]X[_t,0](s)

X[_t,0](s'),

(9)

s,s' E J

Define Kt:J x J - R by letting Kt(s,s') = [1 +

min (s,s')]X[_t,01(s)X[_t,01(s')

s,s' E J

(10)

t i.e.

Kt = E C2(c,C), it is clear from (8) that Kt E C ®7r C and I'll

S(Kt)

wt

(11)

.

ti

In order to calculate lim

8(Kt), we shall obtain a series expansion of Kt.

t-'0+ We appeal to the following classical technique.

Note that Kt is continuous; so we can consider the eigenvalue problem 0

s E J

-r Kt(s,s')E(s')ds' = Xe(s)

(12)

Since the kernel Kt is symmetric, all eigenvalues A of (12) are real.

Using

(10) rewrite (12) in the form

t fO

-t

+ fO

-t

(s')ds' = Ate(s)

XE(s) = 0 Therefore

90

s E [-t,0] (i)

s E [-r,-t)

(ii)1

(13)

S

t JO

t

F(s')ds' + f

-t

s'E(s')ds' + s

E(s')ds' = Ate(s)

D

fs

S E [-t,0]

(14)

Differentiate (14) with respect to s, keeping t fixed, to obtain D

(= fs')ds' at &'(s),

s E (-t,0]

(15)

s E (-t,0]

(16)

S

Differentiating once more,

-&(s) = At

"(s)

,

When A = 0, choose ED:J - R to be any continuous function such that E0(s) = 0 for all s E [-t,0] and normalized by

-t

J_r0(s)2ds=1.

Suppose A # 0.

Then (13)(ii) implies that

c(s) = 0 for all s E [-r,-t)

(17)

In (14) put s = -t to get &(-t) = 0

(18)

In (15) put s = 0 to obtain '(0) = 0

(19)

Hence for A # 0, (12) is equivalent to the differential equation (16) coupled with the conditions (17), (18) and (19). Aet-a'iis

+

Now solutions of this are given by

A2e-t'ia-his,

s E [-t,0]

c(s) = {

(20)

s E [-r,-t),

0

i = -T

= A2 = 1, say.

Condition (19) implies immediately that A

From (18) one gets

1

e-t-x-iit

+

etx-iit

= 0.

Since the real exponential function has no zeros, it follows that a' cannot be 91

imaginary i.e. A > 0.

Being a covariance function, each kernel Kt is non-

negative definite in the sense that

JJ JJ

Kt(s,s')E(s)C(s')ds ds' > 0 for all E E C.

Using (18), we get the eigenvalues of (12) as solutions of the equation

2 cos [ 1 (-t)] = 0 VTF Therefore the eigenvalues of (12) are given by 4t

k = 0,1,2,3,...

(21)

and the corresponding eigenfunctions by

k(s)=(t)

X[-t,03(s) cos

(2k+l

2T -

s],

s E J, k = 0,1,2,...

(22)

after being normalized through the condition Ck(s)2ds = 1, k = 0,1,2,...

J J

Now, by Mercer's theorem (Courant and Hilbert [10] p. 138, Riesz and Sz-Nagy [68] p. 245), the continuous non-negative definite kernel Kt can be expanded as a uniformly and absolutely convergent series 00

Kt(s,s') =

Xk &t (s) Ck(s'),

E

s,s' E J

(23)

k=0 E

8

k=0

Tr (2k+I)Z

cos

[ (2k+1)uts ]

cos

[ (2k+1)"SI

2t

]

s,s' E [-t,0]

{

(24)

s E [-r,-t) or s' E [-r,-t)

0

But from the definition of Kt, one has Kt(0,0) = 1 for every t > 0.

Thus

putting s = s' = 0 in (24) we obtain E

k=0

8

2-

1

(25)

it (2k+1) 2k+1)

From the absolute and uniform convergence of (24), it is easy to see that Kt can be expressed in the form 92

Kt = k=E 0

8 11-t n2(2---Z Ek ® Ck

(26)

where

k(s) = cos

)ns]X[-t,0](s).

C( 2k

S E J.

Note that the series (26) converges (absolutely) in the projective tensor product norm on C ®n C.

Hence we can apply S to (26) getting from (11) the

equality

8 E

z

,

k=0

7r

t Wk a Y ''

(2k+1) ti

=

E

k=0 But (I kJI
, k = 1,...,m, are independent one-dimensional Brownian

Then

motions.

n

EB(A ° wt, A ° wt) = E

S'J((A ° wt)', (A ° wt)J)

Z

t,j=1 m E = E ''j-1

m

k*

w

t

, k

i

E w h=1

h* t

)

m

n E i

E (k=1

Sij

E

,j =1

n

m

E

E k=1

i,j=1

ES'3(wt*, wt*)

k , h=1

k*

k*

Letting t - 0+ and using (7)' gives

m lim

ES(A°wt,A°wt) =

t-,0+

n

E

E

k=1

i,j=1

B1-J(X{0},X{0})

m

E k=1

S(A(ek)X{0}' A(ek)X{0}) = trace

(in x in) o (A x A)].

To obtain the final statement of the lemma, take A = G(n) and note that the last trace term is independent of the choice of basis in Rm.

o

Let V(S) c Cb be the domain of the weak generator S for the shift semigroup {St}

t>O

of §2.

We can now state our main theorem which basically says

that if 0 E V(S) is sufficiently smooth, then it is automatically in V(A). Furthermore, A is equal to S plus a second order partia: differential operator on C(J,Rn) taken along the canonical direction Fn.

The following conditions

on a function :C(J,Rn) -R are needed.

Conditions (DA): (i)

0 E V(S);

(ii)

0 is C2;

(iii) DO, D24 are globally bounded; (iv)

D20 is globally Lipschitz on C(J,Rn).

Theorem (3.2): Suppose q:C(J,Rn) --? R satisfies Conditions (DA).

94

Then

D(A) and for each n E C(J,Rn)

A(4)(n) = S( )(n)+(DO(n)oin)(H(n))+}trace[D2 (n)o(inxin)o(G(n) xG(n))] 2

where Di(n), D fi(n) denote the canonical weakly continuous extensions of DO(n) and D20(n) to C(J,Rn) ® Fn and in:Rn - Fn is the natural identification

v i.--> vX{0} .

is any basis for Rn, then

Indeed if

n

AM(n) = S($)(n)+DO(n)(H(n)X{0})+7j

E C(J,Rn) and let nx be the solution of the SRFDE (I) through

Suppose 0 satisfies (DA).

n.

Since 0 is C2, then by Taylor's Theorem (Lang [47]) we have 116

(nxt) - ON = (nt) -O(n)+ D$(nt)(xt n

- nt) + R2(t)

a.s.,

t > 0,

where

R2(t) = j1 (1-u)D24[nt + u(nxt - nt)](nxt-nt,nxt-nt)du

a.s.

(28)

Taking expectations, we obtain

t E[O(nxt)-O(n)] = t St(4)(n)-4(n)] + t DO(nt)(nxt-nt) + t R2(t) (29) Since 0 E D(S), then

lim !-[S M(n) - 4(n)] = SM(n)

t-,0+t

t

In order to calculate lim

t-'0+

t

(30)

t

1 E[O(nx) - (n)], one needs to work out the

following two limits

lim u0+ t ED4(nt)(xt lim

1

t-,0+t

- nt)

ER (t) 2

(31)

(32)

We start by considering (31).

From Lemma (3.3), there exists a K > 0 such

that

Il t E(nxt - nt) II

.

K

for all t > 0. 95

Hence It EDO(n t)(nxt - nt) - t ED4(n)(nx t -

ID$(nt)[tE(xt - nt)]

-

nt)I

DO(n)[t E(nxt - nt)]I

for all t > 0.

K IID$(nt) - DO(n)II

Let t - 0+ and use the continuity of DO at n to obtain

lim t

EDO(nt)(nxt - nt) _ 1imm+ t ED$()(nxt - nt

= D$(n)H(n)Xf0))

(33)

by Lemma (3.3).

Secondly, we look at the limit (32).

Observe that if K is a bound for H

and G on C(J,Rn) and 0 < t < r, then ti

4

EIIIt - ntll

< 8E

IJt+s

= E

sup SE[-t,0]

n

n 4 H(xu)du + jt+s G(xu)dw(u)I

0

t+s t+s n 4 n 4 + 8E sup H( xu)duI G( x )dw(u)I sup If Ij sE[-t,O] 0 0 u SEC-t,0]

< 8K4t4 + 8K2t

ft

Ell G(nxu)

114du

0

some K2, K > 0,

< K(t4 + t2),

(34)

Furthermore, if

where we have used Theorem (1.8.5) for the Ito integral. u E [0,1] and 0 < t < r, then 1ED20(n

t

t+u(nx-n t t))(nxt -n

< t E IID2$(nt < [E

IID241 (nt +

+

t t

nx -n )

t,

u(nxt-nt)) -

-

D2O(w ) II

t

t

jED2O(n)(nx -n

t,

II nxt-ntII2

u(nxt-nt))-D2m(n)II2]1/2[_! Eli nxt ntlI4]1/2

t

< K1/2(t2+1)1/2[E IID24(nt+u(nxt-nt)) - D20 (n)II271/2. 96

t t

nx -n )I

(35)

But D2$ is globally Lipschitz, with Lipschi.tz constant L say; so

-

E II D2$(nt+u(nxt-nt)

D2$(n) II 2 < L2E( II nt-n II +

II nxt-ntll) 2

< 2L2 Ilnt - nll2 + 2L2[Ellnxt - ntll4]1/2 < 2L2 lint - n112 + 2L2k1/2t(t2 + 1)1/2 because of the inequality (34).

lim t

Letting t - 0+ in (35) and (36), we obtain

ED2$(nt + u(nxt - nt))(nxt -

= lim ED2$(n)(nxt t-)-0+

uniformly in u E [0,1].

(36)

-

nt'

nxt

-

nt

nt, nxt - nt)

(37)

From this and Lemma (3.5), one gets 1

lim

t_0+

t 1

ER (t) = 2

1 ED2$(n)(nx -n ,nx -n )du

0

_ 7 E

D

t t t t

t

(1-u) lim

J

n)(G(n)(e])X{0},G(n)(e])X{0}).

(38)

j=1

Since $ E U(S) and has its first and second derivatives globally bounded on C(J,Rn), it is easy to see that all three terms on the right hand side of (29) are bounded in t and n.

The statement of the theorem now follows by

letting t -.,0+ in (29) and putting together the results of (30), (33) and (38).

a

It will become evident in the sequel that the set of all functions satisfying Condition (DA) is weakly dense in Cb.

Indeed within the next section

we exhibit a concrete weakly dense class of functions in Cb satisfying (DA) and upon which the generator A assumes a definite form.

M.

Action of the Generator on Quasi-tame Functions

The reader may recall that in the previous section, we gave the algebra Cb of all bounded uniformly continuous functions on C(J,Rn) the weak topology induced by the bilinear pairing ($,u).

i

'

JC(J,Rn)

$(n)du(n), where

$ E Cb and u runs through all finite regular Borel measures on C(J,Rn).

More-

over, the domain of strong continuity Cb of {Pt}t>0 is a weakly dense proper 97

subalgebra of Cb.

Our aim here is to construct a concrete class Tq of smooth functions on C(J,Rn), viz. the quasi-tame functions, with the following properties: (i)

Tq is a subalgebra of Cb which is weakly dense in Cb;

(ii)

Tq generates Borel C(J,Rn);

(iii) Tq c 12(A), the domain of the weak generator A of {Pt}t>O; (iv)

E Tq and n E C(J,Rn), A(4)(n) is a second-order partial

for each

differential expression with coefficients depending on n. Before doing so, let us first formulate what we mean by a tame function. A mapping between two Banach spaces is said to be Cp-bounded (1 < p < co) if

it is bounded, Cp and all its derivatives up to order p are globally bounded; e.g. Condition (DA) implies C2-boundedness; and C3-boundedness implies (DA)(ii), (iii), (iv).

Definition (4.1)

(Tame Function):

A function 0:C(J,Rn) - R is said to be tame if there is a finite set {s1,s2,...,sk} c J and a C°'-bounded function f:(Rn)k -* R such that

O(n) = f(n(s1),...,n(sk)) for all n E C(J,Rn).

(*)

The above representation of 4 is called minimal if for any projection p:(Rn)k i (Rn)k-1 there is no function - R with f = gop; in other g:(Rn)k-1

words, no partial derivative Djf:(Rn)k -- L(Rn,R), j = 1,...,k, of f vanishes identically.

Note that each tame function admits a unique minimal represen-

tation.

Although the set T of all tame functions on C(J,Rn) is weakly dense in Cb and generates Borel C(J,Rn), it is still not good enough for our purposes, due to the fact that 'most' tame functions tend to lie outside Cb (and hence are automatically not in D(A)).

In fact we have

Theorem (4.1): (i)

The set T of all tame functions on C(J,Rn) is a weakly dense subalgebra of Cb, invariant under the shift semigroup {St}t>O and generating

Borel C(J,Rn). 98

Let 4) E T have a minimal representation

(ii)

ON = f(n(s1),...,n(sk)) where k > 2.

Proof:

n E C(J,Rn)

Then 4 ¢ Cb.

For simplicity we deal with the case n =

1

throughout.

it is easy to see that T is closed under linear operations.

(r

the closure of T under multiplication.

We prove

Let 01,02 E T be represented by

01(n) = fl(n(sl),...,n(sk)), 02(n) = f2(n(sj'),...,n(sm )), for all

n E C(J,R) , where fl :Rk -' R, f2:Rm -' R are C°°-bounded functions and

s1,...,sk, sj,...,sm E J.

Define

f12:Rk+m

i R by

f12(xl,...,xk, x,..... xm) = fl(xl,...,xk)f2(xj,...,xm) 00

for all xl,.,.,xk, x,...,xr'1 E R.

Clearly f12 is C -bounded and for all n E C(J,R).

Thus

E T, and T is a subalgebra of Cb.

It is immediately obvious from the definition of St that if 0 E T factors through evaluations at s1,...Isk E J, then St(0) will factor through evaluations at t + sJ < 0.

So T is invariant under St for each t > 0.

Next we prove the weak density of T in Cb.

Let T° be the subalgebra of

Cb consisting of all functions 4:C(J,R) - R of the form

ON = f(n(s1),...,n(sk)), n E C(J,R)

(1)

when f:Rk -' R is bounded and uniformly continuous, s1,...Isk E J.

Observe

first that T is (strongly) dense in To with respect to the supremum norm on Cb.

To see this, it is sufficient to prove that if c > 0 is given and

f:Rk -' R is any bounded uniformly continuous function on Rk, then there is a

Cam-bounded function g:Rk -' R such that If(x) - g(x)l < e for all x E Rk.

We

co

prove this using a standard smoothing argument via convolution with a C function (Hirsch [32], pp. 45-47).

d > 0 such that If(x1) - f(x2)I O with the properties that supp h c B(0,6) and JR kh(y)dy = 1, 99

the integral being a Lebesgue one on Rk.

Define g:Rk - R by

g(x) = JRk h(y)f(x-y)dy = J B(0,6) h(y)f(x-y)dy,

x E Rk.

(2)

By choice of 6 and property of h, it follows that If(x) - g(x)1 =

< for all x E Rk.

IJRk h(y)f(x)dy

-

Jk h(y)f(x-y)dyl

JB(O,d) h(y)If(x) - f(x-y)Idy < e I Rk h(y)dy = e

To prove the smoothness of g, use a change of variable

y' = x-y in order to rewrite (2) in the form g(x)

for all x E Rk.

h(x-y')f(y')dy'

J

(3)

B(x,d)

Now fix x0 E Rk and note that B(x,6) c B(x0,2d) whenever x E B(x0,d). Therefore for all x E B(x0,d).

h(x-y')f(y')dy'

g(x) = - J

(4)

B(x0,2d)

Since f is continuous and the map x - h(x-y') is smooth, it follows from (4) that g is C** on B(x0,6) and hence on the whole of Rk, because x0 was chosen arbitrarily.

Indeed g and all its derivatives are globally bounded on Rk

for IIDpg(x)II < J

If(y')Idy'

IIDph(x-y')II

B(x0,2a)

< V.

sup k IIDph(z) II. sup

k

If(z) I = N, say,

volume of a ball of radius 26 in Rk. in Cb.

< ... < S

Let 11 k:-r = s1 < s2 < S 3

V = J

dy' is the B(0,26) Secondly we note that Tois weakly dense

for all X E Rk, where N is independent of x0 and

k

= 0, k = 1,2,..., be a sequence

of partitions of J such that mesh lIk -". 0 as k -

Define the continuous

linear embedding Ik:Rk - C(J,R) by letting Ik(v1,v2,...,vk) be the piece-wise linear path (s-s 1) (s -s) vi_1, vi + Ik(vl,...,vk)(s) = s-sj-1 si_sj-1 100

S E CSJ-1,sj]

joining the points V1,...,vk E R, j = 2,...,k.

R N,

--8t

st

s,-0

s,_,

sa

Denote by sk the k-tuple (st,...,sk) E Jk, and by. Psk the map

C(J,R)

Rk

n F---> (n(s1),...,n(sk)). Employing the uniform. continuity of each n E C(J,R) on the compact interval

J, the reader may easily check that

lim (Ik °Psk)(n) = n

in C(J,R).

(5)

k-*w

Now if 4 E Cb, define k:C(JJO - R by

k =

is bounded and uniformly continuous, so is

410 1 k°psk, k = 1,2,...

o Lk: Rk

R.

.

Since

Thus each

Finally T° and lim 4k(n) = (n) for all n E C(J,R), because of (5). k-' note that II0II IICb for all k > 1. Therefore 0 = w-lim 0k and To is k Cb < 110

k E

--

weakly dense in Cb.

From the weak density of T in T° and of T° in Cb, one

concludes that T is weakly dense in Cb. Borel C(J,R) is generated by the class {$ 1(U):U c R open, 0 E T}. any finite collection sk = (s1,...,sk) E Jk let P

sk

For

C(J,R) + Rk be as befor

CCO

Write each 0 E T in the form 0 = f ° Psk for some

-bounded f:Rk - R.

It is 101

N

aO

well-known that Borel C(J,R) is generated by the cylinder sets

{psk-1(U1X... XU ):U k

E R open, i = 1,...,k, sk=(s1,...,sk) E Jk i

k = 1,2,...) (Parthasarathy [66] pp. 212-213).

Moreover, it is quite easy to see that

Borel Rk is generated by the class

{f 1(U)

U c R open, f:Rk -' R COO-bounded}

:

(e.g. from the existence of smooth bump functions on Rk). set U c R,

(U) = psk [f1(U)].

But for each open

Therefore it follows directly from the

above that Borel C(J,R) is generated by T. (ii)

Let 4) E T have a minimal representation

(s1,...Isk) E Jk is such that -r < s1 < S

2

t= f - Psk where sk =

< ... < S

k

< O, f:Rk

R is Coo-

Take 1 < jO < k so that -r < sj0 < 0. Since the represen-

bounded and k > 2.

tation of 0 is minimal, there is a k-tuple (xl,...,xk) E Rk and a neighbourhood [xj0-E0,xj0 + EO] of xj0 in R such that Djof(xl,...,xj0_1,x,xj0+1,...,xk) # 0 for all x E [xj

-

E0, xj0 + EO], with EO > 0.

Define the function

0

g:[xj0-E0, xj0 + E0]

R by

g(x) = f(xl,...,xj0_1x,xjo+l,...,xk) for all x E [x j0-EO,xjo+EO + EO] and g is a C00 diffeomorphism

Hence Dg(x) # 0 for all x E [xj0-E0, xj 0

onto its range. Ix' - x"I

Therefore, there is a A > 0 such that < AIg(x') - g(x")I for all x', x" E [x.0-EO,x.0 +EO]

Pick d0 > 0 so that d0 < are mutually disjoint.

E0

(6)

and the intervals (sj-260, sj+260), j =1,2,...,k,

In the remaining part of the argument we may assume,

with no loss of generality, that all integers n are such that

< d0.

Con-

struct a sequence {nn} in C(J,R) looking like the picture opposite

103

Viz.

r

sj-60 < s < sj+60.

x

J 0 JO

i

- d xj(s-sj-60) + xj

sj+60 < s < sj+260, j 0 jo

0

xj(s-sj+60) + xj

X11

sj-260 < s < sj-60.

j 0 j0

0

x

1

60

sj0 +60<s<s.0 +260

-6 )+x

(s-s

j0

j0

j0

0

(7)

n n(s) =

d xj (s-sj +60)+xj 0

0

0

0

0

0

sj-60<s<sj-I,sj+n<s<sj

xj

0

0

0

+c

ne0(s-s. ) + xj 0

0

0

0

0

0

0

+

< s < sj

S.

+e0

+60

sj - n < s < sj

0

-ne0(s-sj )+xj

0

0

n

otherwise

0

Suppose, if possible, that lim

sj -260 < s < sj -60

E Cb.

St($)(nn) = lim

Then

f(nn(t+sl),...,nn(t+sk))=f(nn(sl),....nn(sk))

t-.0+

uniformly in n.

But, for j 0 j0 and 0 < t < 60, nn(t+sj) = xj for all n.

Therefore f(x1,x2,...,xj0-l,nn(t+sj0).xj0+l.... .xk

lim

= f(xl,...,xj0-i.

+ e0.

xj0+l"...xk)

xj0

uniformly in n; i.e. for any c > 0, there is a 0 < 6 < 60 such that Ig(nn(t+sj0))

-

Now suppose 0 < t < 60. then

Inn(t+sj0

)

-

g(nn(sj0M < e for all n, all 0 < t < 6.

If

n
O

invariant under St, 0 < t < r, due to the semigroup property.

So let 0 be

Then

as above and t E [O,r].

0

r0t-r

f(n(s))g(s-t)ds + f(n(0)) Et g(s)ds,n(O))

St(4)(n) = h(1

ti

for all n E C(J,R'n). Define g:J +R by

t-r<s 1; e.g.

107

Also choose a sequence {dm}m=1 of Coo bump functions on Rn such that

-1

1xI<m

6m(x)

lxi >m+1

0

and 1dm(x)I < 1 for all x E Rn.

Define the sequence {fm}m=1 of C°°-bounded

functions on Rn by fm(x) = 6m(x)x for all x E R. n E C(J,Rn), Ifm(n(s)) gT(s)I < In(s)l

Then for each m > 1 and

for all s E J.

So by the dominated

convergence theorem 0

n(s.) = lim J

J

nt-- -r

fm(n(s))gm(s)ds,

j = 1,...,k

(12)

J

Therefore fi(n) = lliimm

f(J0

f (n(s))gm(s)ds,..., J0

fm(n(s))gk(s)ds)

(13)

Denoting the expression under the limiting operation in (13) by %(n), it is

clear that% E Tq for each m and 1%(n)1.4 sup {If(z)j: z E (Rm)k} for all n E C(J,Rn).

So Tq is weakly dense in T U Tq, and hence in Cb.

Finally we prove that Tq generates Borel C(J,Rn).

For this purpose it

is enough to show that the a-algebra a(Tq) generated by Tq coincides with that generated by the family of evaluations 108

psk:.C(J,Rn) + (Rn)k, sk = (s1,...,sk), sJ E J, j = 1,...,k, k=1,2,...

But this is again generated by sets of the form {n:n(s is closed.

E C} where C c Rn

Now each closed set in Rn is a countable intersection of comple-

ments of open balls, so we need only prove that if B c Rn is any open ball and B' is its complement, then {n:n(sj) E B'} E a(Tq) for any sj E J.

Suppose

B has radius b > 1 and for each integer p > 0 let Up be the complement of the concentric closed ball of radius b - 1 .

Let the sequences {f }m=1 and

{gy}m=1 be as before, so that (12) ispsatisfied.

Let Wm E Tq be given

by 0

IPM(n) = j_r f (n(s))(s)ds,

m > 1, n E C(J,Rn).

We contend that n p=1

{n:n(s.) E B'} =

To see this, let n(s lim

E B'.

Iim inf {n:W (n) E U } m p m

(14)

Then n(sj) E Up for all p > 1.

Since n(sj) _

Wm(n), then for each p there is an m0 > 0 such that Wp(n) E Up for all

m-

Hence, for each p > 1, n belongs to lim inf {n:Wm(n) E Up} i.e. n m belongs to the set on the right hand side of (14). Conversely, let n be in

m > m0.

lim inf {n:Wp(n) E Up} for each p > 1. Then for every p > 1, there is an m m0 > 1 such that Wm(n) E Up for all m > m0. Taking m + - gives n(sj) E Up CO

But B' =

for all p > 1 i.e. n(s.) E n p=1

Thus

As the sets {n:Wm(n) E Up} are clearly in a(Tq),

our contention is proved.

it follows directly from (14) that {n:n(s a(Tq) = Borel C(J1Rn).

Up, so n(s.) E B'. 1

E B'} E a(Tq).

Therefore

o

The final result in this chapter asserts that every quasi-tame function is in the domain V(A) of the weak generator A of {Pt}t>0.

Theorem (4.3):

Every quasi-tame function on C(J,Rn) is in the domain of the

weak generator A of {Pt}t>0.

Indeed if $ E Tq is of the form

4(n) = h(m(n)), n E C(J,Rn), where

109

0

m(n)

I0

f-r f1(n(s))g1(s)ds.... ...,

fk-1(n(s))gk-1(s)ds,n(0)),

then k-1 E D.h(m(n)){f.(n(0))g.(O)-fj(n(-r))g.(-r) J j=1 0 -

J

-r

Dkh(m(n))(H(n)) + 1 trace [D2h(m(n))°(G(n) x G(n))] for all n E C(J,Rn).

Here again Djh denote the partial derivatives of

h:(Rn) k -' R considered as a function of k n-dimensional variables.

Proof:

To prove that Tq c D(A), we shall show that each 0 = hom E Tq satis-

fies Conditions (DA) of §3. is Cam.

First, it is not hard to see that each 0 E Tq

Also by applying the Chain Rule and differentiating under the

integral sign one gets f0

Df1(n(s))(E(s))g1(s)ds,...,

D4(n)(E) = Dh(m(n))(1_ r

I°- r

Dfk-l(n(s))(E(s))gk-1(s)ds, E(0)),

D24(n)(E1,E2) = D2h(m(n))(Dm(n)(E1), Dm(n)(E2))

+ Dh(m(n))(JO

D2f1(n(s))(E1(s),E2(s))g1(s)ds,..., r 0 D2fk-1(n(s))(E1(s),E2(s))gk-1(s)ds,0).

J_r for all n,

E1., E2 E C(J,R').

Since all derivatives of h, f.,

1

j < k-1, are bounded, it is easy to

see from the above formulae that DO and D20 are bounded on C(J,Rn). induction it follows that 0 is C°°-bounded.

(iv) are automatically satisfied. of Theorem (4.2)(i).

Hence Codnitions (DA)(ii), (iii),

Condition (DA)(i) is fulfilled by virtue

From the above two formulae we see easily that the

unique weakly continuous extensions D4'(n), D m

110

By

)

of D4(n) and D24(n) to

C(J,Rn) ® Fn are given by

Dh(m(n))(O,...O,v) = Dkh(m(n))(v),

D2h(m(n))((O,...,O,v1), (0,...,0,v2))

D2h(m(n))(v1,v2),

=

for all v, v1,v2 E Rn.

The given formula for A(4)(n) now follows directly

from Theorem (3.2) and Theorem (4.2).

Definition (4.3)

o

Say n° E C(J,Rn) is an absorbing state for

(Ihynkin [16]):

the trajectory field {nxt:t > 0, n E C(J,Rn)} of the stochastic FDE(I) if P{w:w E 0, n0xt(w) = n°}

1 for all t > 0, where we take a =

=

here, i.e.

p(0,n°,t, {n°}) = 1 for all t > 0.

The following corollary of Theorem (4.3) gives a necessary condition for n° E C(J,Rn) to be an absorbing state for the trajectory field of the stochastic FDE (I).

Corollary:

Let n° E C(J,Rn) be an absorbing state for the trajectory field

of the stochastic FDE (I). (i)

n°(s)

=

Then

n°(0) for all s E J i.e. no is constant;

(ii) H(no) = 0 and G(n°) = 0.

Proof:

Let no E C(J,Rn) be an absorbing state for {nxt:t > 0, n E C(J,Rn)}.

For each t > 0 and s E J, define the Ft-measurable sets

0

Qt = {w:w E 9, n xt(w) = no} 0t(s) = {w:w E 52,

n

0

xt(w)(s) = n°(s)}.

Then Sgt c Stt(s) for all t > 0, s E J and since P(SZt) = 1, it follows that

Suppose if possible that there exist

P[Qt(s)] = 1 for t > 0, s E J.

s1,s2 E J such that n°(s1) # n°(s2).

-r< s1 < s2 < 0.

For each w E S2,

Without loss of generality take o xs2-s ()(sl) = n°(s2) # n°(s1) and so 1

(s1) =

Ds 2-s 1

0.

Hence P[S2s2-s (si)] = 0, which contradicts P[Qt(s)] = 1, 1

111

So n° must be a constant path.

t > 0, s E J.

This proves (i).

To prove that n° satisfies (ii), note that the absorbing state no must

satisfy A(O(n°) = 0 for all 0 E D(A) (Dynkin [16], Lemma (5.3), p. 137). In particular A($)(n°) = 0 for every quasi-tame 4':C(J,Rn) -IR. that since

n

is constant then so is the map t

every 0 E D(S).

Take any A:Rn

Note first

- nt and so S(q)(n°) = 0 for

R Coo-bounded and define the (quasi)-tame

function P:C(J,Rn) - R by W(n) _ A(n(0)) for all n E C(J,Rn).

Then by

Theorem (4.3), W E D(A) and

A(W)(n) = DA(n(0))(H(n°)) +

trace [D2A(n(0))0(G(n°) xG(n°))] =0

2 The last identity holds for every Cam-bounded A:Rn - R.

Choose such a A with

the property DA(n°(0)) = 0 and D2a(n°(0)) = , the Euclidean inner product on Rn e.g. take A to be of the form M(v) = Iv - n°(0)12 in some neigh-

bourhood of n(0) in Rn.

Then

Y trace [D2a(n°(0)) ° (G(n°) x G(n°)] =

for any basis

of Rm

for every Cam-bounded A:Rn ; R.

E j=1

Therefore G(n°) 0.

IG(n°)(ej)12 = 0

=0

Thus

Now pick any C -bounded A such that A(v) _

for all v in some neighbourhood of n°(0) in Rn.

Then

Da(n°(0))(H(n°)) = IH(n°)12 = 0; so H(n°) = 0 and (ii) is proved.

o

Remarks (i)

We conjecture that conditions (i) and (ii) of the corollary are also

sufficient for no to be an absorbing state for the trajectory field {nxt:t > 0, n E C(J,Rn)}.

It is perhaps enough to check Lemma (5.3) of

Dynkin ([16], p. 137) on the weakly dense set of quasi-tame functions in D(A).

(ii) An absorbing state no corresponds to the Dirac measure dno being invariant under the adjoint semigroup {Pt} associated with the stochastic t>O FDE M. Thus a necessary condition for the existence of invariant Di-rac measures is that the coefficients H, G should have a common zero (cf. §111.3).

112

V Regularity of the trajectory field

§1.

Introduction

Given a filtered probability space (Q,F,(Ft)0.< ,P) our main concern in this chapter is to look at various regularity properties of the trajectory field {nxt:t E [0,a], n E C(J,Rn)} generated by the autonomous stochastic FDE: dx(t) = H(xt)dt + G(xt)dw(t)

0 < t < a

l

(I)

J

x0=nEC=C(J,Rn) with coefficients H:C w.

Rn, G:C

L(Rm,Rn) and m-dimensional Brownian motion

A version of the trajectory field of (I) is a measurable process

X:S2 x [0,a] x C - C such that n E C.

nxt a.s. for all t E [0,a] and

In order to deal with the question of the existence of versions of

the trajectory field with reasonable sample function behaviour one needs to isolate two qualitatively different examples of stochastic FDE's.

The first

of these is when the diffusion coefficient G is independent of the past history, in which case we show that the trajectory field has a version whose sample functions are almost all compactifying (See Theorem (2.1)).

In the

second example, where G incorporates explicit dependence on the past (e.g. as in stochastic delay-equations), we shall see that all versions of the trajectory field are almost surely highly irregular (See §3).

Such erratic

behaviour is essentially due to the Gaussian-type nature of delayed diffusions coupled with infinite-dimensionality of the state space C.

Under these con-

siderations we are lead naturally to investigate regularity in probability of the trajectory field for the general system (I).

This is done in §4 where

the compactifying nature of the trajectory field is still shown to persist, though in a distributional sense, as in Theorem (4.6). It is perhaps worth noting here that for ordinary stochastic DE's (r = 0) satisfactory results on the sample function behaviour of the trajectory field are known.

See Remark (i) of §3.

113

Stochastic FDE's with Ordinary Diffusion Coeffici.ents

§2.

Consider the stochastic FDE 0 < t < a

dx(t) = H(xt)dt + g(x(t))dw(t)

1

(II)

J=[-r,0].

x0 = n E C= C(J,Rn),

J

The coefficients H:C - Rn, g:Rn -' L(Rm,Rn) are Lipschitz maps and w is

m-dimensional Brownian motion adapted to the filtered probability space

(c,F,(Ft)O 0 such that if t1,t2 > r and

It1-t21

< 6, then

- ht 11 Ca <E/2.

Ilht 1

114

2

Now if s1,s2 E J, t >r,

ti

and h E Ca([0,a],R), we have

i.e.

I I ht I I

Ca

< II h I I

for all t > r. Ca

If t1,t2 > r and Iti-t2I < 6, then one has

Ilht -ht 1

< Ilht -ht

II

2

CO'

1

II

i

C

< 2 IIh-h°II

a

Ca

+ Ilht -ht 1

II

2

C

a+

Ilht -ht 2

II

2

Ca

+ IIht1-ht2II < e.

Thus we need only prove the lemma for h E C1([O,a],R).

If so, let

r < t1 < t2 < a, -r < s1 < s2 < 0 and consider s2

(ht -ht )(u)du

(ht -ht )(s2)-(ht -ht )(s1) = 2

1

1

2

/

s1

1

2

fs2 [h'(t1+u)-h'(t2+u)]du. si

By the uniform continuity of h and h', if c > 0 is given there exists 60 > 0

such that Ih(u) - h(v)I< a/i+r1-a and Ih'(u) - h'(v)I< C/i+r1-a whenever Hence if

u,v E [0,a], Iu-vl < d0.

It1-t21

< d0, then

I(ht1-ht2)(s2) - (ht1-ht2)(si)I < Ei_a Is2-s1l So I- a Ilh tl

-ht

=

II

2

Ca

)(s )I.I(s -s sup I(h h h )(s )-(h t1 t2 2 1 2 t1 t2 1 s1,S2EJ

si
0 such Co,,

that

11&

IIDF(z(u)) - DF(y(u))II< K1Iz(u)-y(u)I for all u.E [0,1].

Letting K2 = sup{IIDF(v)II: v E RP, IvI < 2}, we get

116

IE(s1,s2)I < K2[ IK1 - Y1Ca - II&1 - &211C] Is1-s21a + K1 IIE1 - E211 C J 1 Iz(u)-y(u) Idu 0

< K2[II&1 -E21ICa- IIE1 - E211 C] Is1-s21a

+ K1IIE1 - C211C ID

< K2[I1

1

(1-u)1&2 (s1)-&2(s2)I}du

-&211 Ca - 11&1-&211

C]Isi-s21

+ Ki

Isi-s2Ia

11&211 Ca)

1s1-s21at

< K3 II&1 - C211 Ca

where

K3 = max (K2, 2K1) . The last estimate clearly implies that there is a K4 > 0 such that

IIF(E1) - F(C2) II Ca < K4 II&i - &211 Ca for all C1,E2 E Ca(J,Rp) with

a
0 such that

I4(s,x,t) - (s,x',t')I < K5[Ix-x'I + It - t'I] for all s E [0,1]. Therefore

120

ID2X(X,x,t)(y) - D20(A,x',t')(y)I

o ID20(s,x,t)(y) - D20(s,x',t')(y)Ids K2K3 J0

+ K1K4K5 Iyl [Ix-x'I + It - t'I]

for all A E [0,1].

By Gronwall's lemma,

I[D2W,x,t) - D24(A,x',t')](y)I < K1K4K5IYI[Ix-x'I+It-t'I]e for all A E [0,1].

KKA 2 3

In particular,

IID2F(t,x) - D2F(t',x')II = IID2m(1,x,t) - D24(1,x',t')II

< K1K4K5 e

2 3

for all x, x' E B1, t,t' E B2. Rm x Rn.

[Ix-x'I + It - t'I]

Hence D2F is Lipschitz on bounded sets in

Also D1F = goF is Lipschitz on bounded sets in Rm x Rn.

Therefore

so is DF. Theorem (2.1): a C2

In the stochastic FDE (II), suppose H is Lipschitz and g is

map satisfying the Frobenius Condition.

Then the trajectory field

{nxt:t E [0,a], n E C(J,Rn)} has a version X:St x [O,a] x C(J,Rn) _ C(J,Rn)

having the following properties.

For any 0 < a < }, there is a set SZa c St

of full P-measure such that, for every w E Sla, (1)

the map

(ii)

for every t E [r,a] and n E C(J,Rn), X(w,t,n) E Ca(JRn);

(iii) (iv)

x C(J,Rn) _ C(J,Rn) is continuous;

x C(J,Rn) - Ca(J,Rn) is continuous; for each t E [r,a], X(w,t..):C(J,Rn) _ Ca(J,Rn)

is Lipschitz on every

bounded set in C(J,Rn), with a Lipschitz constant independent of t E [r,a]. In particular each X(w,t,.) : C(J,Rn) - C(J,Rn) is a compact map.

Proof:

Suppose H is Lipschitz and g is a C2 map satisfying the Frobenius

Condition.

Employing a technique of Sussman ([74]) and Doss ([12]), we show that a

121

version X:2 x [O,a] x C(J,Rn) - C(J,Rn) of the trajectory Oxt:t E [O,a],

n E C(J,Rn)} of (II) can be constructed, sample-function-wise, by first solving a suitably defined random family

x C(J,Rn)

Rn of

retarded FDE's, and then deforming their semi-flows into X using the flow F ti

of g parameterized by Brownian time w(t).

The random family H and the flow

F are sufficiently regular to guarantee the required sample function properties of X.

Indeed, by Lemma (2.3), we let F:Rm x Rn

Rn be the C2 flow of g, viz:

D1F(t,x) = g(F(t,x)) (5)

F(O,x) = x

for all t E Rm, x E Rn.

For any C E C(J,Rm) and n E C(J,Rn), define the map Fo(C,n):J + Rn by [F-(C,n)](s) = F(E(s),n(s)) for all s E J.

For each t E Rm, F(t,.) is a diffeomorphism of Rn; so for any x E Rn the linear map D2F(t,x):Rn - Rn is invertible. Define the 'Brownian motion' w°:0 - C([-r,a],Rm) by w(w)(t) - w(w)(0) w°(w)(t)

for a.a. w E 0.

t E J

0

Recall that wt:c -

t E [O,a] (6)

C(J,Rm) is the slice of w° at t E [O,a],

i.e. wt(w) E C(J,Rm) for a.a. W E 92.

We now define a random family of retarded FDE's on Rn by letting ti

H:[O,a] x C(J,Rn) x Q ; Rn be the map 116

H(t,n,w) = [D2F(w°(w)(t),n(O))]-{H[F-(wt(w),n)] 1

-

trace (Dg[F(w°(w)(t),n(O))]og[F(w°(w)(t),n(O))])}

for a.a. W E 2, t E [O,a], n E C(J,Rn). We shall show that, for a.a. W E Q, there is a map n&(w):[-r,a] , Rn continuous on J and C1 on [O,a] which is a solution of the RFDE:

122

(7)

0 0 (20)

But for a > t > 0 we have

dnV -)(t) = and

126

trace

So (20) yields

a > t > 0.

I trace

(21)

Now in (5), keep x fixed and differentiate with respect to t E Rm; then D2F(t,x) = Dg(F(t,x)) o D1F(t,x) = Dg(F(t,x))og(F(t,x))

for all t E Rm, X E Rn.

In particular,

trace D2F(wo(w)(t),nE(w)(t))

=

Dg[F(w (w)(t), TE(w)(t))] o g[F(wo(w)(t),'E(w)(t))]

ti

ti

= Dg[x(w,t,n)] o g[x(w,t,n)]

(22)

for t E [0,a], and a.a. W E 0.

From (7) we see that

D2F(wo(w)(t),%(w)(t)){H(t,%t(w),w)} = H[F o (wo(w),ntt(w))]

-

trace (Dg[F(wo(w)(t),%(w)(t))] o g[F(wo(w)(t),n&(w)(t))])

=

-7 trace Dg[x(w,t,n)] o g[x(w,t,n)], 0 < t < a

(23)

using the fact that 1XI

T

(

) )

F o (wo(w),

(24)

a.s., for all t E [0,a], n E C(J,Rn). ti

From (21), (22) and (23), it follows that, for each n E C(J,Rn),

127

is a solution version of the stochastic FOE 0 < t < a

(25)

Moreover, x(w,s,n) = F(O,nE(w)(s)) = F(0,n(s)) = n(s) for all s E J, a. a. W E S. ti

is the unique solution of the stochastic FDE (II),

Therefore

and the random field X:S2 x [0,a] x C(J,Rn) - C(J,Rn) a.a. W E SZ, t E [0,a], n E C(J,Rn)

X(w,t,n) =

is a version of the trajectory {nxt:t E [O,a], n E C(J,Rn)} of (II), viz. nxt a.s., for all t E [0,a], n E C(J,Rn). We now verify that X has the stated regularity properties.

For a.a. w E S2,

w°(w) is continuous on [-r,a]; hence the map > C(J,Rm)

[O,a] t

wt (w)

i

is continuous for a.a. w E 0.

Also by continuous dependence of solutions

to RFDE's on the initial condition, it is clear that the semi-flow

[O,a] x C(J,Rn)

C(J,Rn)

(t,n) i- > n t(w) is continuous for a.a. W E 0 (Hale [26]).

Therefore, composing with the

continuous flow F, it is easy to see that, for a.a. W E 9, [0,a] x C(J,Rn)

(t,n)

.

C(J,Rn)

i----> X(w,t,n) = Fo(wt(),nct())

is also continuous.

since 0 < a < For the rest of this proof, fix w E Sta. Let r < t< a and n E C(J,Rn). Let 0a = {w:w E S2, w(w) E Ca([O,a],Rm)}. Then P(fa) =

128

1

Since w°(w) E Ca([-r,a],Rm), then wt (w) E Ca(J,Rm). C1

,

and since nEt

But F:Rm x Rn - Rn is

is C1 on J, it follows that the composite map X(w,t,n) _

(M )

Fo(wt(w), r'C (w)) must be in Ca(J,Rn).

This proves (ii) of the theorem.

We now go on to prove assertion (iii) of the theorem.

First we establish

the continuity of the map Ca(J,Rn)

[r,a] x C(J,Rn)

(26)

%t(w)

(to) 1--

Fix any n E C(J,Rn) and let r < t1 < t2 < a, -r < s1 < s2 < 0.

net

()](s2) - [n&t2 (w) - nEt (w](sl)

[nEt2 (w) -

1

1

= I{n,(w)(t2+s2) - n (w)(t2+s1)}-{TIE(w)(tl+s

tl+s2

t2+s2 ti = If 't2+s1

t1+s1

nEs+t

(w),w)dul

n

H(s+t1' &s+t (w),w)dsl

(w),w)ds 2

s1

u

fS2

H(s+t2'

n&(w)(tl+sl)}1

,,

H(u, n

H(u,n&u (w).w)du -

s2 ,

= JJ

Then

sl

1

s2 ti

n

ti

< Js

n

s+t2(M).w) - H(s+tl. Cs+t2(w).w)Ids

IH(s+t2, 1

+ J sl

TI IH(s+tl,n& "' s+t ()'w) - H(s+tl, &s+t (w),w)Ids

2

(27)

1

Now by continuity of the slicing map the set {n&t(w):t E [0,a]} is compact in C(J,R").

Furthermore, each map

is Lipschitz in each bounded set of

C(J,Rn) with a Lipschitz constant independent of t E [0,a]; in particular

the map C(J,Rn)

> C([O,a].Rn) ti

ti

is continuous.

Hence the set

E [0,a]} is compact in

129

C([O,a],Rn), and therefore equicontinuous due to Ascoli's theorem i.e. given e > 0, there is a 6 > 0, depending only on n and w, such that, if It1-t21 < 6 , then IH(s+t2,

for all s E J.

1Es+t2

(w),w) - H(s+t1, nEs+t (w),w)I < e 2

Indeed because of the uniform continuity of

[0,a]

C(J,Rn)

t I--- nEt(w)

(28)

we may suppose that the above 6 is so chosen that II11Es+t1(w) nEs+t2 w)IIC < e for all s E J, whenever

It1-t2I < 6.

Denote by Li(w) >0 the Lipschitz constant of

on the compact set {nEt(W):t E [0,a]}. It1-t21 < 6

Then (27) gives, for

,

t2(w) - 'Et1(w)](s2) - [nEt2(w) - nEt (w)](s1)I

I['

1

S

< cls1-s21 + L1(w)

fS1 IInEs+t

(w) - nEs+t1 (w)IICds 2

ti

< elsl-s21 + ELi(w)Is1-s21 Therefore

I

I "Et2(w) - nEti (w) II ca N(w) is continuous.

(30)

In fact we shall prove next that the above map is Lipschitz

on bounded sets of C(J,Rn) (with Lipschitz constants independent of t E[r,a]). Then

Let n E C(J,Rn).

In,(w)(t) I < IIn II C + J0 IH(u,n&u(),w) Idu, t E [0,a] 11

+ K(w)t + K(w) Jt

< IIn II

II Cdu,

t E [0,a],

0

due to the linear growth property (18)'. II

nyw) III < IIn II C +

Therefore

K(w)t + K(w) J0t

II Cdu,

t E [0,a]

Applying Gronwall's inequality, we obtain ti II

nCt(w)

II

C < [IIn 11 C +

K(w)t]eK(w)t,

Let B c C(J,Rn) be any bounded set.

t E [0,a], n E C(JRn)

(31)

By (31) the set

B(w) _ {nCt(w):t E [0,a], n E C(J,Rn)} is also bounded in C(J,Rn).

Let

ti

L2(w) > 0 be the Lipschitz constant of t E [r,a] and -r < s1 < s2 < 0.

I[n1Ct(w) -

[n1Ct(w)

-

(w),w)du

t+s1 s2

u

[ti

n1

H(t+s,

IJ

-

n2Ct(w)](s1)1

- rt+s2 ,

Ti 1

H(u,

=

Take any n1,n2 E B,

Then

n2Yw)](s2)

rt+s2 ti

on B(w).

H(u,

J

t+s1 ti

Ct+s(w),w) - H(t+s,

s1

n2

C (w),w)dul u

n2

Et+s(w),w)]dsl

fS2

< L2(w)

11n1Ct+s(w)

n2Ct+s(w)IIC ds

(32)

s1

131

But again, II

n1Et.(w) -

n2CtI() IIC < IIn 1-n2IIc+L2(w) J0IIfuu(w)2u(w)IIcdu

for all t' E [O,a]; so by Gronwall's lemma, one gets

ti

ti

n1

II

EtI() -

for all t' E [O,a].

L2(w)t'

n2

Et'(w)IIC < IIn1-n2IlCe

L2(w)a

< IIn1-n2IICe

(33)

Therefore (32) gives

[n1Et(w)-n2Et(w)](s2)-[n1&t(w)-n2Et(w)](s1)I

L2(w)8,

< IIn1-n2IIC e

L2(w)-Is1-s21

From (33) and (34), we deduce that II n1&t(w) - n2Et(w) II

Ca

(34)

ti

< II

n1Ct(w)-n2Et(w)

ti L (w)a II c+ rt-a L2(w)e 2 IInj n2IIC

ti e

'L 2M] IIn1-n2IIC

(35)

This last inequality clearly shows that the map (30) is Lipschitz on B uniformly with respect to t E [r,a].

Now W E %, so the map

[O,a]

> Ca(J)Rm)

t I---) wt(w) is continuous (Lemma (2.1)).

By continuity of (26), the map Ca(J,Rm+n)

[r,a] x C(J,Rn)

Ca(J,Rm) x Ca(J,Rn) =

(tin) I-- (wt(w), is also continuous.

t())

Thus, composing with the continuous map

F':Ca(J,Rm) x Ca(J,Rn)

Ca(J,Rn)

(,Ti) CI> FoR,n)

132

(36)

(Lemma (3.2)), one gets the continuity of the sample function X(w,.,.):.[r,a] x C(J,Rn)

>

Ca(J,Rn)

F°(wt(). %t(w)).

(tin)

To prove the final assertion (iv) of the theorem, fix t E [r,a] and compose the isometric embedding

> Ca(J,m) x Co`(J,n)

Ca(J,Rn) T1

> (wt(),n)

i

with the map (30), deducing that the map

C(J,Rn)

Ca(Jm) x Co`(JRn)

n r---> (wt(). TIEt()) is Lipschitz on each bounded set in C(J,Rn) with Lipschitz constant independent of t E [r,a].

Applying Lemma (3.2) once more shows that

X(w,t,.):.C(J,Rn) - Ca(J,Rn) is Lipschitz on every bounded set in C(J,Rn),

with Lipschitz constant independent of t E [r,a].

The compactness of

C(J,Rn) is then a consequence of the last statement together with the compactness of the embedding Ca(J,Rn) Ascoli's Theorem).

---> C(J,Rn) (viz.

a

The reader may easily check that the construction in the above proof still works when the drift coefficient is time-dependent i.e. for the stochastic FDE

dx(t) = H1(t,xt)dt + g(x(t))dw(t)

0 < t < a (III)

x0

= n E C(J,Rn)

With H1:[0,a] x C(J,Rn) _Rn, g:Rn -> L(Rm,Rn) given maps.

Corollary (2.1.1): map (t,n)

Indeed one has

Suppose g is a C2 Lipschitz map satisfying (Fr), and the

F.-> H1(t,n) in the stochastic FDE (III) is continuous, Lipschitz

in n over bounded subsets of C(J,Rn) uniformly with respect to t E [0,a]. Assume also that H1 satisfies a linear growth condition:

133

IH1.(t,n)I < K(1 + 11nIIC), n E C(J,Rn), t E [O,a], for some K > 0.

(37)

Then all the conclusions of Theorem (2.1) hold for (III).

Unfortunately, it is not possible to deal with time-dependent diffusions as above, except perhaps in the following rather special case:

Corollary (2.1.2):

In the stochastic FDE

dx(t) = HI(t,xt)dt + g1(t,x(t))dw(t),

0 < t < a (IV)

x0

=

}

n E C(J,Rn)

let H1:[O,a] x C(J,Rn)

Rn satisfy the conditions of Corollary (2.1.1) and

g1:[O,a] x Rn + L(Rm,Rn) have a C2 extension 51:R x Rn -> L(Rm,Rn) with D51

bounded and fulfilling the Frobenius condition {D 291(t,x)[91(t,x)(v1)}(v2) = {D 2g1(t,x)[91(t,x)(v2)]}(v1)

(38)

Then the trajectory field

for all t E R, X E Rn, v1,v2 E Rm.

{nxt*t E [O,a], n E C(J,Rn)} of (LV) has a version X:c2x[O,a] x C(J,Rn) +C(J,Rn)

with all the properties stated in Theorem (2.1). Proof:

To eliminate explicit time-dependence in the diffusion coefficient

of (IV) we borrow the following technique from deterministic ODE's.

Embed

(IV) in a stochastic FDE of type (III) but defined on R x Rn, viz. dy(t) = H(t,yt)dt + g(y(t))dw(t) y0

= (0,n) E C(J,R x R

Indeed let p2:C(J,R x Rn)

a > t > 0 (III)

n )

J

C(J,Rn) be the natural continuous linear pro-

jection p2(Y,n) = n,

Y E

C(J,R), T1 E C(J,Rn).

Define H:[O,a] x C(J,R x Rn) - R x Rn, g:R x Rn ; L(Rm,R x Rn) by H(t,E) = (1,H1(t,P2(C))), t E [O,a], C E C(J,R x Rn g(t,x) = (O,g1(t,x)),

134

t E R, X E R n

By virtue of (38), it is easily seen that g and HA' satisfy all the conditions of Corollary (2.1.1).

Hence there is a continuous version

Y:S2 x [O,a] x C(J,R x Rn) -' C(J,R x Rn) of the trajectory field

ti { yt:t E [O,a], C E C(J,R x Rn)} for (I.I.I.) which satisfies Theorem (2.1).

Define X:S2 x [O,a] x C(J,R")

C(J,Rn) by

X(w,t,n) = p2[Y(w,t,(O,n))], w E Pa (0 < a < n E C(J,Rn).

), t E [O,a],

We claim that X is a version for the trajectory field

{nxt:t E [O,a], n E C(J,Rn)} of (IV).

To see this define the process y on

[-r,a] by (t,nx(t))

t E [0,a]

(0,n(t))

t E J

y(t) _ {

where nx is the unique solution of (IV). dy(t)

Then, for t E (0,a),

= (dt, dnx(t))

= (dt, H1(t,nxt)dt + g1(t,nx(t))dw(t))

= (1,H1(t,P2(yt)))dt + (0,g1(Y(t)))dw(t) =

H(t,yt)dt +

g(y(t))dw(t)

i.e. y is a solution of (III) and by uniqueness one gets a.s. Y^ = (O,n)y, the solution of (III) at (0,n) E C(J,R x Rn). and so

Therefore

P2(yt) = nxt a.s. for t E [0,a], n E C(J,Rn).

yt a.s.

This proves

our claim, and it is then easily checked that X is the required version satisfying Theorem (2.1).

o

Remark Condition (38) is apparently rather strong; it is satisfied by C2-limits of

mappings [O,a] x Rn -' L(Rm,Rn) of the form (t,x) . A(t)g(x) where X:[O,a] - R is such that A(0) # 0 and g:Rn -' L(Rm,Rn) is of Frobenius type.

It is not known to me whether this condition, or the original Frobenius condition for that matter, can be relaxed with Theorem (2.1) still true. However, see the next section on delayed diffusions (§3), and also Corollaries

(2.1.3) and (2.1.4) below.

135

Corollary (2.1.3):

In the stochastic FOE (LI.L) suppose H1 and g satisfy all

the conditions of Corollary (2.1.1), with X:S2 x [0,a] x C(J,Rn) _ C(J,Rn) the version of the trajectory field of (LLI.) obtained therein.

Assume also that

H1 has a partial derivative D2H1(t,n) In the second variable such that (t,n)

--> D2H1(t,n) is continuous.

Then for all w E QM (0 < a < }) and

t E [0,a], the map X(w,t,.): C(J,Rn)

> C(J,Rn) X(w,t,n)

n

is C1.

Let its derivative be D3X(w,t,n), and for any C E C(J,Rn) define

the process a:S2 x [-r,a] -> Rn by rD3X(w,t,n)(C)(0)

0 < t < a

a(w,t) = j

l

t E J,

C(t)

W E. Sta

Then a E t°(S2,C([-r,a],Rn)) and is the unique solution of the linear stochastic FDE da(t) =

0 0 we have

> N) < P{w.w E Q, llnxt(w) II C > N/2}

P{w:w E S2, Ilnxt(w) II Ca

+ P{w:w E 12.

Ir'x(w)(t1) nx(w)(t2) I

sup

N

>7} C.

O 0 and choose

M, NE > 0 sufficiently large so that B c {n:n E C, II nllC

< }E.

NE

, and the map

Therefore sup {p(O,n,t,C-,IZE):n E B, t E [r,a]} < is < c (t,n)

i ->

sets.

o

takes bounded sets into relatively weakly * compact

Denote by Mp(Ca) the space of all Borel probability measures on Ca(J,Rn) We shall show below that for t > r

(0 < a < }) given the weak * topology.

the transition probabilities p(O,n,t,.) E Mp(Ca) for 0 < a < i i.e. they are all supported on Ca rather than C.

This will follow easily if we prove that

Borel Ca is a sub-Q-algebra of Borel C.

Indeed if Borel Ca a Borel C, then

by Theorem (4.4) there is a set 52a E Ft of full P-measure such that nxt(w) E Ca

for all w E a and so

n

x

t

By separability,

E ro(S2,Ca;Ft) for all t > r.

Borel Ca is generated by a countable collection of closed balls; so for Borel Ca to be contained in Borel C we need only check that every closed ball {n:n E Ca,

Iln-nollCa < d} in Ca belongs to Borel C.

For simplicity,

and without loss of generality, it is sufficient to show that the closed unit ball Ba = {n:n E Ca, IIn1ICa < 1} in Ca is a Borel set in C.

Let {si

be

Then

an enumeration of all the rationals in J = [-r,0].

I(s .)-n(sa)I

Ba = {n:n E C, sup In(si)I + i>1

< 1}

sup j,k>i j#k

Is

k

In(s

= {n:n E C,

sup [In(si)I + i,j,k>l j#k

s

-n( s

k

)I

] < 1}

Is.-ska

For each i,j,k > 1, j # k, define the function

ijk:C - R>O by

HSs j )-n(a )I for all Ti E C.

H= I(si)I +

Isj-Ski Then each ijk is continuous and hence (Borel C, Borel R)-measurable. fore B. =

There-

{n:n E C, $ijk(n) < 1} E Borel C. n i,j,k>1 j#k 157

Recall that the spaces L°(c,C) and c(S2,Ca) carry the pseudo-metrics do, da

defined by d0(01,02) = inf [e + P{w:w E 2, I1e1(w)-02(w)IIC > e}] e>O

for all 01,02 E c°(52,C) and da(61,02) = inf [e + P{w:w E S2, 1161(w)-02(w)II e>O Ca

> e}]

Under these pseudo-metrics and a global Lipschitz

for all 01,02 E CO(S2,Ca).

condition on the coefficients H, G of the stochastic FDE (I), we get our last main result concerning the regularity of the map n ---> nxt into t°(c,C) or

Theorem (4.7):

Suppose H:C - Rn and G:C - L(Rm,Rn) in (I) are globally

Lipschitz with a Common Lipschitz constant L > 0. integer greater than

Let 0 < a < J and k be any

Then there are positive constants

.

C4 = C4(L,m,n,a), C5 = C5(a,L,m,n,a) such that for any

C3 = C3(k,a,L,m,n,a),

ni,n2 E C we have

P{w:w E 0,

I(nlx(w)- 2x(w))(t 1 )-(n1x()-n2x(w))(t 2 )I

sup t1,t2E[0,a]

It1 -

t2Ia

> N}

t1 0 t2 IIn l -n 2 II Ck

P{w:w E

2, Il n

1xt()- n 2xt(w) II a Al

(5)

N>0

, 3

-IIn1-n2II 2k ,


0, t E [r,a],

N

(6)

n do(

1

(7)

2k/2k+1,

xt, 2xt)

t E [O,a],

C4 IIn 1-n211 C

and

da( 2xt,

2

2k/2k+1,t E [r,a].

x ) < C5 I1 ni-n2IIC t

In particular the maps C

T1

158

r°(S2,C),

nxt

t E [0,a], C --> £C°(52,Ca), t E [r,a] n

1

;.

nxt

(8)

are uniformly Holder continuous with Holder exponent (7

-a)-1

Furthermore

the maps

[O,a] x C - Mp(C), [r,a] x C - MP(Ca) (t,n) -->

(to) F--> are continuous.

Proof:

Define the measurable process

Let n1, n2 E C and 0 < a < }(t - k

y:S2 - C([-r,a],Rn) by Y(w)(t) = n1x(W)(t) -

t E [-r,a], a.a. W E St.

2x(W)(t),

Then, for 0 < t1 < t2 < a, we have

22k-lEIt2

[H(n1x)-H(n2x)]dul2k u u

Jt

2

1

1

t2 + 22k-1Elf

[G(n 1xu)-G(n

2xu)]dw(u)l2k

t1

t


N}
0.

Hence (5) is satisfied.

To prove the inequality (6), let r < t < a, N > 0 and consider P{w:w E ,, Ilnlxt(w)

-

a > N}

n2xt(w)II C

< P{w:w E 0, Ilnlxt(w) - n2xt(w)IIC

>

2}

In1x(w)(t+s )-n2x (w)(t+s2)l

+ P{w:w E S,

sup s1,s2EJ si

160

i S2

Is1 -

s2a

>

N}

-2k 3

2k

< NW

E IlytllC

+ 2 Zk

m2UCk

II n1

2k 3

2k+

22kK5. MW

2

2k

-

IIn1-n2IIck

_

where we have used Chebyshev's inequality (Theorem (L.2.4)) and

(9);

K7 = K7(k,a,L,m,n,a) = 22kK5 + 22kC3 is independent of n1,n2, N and t E [r,a]. This proves (6) with C3 replaced by K7.

For the inequality (7), note that, if t E [O,a], then

do(nlxt,n2xt) = inf [c + P{w:w E St, IInlxt(w) n2xt(w) IIC e>O

O

IIn1-n2II2k]

+

c>O

Define the Coo function f:R>0 - R by K

f(c) = c + - 5 IIn1-n2IIck for all e > 0. Then f attains its absolute minimum value at CO > 0 where f'(c0) = 0, f"(e0) > 0

i.e.

2k V 1

i.e.

-

Zk+1

IIn1-n2IIC k = 0

c02k+1 = 2k K5 IIn1-n2IIc k

Therefore

do(n1xt,n2xt) < inf f(c) = f(c0) = c0(1 + c>0 =

2k

1

1

K52T (2k)

(1 +

IIn1-n2"Ck+'f

)

1

Hence (7) holds with C4 =

,a).

161

Choose and fix an integer k0 = k0(a) such that k0 >

a .

Then by a similar

argument to the above one, we can prove that da(nixt.r)'xt)

inf [e + e>0

1

where C5 = (C3)

0

e

2k0 2k +1

II n1-n2II 2k0] = C51In1-n2IIC

0

C

1

(2k0)

0

(1 +

0

C5(k0(a),a,L,m,n,a) = C5((x,L,m,n,a).

= 0

Since

2k0

> (4 - a)-1, then

2k0+1 2k

I I n-r

IIn1-Ti 2II

0

II ^21

if I1n1-n211C > 1

C

1


L°(2,C), (t,n)

>

E°mCa)

[r,a] X C (t,n)

nxt

nxt

and the continuity of the maps

t°(S2,C)

-p Mp(C),

0 i->

t°(c2,Ca)

Poe-1

-->

0

Mp(Ca)

po0-1

(See Elworthy [19], Appendix C, p. 300). This completes the proof of the theorem.

o

Remark (4.2)

Let Pt:M(C) - M(C), 0 < t < a, be the adjoint semigroup associated with the stochastic FDE (I) ((§III.3)). P E M(C).

If r < t < a, then PtpE M(Ca) for all

In fact each map PP:M(C) - M(Ca), r < t < a, is continuous linear

and the space of probability measures Mp(Ca) on Ca is invariant under the adjoint semigroup Pt for t > r.

As a consequence of the above theorem, one can always select measurable

162

versions 2 x [O,a] x C -* C, Q x[r,a] x C - Ca of the trajectory field

{nxt:t E [O,a], n E C}, {nxt:t E [r,a], n E C} for 0 < a < }.

The proof of

this result was shown to me by K.D. Elworthy; I am very grateful to him.

Corollary (4.7):

Suppose in the Stochastic FDE (I) the coefficients H, G

are globally Lipschitz on C.

Then the trajectory field {nxt:t E [0,a],n E C}

has a measurable version Q x [O,a] x C + C, and for any 0 < a < } the segment

{nxt:t E [r,a], n E C} admits a measurable version 0 x[r a] x C - Ca.

Proof: Since the constant C4 in (7) is independent of t E [O,a], it follows, from the continuity of the map

[0,a] -> c2(c,C)

t )-;$

cow,C)

nxt

for each n E C, that the map Y:[0,a] x C

(to)

; t°o,C)

t-- nxt

is jointly continuous (in the metric d0) and hence Borel-measurable.

But

[0,a] x C is complete separable metric in the product metric; so by a result of C. Moore (Cohn [9]), the map Y has separable range in t°(Q,C).

Applying

a general theorem of Cohn we immediately get an (F ® Borel [0,a] ® Borel C, Borel C)-measurable version X:c2 x [0,a] x C + C for Y.

(Theorem (1.5.2)).

A similar argument works for the continuous map

[r,a] x C -> L°(Q,Ca)

(t,n) r- >

nxt

noting that Cohn's theorem above does hold for processes with values in the

complete separable Banach space C.

o

Remark (4.3) (i)

I do not know if the trajectory field {nxt:t E [0,a], n E C} has a C-

valued separable measurable version.

If one exists, then by the Borel-Cantelli 163

1

Lemma it is not hard to see that this separable measurable version X will have the property that there is a set Sa E F of full P-measure, chosen

independently of n E C, so that, for all w E "a and any n E C, X(w,t,n) E C, X(w,t,n) E Ca if t E [O,a] and t E [r,a] respectively. (ii)

Both Theorem (4.7) and its corollary hold for random coefficients

H:Sl x C -, Rn, G:SE x C - L(Rm,Rn) satisfying the conditions: and G(.,n) are F0-measurable;

(a)

For each n E C,

(b)

There exists L > 0 such that

IH(w,n1)-H(w,n2) I < L IIn1-n2IIC,IIG(w,n1)-G(w,n2) II< L 11n1-n2IIC

for a.a. w E.

164

Q,

all n,02 E C.

VI Examples

§1.

Introduction

In this chapter we illustrate our general results of Chapters II,III,IV and V on various examples of stochastic FDE's.

These examples include stochastic

delay equations (93) and linear FDE's which are forced by white noise (§4). The latter class of stochastic FDE's corresponds to equations of the form

dx(t) = H(xt)dt + g(t)dw(t),

t > 0

x0=nEC EC(J,Rn) where H:C - Rn is a continuous linear drift and g:R - L(Rm,Rn) is locally integrable.

Section 4 is joint work of the author with Henrich Weizskcker

and Michel Scheutzow :

the asymptotic behaviour of trajectories to the above

equation is analysed along the stable and unstable subspaces in C of the deterministic linear system dx(t) = H(xt)dt,

t > 0.

Applying well-known results of J. Hale ([26] pp. 165-190) we find that, if g is constant, the forced system is globally asymptotically stochastically stable (§3) whenever the unforced linear system is globally asymptotically stable.

The stochastically stable case corresponds to the existence of a

limiting invariant Gaussian measure under the stochastic flow.

On the other

hand, when g is periodic and the unforced system is asymptotically stable, the transition probabilities converge to a periodic family of Gaussian measures on C.

§2.

Stochastic ODE's

These correspond to the case when our system has no memory i.e. r = 0, thus J = {0} and C(J,Rn) is just the Euclidean n-dimensional space

n.

Our basic

stochastic FDE (I) of Chapter II then takes the simple differential form

165

t > 0

dx(t) = g(t,x(t))dz(t),

(I)

x(0) = v E C2(St,Rn) Such stochastic ODE's were first studied by Ito ([36]) in 1951 and has since been the subject of intensive research.

Indeed there are now excellent texts

on the subject such as the works of Gihman-Skorohod [24], Friedman [22], Stroock-Varadhan [73]

on Euclidean space,and Ikeda and Watanabe [35] and

Elworthy [19] on differentiable manifolds.

We therefore make no attempt to

give any account of the behaviour of trajectories to (I), but content ourselves by noting that Theorems (11.2.1), (11.2.2), (II.3.1), (III.1.1) (111.2.1), (111.3.1) are all well-known to hold for stochastic ODE's of type (I) or its autonomous versions dx(t) = g(x(t))dz(t)

,

t > 0

x(0) = v

dx(t) = h(x(t))dt + g(x(t))dw(t),

t > 0

x(0) = v

for z a continuous semi-martingale, w m-dimensional Brownian motion and coefficients h:Rn - Rn, g:Rn

L(Rm,Rn).

The associated semigroup {Pt}t>O

on Cb(Rn,R) for (III) is, however, always strongly continuous in the supremum norm of Cb(Rn,R).

A strong infinitesimal generator A:D(A) c Cb -). Cb can

therefore be computed to get A(4,)(v) = D4(v)(h(v)) + } trace D2O(v) o (g(v) x g(v)), for c C2-bounded on Rn.

v E Rn

(*)

The reader may check that this agrees formally with

the conclusion of Theorem (IV.3.2) (or that of Theorem (IV.4.3)). and [24] for a classical derivation of formula (*).

See [22]

The trajectory field

{vx(t):t > 0, v E Rn} of (III) has a measurable version consisting a.s. of diffeomorphisms in case h and g are sufficiently smooth (Elworthy [19], Kunita [46], Malliavin [51]).

Furthermore, if h and g are linear maps, then

selecting a separable measurable version X:Q x R>0 x Rn _ Rn of the trajectory field implies that for a.a. W E S2 the map X(w,t,.):R

jection for all t > 0.

Rn is a linear bi-

This sharply contrasts the corresponding behaviour

for linear delayed diffusions of §(IV.3).

166

n

§3.

Stochastic Delay Equations

) be a filtered probability space with z:c

Let (S2, F, (Fd t>0 9

->

C(R>,Rm) a

continuous Rm-valued martingale adapted to (Ft)t>0 and satisfying ) < K(t2-t1)

E(z(.)(t2) -

1

1

I

) < K(t2-t1)

t 1

a.s. whenever 0 < t1 < t2, for some K > 0. Suppose hj:Rn + Rn, i = 1,...,p, gi:Rn - L(Rm,Rn), i = 1,...,q are Lipschitz maps.

Let there be given p + q delays as random variables

rj:S2-

R>0

j = 1,...,p, di.Q - R', i = 1,...,q, which are P-essentially bounded and Define

F0-measurable. r =

(essup r.(w),

max

W

7 0 be a common Lipschitz constant for all the hj's and gi's.

Then,

if W11W2 E C2(S2,C(J,Rn)), we have

167

Ilh(W1)-h(W2) 1Ir2(p.Rn)

< P

E

Ihj[w1(w)(-rj(w))-h.[P2(W)(-r.(W))]I2 dP(W)

J

j=1

Q

p E

p L2

J

j=1

< p2 L2

0

1k)1

lip (w)(-r.()) - W (w)(-r.(w)12 dP(W)

-

2

J

1

3

w2112 2

r (,C)

Similarly g is Lipschitz with Lipschitz constant qL.

It remains now to

verify the adaptability condition E(iii) of §(II.1).

To do this it is

11

sufficient to show that for each measurable, for t > 0.

It,

E L(Q,C;Ft), 2 h('P) and g(W) are Ft-

2 n Let W E L (2,C) be Ft-measurable, and P:J X C -> R

the evaluation map (s,rl) Ea n(s), s E J,n E C.

Then p is continuous, and

since each rj is F0-measurable, it follows that

W

>

h('P)(w) _

[P(-r .(W),w(W))]

h

j=1

j

J

ti

is F -measurable. ti

Therefore h(W) E r2(D,Rn;F ); similarly

t

g(W) E 92(SI,L(Rm,Rn);Ft).

Hence all the conditions of Theorem (11.2.1) are

satisfied and so a unique strong solution

ex

E C (c,C([-r,a],Rn)) of the

stochastic DDE (IV) exists with initial process e.

The trajectory

{ext:a > t > 01 is defined for every a > 0, is (Ft)-adapted and has con-

Moreover each map

tinuous sample paths. Tt:c2(c,C;F0)

`

-'

e

c2(c,C;Ft),

t > 0,

ext

is globally Lipschitz, by Theorem (11.3.1). Now suppose in (IV) that z is m-dimensional Brownian motion w adapted to

(F t)0., where each Ft is suitably enlarged by adding all sets of measure zero, i.e. we get the stochastic DDE of It; type q

p

dx(t) =

Z

j=1

h.(x(t-r .))dt + J

x0=nEC(J,Rn) 168

J

E

i=1

g (x(t-d.))dw(t),

t > 0

(V)

with deterministic initial condition n.

Note that the coefficients in (V)

ti

can also be viewed as F0 ® Borel C measurable maps H:S2 x

G:Q x C(J,Rn) ---> H(w,n) =

n

L(Rm,Rn) given by p E

ti

n

C(J,R ) ---> R

j=1

q

ti

h(n(-r.(w)))

,

G(w,n) =

i

E i=1

1

1

So (V) becomes the stochastic RFDE with random

a.a. w E sl, all n E C(J,Rn).

coefficients:

ti

ti

0 < t< a

dx(t) =

x0=n Because of the randomness in the coefficients of (V)' we cannot apply the results of Chapter III to obtain a Markov property for the trajectory field of (V).

However, if the delays rj, di are all independent of the Brownian 0 < t < a}, we shall show that the trajectory field

a-algebra Fa E { n xt:0 < t < a,

n E C} of (V) is generated in a natural way by a random

family of continuous time-homogeneous Feller processes on C, each of which adapted to the Brownian filtration

u < t})0 6)

C

< E + d IIn IIC EIrO-rkl2 < E + 4IIn II2 E, because k > k0.

(5)

Similarly, for k > k0, we get

EIn(u-rk)-n(u-r0)I2X(rk

> u)

IU)

dP

k k1>6

< E + s IIn III E IrO-rkl2 < (1 + 4IIn I102)E Combining

dP

k1u)

(6)

(4), (5) and (6), one gets k0.

(7)

Note that k0 is independent of u E [O,a].

A similar argument applied to the last integrand in (1) yields for every

E>0, a k 0 > 0 such that Elnx(u-dk) nx(u-d0)I2 for all u E [O,a], k > k0. (7) and (8) to obtain

176


0, there is a k0 = k0(e,n) > 0 such that

J w'ES2

E

sup Inxk,w,(v) -r 1X(w',w) as k'-- - for a.a. w,w' E S2. Note also that the subsequence{nxk }k'=1 of {nxk}k=1 converges in L2 to nx as In view of Lemma (3.1), write nxt" nxt,,w, in the form 179

n

nxk(M ) t

=

k

,

nxt ,k (w) X m,.k+(w),

E

m,=1

ni nxkw1(w) t

nxml'k'()X

E

=

m ,k ,(w')

t

m'=1

nk,

k

for a.a. w,w' E 11, t E [0,a], where 0m

E

FD,

rk, =

m'=1

nk,

dk, _

E m=1

rm

E

k

,

X Stm

, k"

k' and

dm''k x f?'

dxml'ke(t) = h(xm" V(t-rm',k'))dt + g(xm.k'(t-dm" ki))dw(t)

(12)

0O Proof:

184

Take r = rj(w'), d0 = di(W') for a.a. w' E Q in (VII).

Then by

hypothesis, lim

u(w'), a.a. w' E S2, where {u(w'):w' E S2} is

ta random family of probability measures on C.

If {nxt:t > 0, n E C} is the

trajectory field of (V), then the identity {Po(nxt)-1}(B) =

P(w',O,n,t,B)dP(w'), B E Borel C, J

wES2

the dominated convergence theorem, and Theorem (I.2.2)(v) imply that lim t+00

Po(nx

)-1

t

= u - J

p(w')dP(w') in M (C).

a

P

w' ES2

Remarks (i)

In our next section we shall give some sufficient conditions in the

linear case under which the system with fixed delays (VII) becomes asymptotically stochastically stable.

(ii) We conjecture that the first assertion of Theorem (3.1) also holds if in (V) we allow the delays rj, dill j = 1,...,p, i =1,...,q to be continuous

stochastic processes, each independent of w. (iii) Although the trajectory field {nxt:t > 0, n E C} of (V) may not in general be isonomous to a Markov process on C, we can still apply the results of Chapter IV to obtain the random family of weak infinitesimal generators

Aw :V(e) c Cb

Cb associated with the family of semigroups {Pt }b0 defined

by the systems p

dx(t) =

q

h.(x(t-r.(w'))dt +

E

j=1

E g (x(t-d1(w'))dw(t) i=1

x0=nEC for a.a. w' E S2.

t > 0 (V(w'))

Using the notation of Chapter IV and the results of Theo-

rems (IV.3.2), (IV.4.2), we observe that when hj, gi are globally bounded the weakly dense subalgebra Tq of all quasi-tame functions in Cb is determined independently of the choice of the random parameter w' in V(w'). cular, Tq c V(Aw

In parti-

for a.a. w' E S2. w3o suppose 0 E Tq and using functional

calculus write formally Pw- (4) = etA

a.a. w' E 0 , t > 0.

Then by (14)'

of the proof of Theorem (3.1) we get O(nxt(w))dP(w) = J

J wES2

etAw,(0)(n)dP(w')

PW'($)(n)dP(w') w ES2

Jw ES2

185

where

p S($)(n) +

+ 3

E

E

k=1

i,i =l

E j=1

DO(n)(h(n(-r(w')))X(0})

D O(n)(gi(q(-di(w')))(e

Recall that S is the weak infinitesimal generator of the shift semigroup {St}t>O and {ek}k=1 is any basis forIRm.

To get the distribution P{w:w E Q,

nxt(w) E B} of the trajectory field to (V), approximate each characteristic function XB weakly by a directed family {fit} in Tq; thus W'

t

P{w:w E S2, nx (w) E B} = lim

t

e J W'ES2

to ( )(n)dP(w'). t

As our final topic in this section, we now turn to the question of selecting a measurable version of the trajectory field to the stochastic DDE(V).

If

the coefficients hj, gi are all Lipschitz, it is easy to check that Remark (V.4.3)(ii) is fulfilled and so the trajectory fields {nxt:t E [O,a],n E C}, {nxt:t E [r,a], n E C} admit measurable versions 2 x [O,a] x C i C, 2 x[r,a] x C ; Ca, for any 0 < a < J.

Note here that we do not require the

delay a-algebra FD to be independent of F. Alternatively, and in relation to our Remark (V.3.1)(iii) on delayed diffusions, it is perhaps instructive to indicate here how a measurable version may be obtained for the trajectory field of the one-dimensional quadratic delay equation dx(t) = [x(t-1)]2dw(t)

0 < t
n(-1 + j/k), (u,n) h-->

(w,u) --a

nk(u-1) are continuous and

w(w)(u) is measurable, so the integrals k t0 k w(w)(u)n (u-1)du, (w,n) - Jj0/k w(w)(u)n (u-1)du

(j+1)/k (w,n) F--> J

J/k

depend measurably on the pair (w,n).

From the preceding equality, it follows

that the map t

(w,n) -a

w(w)(u)nk(u-1)(nk)'(u-l)du

J 0

is(F ® Borel C}measurable for every t E [0,1].

Since this indefinite inte-

gral is continuous in t for each w E P09 n E C, it is easy to see that the map t (w,t,n) T-->

w(w)(u)nk(u-1)(nk)'(u-1)du

J 0

is (F ® Borel [0,1] ® Borel C)-measurable.

follows that (w,t,n),->

From the definition of Yk, it

Yk(w,t,n)(s) is measurable for each s E [-1,0].

Thus Yk is (F 0 Borel [0,1] 0 Borel C, Borel C)-measurable.

Moreover, using

integration by parts (Elworthy [19] p. 79), one has a.s. t+s

[nk(u-1)]2dw(u)

{ 188

s E [-t,0]

f0

=

0

s E [-1,-t)

i.e.

ti Yk(-,t,n) = nxt

k

- nt

t E [0,1], n E C, a.s.

,

Since nk(s) -> n(s) as k - - uniformly in s E [-1,0], it follows easily from

Doob's inequality for the stochastic integral (Theorem (1.8.5)) that -ti

nxt

2

nt as k - - in r (c,C).

Hence by the Stricker-Yor lemma

for C-valued mappings we get a measurable version Y:S2 x- [0,1] X C -

C for

the field {nxt-nt:t E [0,1], n E C}.

A very similar argument to the above gives a measurable version for the trajectory field of the one-dimensional polynomial delay equation dx(t) = [x(t-1)1'dw(t),

0 < t
1Rn, given the supremum norm.

ti

extension also by H.

Solving the linear FOE (XI.I.) for initial data in C, we ti

can extend the semi.group {Tt}

t>O

ti

The splitting (1) of C is topological, so the projections IIS:C

Al

to one on C denoted also by Tt:C -> C, t > 0.

+ S are continuous linear maps.

Since dim U
O*

ti

Lemma (4.1):

For each C E C, and t > 0, we have

CTt(E)]u = Tt(CU), [Tt(C)]S = Tt(CS

Proof:

For C E C, the result follows directly from the well-known invariance ti

of the splitting (1) under {Tt}tom.

To prove it for C E C, consider the

following definition of weak continuity for linear operators on C. operator B:C - C is weakly continuous if whenever bounded sequence in C with Ck(s) -> 0 as k ; as k -

A linear

k}k=1 is a uniformly

for each s E J, then B(Ck)(s)

- for each s E J (cf. the 'weak continuity property (w1)' of Lemma

(IV.3.1)).

The Riesz representation theorem implies that every continuous linear map C - U has a unique weakly continuous extension C -i U.

Hence for the first

assertion of the lemma to hold, it is enough to show that IIU°Tt and Tt°lIU are 192

Lt is clear from the definition of

both weakly continuous for all t > 0.

As the composition of weakly con-

IIU:C - U that it is weakly continuous. ti

tinuous linear operators on C is also weakly continuous, it remains to show that each Tt:C - C is weakly continuous.

This is so by the following lemma.

The second assertion of the lemma follows from the first because ti

CTt(C)]S = Tt(E) - ETt(C)]U = Tt(E) - Tt(&U)

= Tt(& -

U)

=

Tt(ES),

t > 0,

C E C.

o

Lemma (4.2): For each t > 0, TtC - C is weakly continuous. Let v(u) be the total variation measure of the L(Rn) -valued measure Ef:R'0 R by y on J representing H. Fix E E C and define Proof:.

Cf(t) = j_r ITt()(s)Idv()(s) + ITt()(0)I, O Now

t > 0.

rt+s

&(0) +

H(Tu(g))du J

Tt(9)(s) =

Vt+s)

t+s > 0

0

-r < t+s < 0

Thus max(-r,-t) (t+s)Idv(u)(s)

Cf(t) = 1_r

(t+s H(Tu(E))duldv(U)(s) + J0 max(-r,-t)

R(0) + 0

t

+ IC(0) + JO H(Tu(C))dul

t max(-r,-t) IC(t+s) Idv(u)(s) + Cv(u)(J) + 1](IE(0) I+ J < J -r 0

Idu)

max(-r,-t)

J -r

IC(t+s) Idv(u)(s) + WOW +1] W0) I 193

ft

f0

+ [v(u)(J) + t](J O ITu(C)(O)Idu +

ITu(&)(s)Idv(u)(s)du)

0

0

-r

t


0,

J_r

and

C = v(u)(J) + 1.

By Gronwall's lemma, we obtain t

Ef(t)


0. Now let

be a uniformly bounded. sequence in C converging pointwise

to 0; then the sequence

is uniformly bounded on [O,t] and kh(t) -, 0

as k -. co for each t > 0, by the dominated convergence theorem.

The last

estimate then implies again by dominated convergence that Ckf(t) -1.0 as k - for each t > 0.

In particular, Tt(Ek)(0) - 0 as k -

for every t > 0.

But

for each s E J, k(t+s)

Tt+s(Ek)(0)

-r < t+s < 0 t+s > 0, k > 1,

so Tt(Ek)(s) -> 0 as k - - for each s E J, t > 0.

o

By analogy with deterministic forced linear FDE's, our first objective is to derive a stochastic variation of parameters formula for the forced linear system (XI).

The main idea is to look for a stochastic interpretation of the

deterministic variation of parameters formula corresponding to non-homogeneous linear systems

dy(t) = H(yt)dt + g(t)dt,

t > 0 (XIII)

YO =n EC (cf. Hale [26] pp. 143-147; Hale and Meyer [29]). To start with, we require some notation. 194

Denote by t:J - L(Rn) the map

A = X{U}I, where I. E L(IR") is the identity n x n matrix. "U

^U

Also, for any

nu

linear map B:C(J,Rn) + C(J,Rn) and any A E

C(J,L(Rm,Rn)),

A(s) = (a1(s), a 2(s),...,am(s)),

s E J

BA = (B(aI),B(a2),...,B(am)) E C(J,L(Rm,Rn))

let

where a.(s) is the j-th column of the n x m matrix A(s) for each 1 < j < m Thus each aj E C(J,Rn) and BA E C(J,L(IR R )).

and s EJJ.

,

If F:[a,b] - C(J,L(Rm,Rn)) is a map, define the stochastic integral

fa F(t)dw(t) by b

[Jb

F(t)dw(t)](s) = J a

F(t)(s)dw(t),

S E J,

a

whenever the Ito integral tb F(t)(s)dw(t) E Rn exists for every s E J. This will exist for example if F is measurable and fa IIF(t)Ik. dt < -.

In case

fa F(t)dw(t) E C(J,Rn) a.s., its transform under a continuous linear

map C(J,Rn) + Rn is described by

Lemma (4.3):

"'ti

Rn be continuous linear and suppose L:C(J,Rn) -'

Let L:C(J,Rn)

Rn is its canonical continuous linear extension using the Riez representation ti

theorem.

Assume that F:[a,b] -> C(J,L(Rm,Rn)) is such that to F(t)dw(t) E b

Then to LF(t)dw(t.) exists and

C(J,R ) a.s. b

m

b ti

F(t)dw(t)) = J

L(J a

LF(t)dw(t) =

E

j=1

a

b ti

L(f(t))dw.(t)

J a

i

a.s., where fj(t) is the j-th column of F(t) and wj(t) is the j-th coordinate of w(t), j = 1,...,m.

Proof:

Represent L by an L(Rm,Rn)-valued measure on J via the Riesz repre-

sentation theorem; then use coordinates to reduce to the one-dimensional case m = n = 1.

Namely, it is sufficient to prove that if u is any finite positive

measure on J and f E C2([a,b] x J,R;dt 0 du), then fl

Jb

a

-r

0

f(t,s)du(s)dw(t) = f

r

Jb f(t,s)dw(t)du(s)

a.s.

(3)

a

195

Suppose first that f = X[a,R]M[y,6]

,

rectangle [a,B] x [y,5] c [a,b] x J.

the characteristic function of the Then (3) holds trivially.

Also, by

linearity of the integrals in f, (3) is true for all simple functions on [a,b] x J with rectangular steps.

Since these are dense in

r2([a,b] x J. dt ® du), we need only check that each side of (3) is continuous in f E C2([a,b] x J,R,dt ® du). Elrb a

But this is implied by the easy inequalities: fb [f0

0

f(t,s)du(s)dw(t)I2 =

f-r

a

u(J)

fb fl _

a

r

-r

f(t,s)du(s)]2dt

lf(t,s) I2du(s)dt = u(J)

111,112

t2

and ElJ0

b f(t,s)dw(t)du(s)I2 < u(J)EJO lJbf(ts)dw(t)I2du(s) -r a

r fa

JO

Jb lf(t,s)I2dt du(s) = u(J) IIfII22.

= u(J) r

r

a

°

Remark

Since f is r2, there is a version of the process s N Jb f(t,s)dw(t) with almost all sample paths in r2(J,R;du). Next we shall need the following result in 'differentiating' a stochastic integral with respect to a parameter.

Lemma (4.4):

Assume that f:[O,a] X [0,a] - L(Rm,Rn) is continuous on

{(t,u):O < u < t < a} with partial derivative (t,u) - at f(t,u) continuous on {(t,u):O < u < t < a}.

Let z be the process

t

f(t,u)dw(u),

z(t) = J

t E [O,a].

0

Then

dz(t) = f(t,t)dw(t) + {JO at f(t,u)dw(u)}dt

viz.

196

v

It f(t,u)dw(u) = It f(u,u)dw(u) + It 0

0

0

f(v,u)dw(u)}dv

(4)

{fo

f or all t E [0,a], a.s. ti

Proof:

Suppose first that w is a process on [0,a] with almost all sample ti

paths piecewise C1. Z(t) =

Define z by

f(t,u)dw(u) =

f(t,u)w'(u)du,

I0t

t I0

t E [0,a],

ti a.s.

Then almost all sample paths of z are differentiable and a.s.

t

IV

z'(t) = f(t,t)w'(t) + IO

for all t E (0,a).

a Tt-

f(t,u)w'(u)du '\j

Thus

dz(t) = z'(t)dt [It

f(t,t)w1(t)dt

f(t,u)w'(u)du]dt

+

=

0

t =

f(t,t)dw(t) + [ IO t f(t,u)dw(u)]dt ti

i.e. (4) holds if w is replaced by the piecewise C1 process w. If w is Brownian motion on Rm, define piecewise linear approximations of Let 11:0 = t1 < t2 < t3 < ... < tk = a be a partition of [0,a].

w as follows.

Define the process w11 in Rm by .

(t-tj), ti < t < tj+1

= w(ti) +

w11 (t)

i

where A t i

= tj+1-tj,

A w = w(tj+1)-w(tj), j = 1,2,...,k. i

Suppose G: $Z x [0,a] -> L(Rm,Rn) is an (Ft)0 0 such that 197

IIG(u1) - G(u2)IIr 2 < jE/,ia whenever u1, u2 E [O,aj,

mesh

lu1-u2I

< S.

Suppose

II < d1. Then k-1 E

a

G(t)W(t)

G(t.)A.w -

j=1

J

J

I

0

k -1

j=1

t

{G(t.) -

= kE1

r J+1 1

j=1

AJw G(t)dt

tj

j=1

J

J

Jtj1

k-1 E

G(t )A w

E

a

J

f

G(t)dt}Aw.

tj

J

Now E(AjwlFt.) = 0, E(IAjwI2IFt ) = Ajt a.s. for j = 1,2,...,k-1 and

J

J

tj+l

J+1

IIG(tj) -

J

2=

G(t)dtII

tj

4

II J

J

tj

{G(tj)-G(t)}dtll

2

jtJ+1


O

possesses

C* defined by the relations

(a,AHC) = (A*Ha,E), C E D(AH), a E D(A*H)T

-al(t),

(A*Ha)(t)

0 < t < r

r0

-

a(-s)du(s), t = 0 ;

J r to

D(A*H) = {a:a E C*, a is C1, a'(0) = J -

a(-s)du(s)}. r

Furthermore, a(AH) = a(A*H) and the spectra are discrete consisting only of eigenvalues with finite multiplicities.

Both a(AH) and a(A*H) are invariant

under complex conjugation; and the multiplicities of the eigenvalues coincide.

Construct U* c C* using the generalized eigenspaces of A*H which

correspond to eigenvalues with non-negative real parts. = d.

Take a basis 0 =

Then dim U* = dim U

for U and a basis

W1

'Pd

for U* such that (Wj.0i) = 6ji, i,j = 1,2,...,d.

The basis (D of U defines a

unique matrix representation B E L(Rd) of AHIU i.e.

AH.D

= OB, A*H T = BT,

where AHD, OB, A*H'V, B'' are all formally defined like matrix multiplication.. 206

Note that the ei.genvalues of B are precisely those A E a(AH) with ReX > 0. The reader should. also observe here that the splitting (1) of C is realized

by the bilinear pairing (10) through the formula &U

for all E E C.

= IP(Y'oC)

The results quoted in this paragraph are all well-known for linear FDE's and proofs may be found in Hale [26]. ti

We would like to extend formula (1i) so as to cover all E E C.

First

note-that the bilinear pairing (10) extends to a continuous bilinear map C* x

C + R defined by the same formula.

So the right hand side of (11)

ti

makes sense for all E E C.

But both sides of (11) are continuous with

respect to pointwise convergence of uniformly bounded sequences in C, because ti

of the dominated convergence theorem. and the weak continuity of 11 :.C - U. ti

As C is closed under pointwise limits of uniformly bounded sequences, (11) ti

holds for all & E C.

In view of the above considerations we may now state the following corollary of Theorem (4.1).

Corollary (4.1.1):.

Define 4) c U, 'Y c U* and B E L(Rd) as above.

{xt:t E [0,a]} be the trajectory of (XL) through n E C.

Let

Define the process

z:St x [0,a] - Rd on Rd by z(t) = ('Y,xt), 0 < t < a. Then xt = Oz(t) for all t E [O,a] and z is the unique solution of the stochastic ODE: dz(t) = Bz(t)dt + 'Y(0)g(t)dw(t), 0 < t < a 1

I

z(0) = ('',n) Proof:.

Use the definition of z., the stochastic variation of parameters

formula and Lemma (4.3) to obtain t

z(t) = (`Y,Tt(n)) +

(`Y,Tt-uAg(u))dw(u)

a.s.

0

for all t E [O,a].

Take differentials and use properties of the bilinear

Pairing and the generator AH of {Tt}O0

ti

°°x0

=

T_u 'Sg(u)dw(u) E C2(St,C),

O

(X1()

J (iv) (v)

If g is constant, then °°x is stationary; If g is periodic with period k > 0, °°x

is periodic in distribution

with period k i.e.

Po0°x(t)-1 Proof:

Po°°x(t+k)-1 for all t > 0.

=

Suppose g and its derivative g' are both globally bounded on R.

We define the process °°x:0 x [-r,oo) - Rn by ti

t J

AS g(u))(0)dw(u), (Tt-u

t > 0

(12)

x(t) ti

f_00 (T_u AS g(u))(t)dw(u).

t E J

To see that °°x is well-defined, we note the existence of the limit

1t

(Tt-u

ti

a.s.

vv

AS g(u))(0)dw(u) = limn ft (Tt-u AS g(u))(O)dw(u)

Indeed the map u F-->

(Tt_u

AS

g(u))(0) is C1 and so integrating by

parts gives the classical pathwise integral t lv

(Tt-u

AS g(u))(0)dw(u) =

-

A (0)g(t)w(t) - (Tt-v

Jv

{(Tt-u

9(v))(0)w(v)

A g(u))(0)}w(u)du 209

a.s. for all v < t.

Now by Hale ([26], p. 187) there are constants M > 0,

a < 0 such that ti

IITt AS g(u) II < Mest 11g(u) II

(13)

But the law of the interated logarithm for Brownian

for all t a 0, u E R.

motion (Theorem (I.8.2)(iv)) implies that there is a constant L > 0 such that for a.a. W E sl there exists T(w) < 0 with the property

IW(w)(t) I

< L(ItI log log Itl)'

for all t < T(w).

(14)

In particular, we have a.s. ti

liml(Tt-vASg(v))(0)w(v)I

ML IIglIC lim es(t-v)(Ivi

M lim

es(t-v) II9(v)IIIw(v)I

log log Ivl)'

V-),-CO

= ML II911C

eat lim

e-av(Ivl log log IvI)i

v+-oo

where

<MLIIgIIC eat lim

Ivi a-sv=0

I1911C = sup { 119(t) II

. t E R}.

Next we consider the existence of the a.s. limit ft

lira

v-1-

V

a

{(T t_o AS g(u))(0)}w(u)du.

To prove its existence, it Is sufficient to demonstrate that ti

t

f -00 a.s.

{(Tt-u OS g(u))(0))w(u)Idu < co

Note first that

ti u {(Tt-u AS g(u))(0)} = -H(Tt-u ASg(u)) + (Tt-u AS g'(u))(0)

for u < t; so

210

ti

I2u {(Tt-u AS 9(u))(0))w(w)(u)I

IIHII

r, s E J.

+

v

0

e-su lw(u) Idu

J v

In the above inequalities, all limits exist a.s. because

for a.a. wESt, e-sulw(w)(u)I

Lue-13u


r. In particular, "xtIIC

lim Ilxt -

a.s.

= 0

tTo establish assertion (ii) of the theorem, note that by cominated convergence and Fubini`'s theorem one gets

EK1= E(J0

e-6uIw(u)kiu)2
r.

M2m

ii S112C

-3

(

-By

IIHII II9IIC+ IIg'IIC)2

e20(t-r)

Now take

K = 3M2 IInSII2 - 3m

(IIHII IIgIIC + II9'IIC)2

a-2gr

>0

3

and a = 28 < 0.

Then

E Ilxt - xtii

< Keat for all t

> r. 213

We next show that °°x is the solution of the stochastic FDE (XV) through ti

0

T_u AS g(u)dw(u).

e = J 111-°,

By our main existence theorem.(Theorem (IL.2.1)), the stochastic FDE(XV) will have a unique solution in r2(52,C([-r,a],,n)) for any a > 0 if e E t2(S2,C).

In fact, we can write x in the form ti

AS

Tt(e)(0) + J 0t

I.

g(u)](0)dw(u).

ETt-u

t > 0

°°x(t) =

(15) t E J

e(t)

provided we can show that 0 E 92(S2,C).

Now the stochastic integral depends

measurably on parameters, so the process ti

0

[T

e(s)' = J

_u

s E J.

AS g(u)](s)dw(u),

has a measurable version 0:12 x J - Rn according to a result of Stricker and

Yor ([72], Theorgme 1, §5, p. 119).

Indeed a i.s sample continuous.

To see

this, use integration by parts to write ti

s

9(s)

AS g(u)](0)dw(u)

= J -00 [T_u+s

ti

s

= AS(0)g(s)w(s)-lim

v-p-

A (0)g(s)w(s)

ti

s J

s

ti

Since g, g' and w are continuous, the processes

a.s. for all s E J.

s I---

ti

J [-H(T_u+sOSg(u))+{T-u+sASg'(u)}(0)]w(u)du (16) v

v

ti

AS

[-H(T-u+s

are sample continuous on J.

AS

g(u)) + {T_u+s

g'(u)}(0)]w(u)du

Thus for a to have continuous sample paths one

heeds to check that the a.s. limit on the right-hand side of (16) is uniform Now, for v < s < 0,

for s E J. s

f v [-H(T-u+s

214

ti

ti

AS

AS

g(u)) + {T-u+s

g'(u)}(0)]w(u)du

JS

-

-00

^L

1\,

[-H(T-u+S AS g(u) + T_u+S A 9'(u)]w(u)dul

eB(-'+s) + M Ilg'II, es(-u+s)}Iw(u)Idu

11H

v

< M( IIHII II911C + IIg IIC ) e Sr J

e-au i

w(u) du

(17)

V

Since

lim f e-Sulw(u)Idu = 0, it follows immediately from (17) that 00

the limit in 00 the right-hand side of (16) is uniform in s E J and so 6(w) E C for a.a. w E Q.

As the map J 3 s -> 6(w)(s) E Rn is Borel-measurable for

a.a. W E S2, it follows that 0 E to(c,C).

Indeed, starting from (16), it is

easily seen that

I0(s)2 < 2 II9II 2 Iw(S) I2 a.s. for all s E J.

+ 2142(11.H 111b

II C + II9' II C )2

a-2Br K1

Taking suprema and expectations gives

E sup I0(s)I2 < 2 II9II2CE sup 1w(s)12+ 2M2(IIHII sEJ

SEJ < 2 11911

E sup Iw(s) I2 - 2m M2( IIHII I19I1C + 119' IIc)2 a-20r sEJ s

0.

Note also that this is a consequence of assert-

ion (ii) of the theorem for t > r.

Since {w(t):t E R} is a Gaussian system (§§I.8(A),(B)), the right-hand side of (16) is also a Gaussian system. process (Theorem (L.8.1)(ii)).

Thus 6:52 x J - Rn is a Gaussian

In a similar fashion °°x:S2 x [-r,-) - Rn,

xt:52 x J -Rn are Gaussian for each fixed t > 0. The stochastic FDE (XV) for x may be derived directly from (15) by taking stochastic differentials as in the proof of the stochastic variation of parameters formula (Theorem (4.1)).

Thus ti

d x(t) = {Uf Tt(0)(0))dt + (AS 9(t))(0)dw(t)

215

+

{[Tt-JAS JO

= {H(Tt(e)) +

g(u)](0)dw(u)}dt

J0t H(Tt-u A

g(u))dw(u)}dt + A (O)g(t)dw(t)

t > 0.

= H(°°xt)dt + o (0)g(t)dw(t),

Let us now check assertion (v) of the theorem. with period k > 0.

Consider°°x(t+k) for t > -r.

Suppose g is periodic on R Then a.s.

t+k

"x(t+k)

=

J_00 Tt+ku

_

Jt

g(u)dw(u)

ti Tt_u' OS g(u'+k)dw(u'+k)

(u' = u-k)

But g(u'+k) = g(u') and {w(u'):u' E R}, {w(u'+k)-w(k):u' E R} are isonomous, so by isonomy properties of the stochastic integral (Lemma (111.2.3)) we get

x(t+k) = J t'O Tt-u' A g(u')dw(u'+k) - t. flt_u.A 9(u')dw(u') = x(t) for every t > -r.

Thus xj[O,oo) is periodic in distribution with period k

x(t)-1

i.e. Po

Po°°x(t+k)-1

=

for all t > -r.

In particular if g is constant, one can take the 'period' k to be arbitrary and so Pa x(t)-1 is independent of t > -r. This completes the proof of the theorem.

Hence °°x is stationary.

o

Corollary (4.2.1): Let H:C - Rn be continuous linear and g:R -). L(Rm,Rn) be C -bounded.

Suppose that the deterministic RFDE

dy(t) = H(yt)dt,

t > 0

(XII)

is globally asymptotically stable i.e. Re x > 0 for every X E a(AH), where

AH is the infinitesimal generator of {Tt}. Then the process

z :St x [-r,00)+ Rn

given by

Jt

CO

xt =

00

Tt-u og(u)dw(u),

t>0

is a sample continuous Gaussian solution of the stochastic FDE 216

dx(t) = H(xt)dt + g(t)dw(t),

t > 0

such that for any trajectory {xt:t > 01 c r2(sl,C) of (XI) there are constants K > 0,

a < 0 with

E JJxt - xt J J 2 < Ke°''t

for al l t > r.

If g is constant, "x is stationary; if g is periodic with period k, x is periodic in distribution with period k.

Proof:

If o(AH) lies to the left of the imaginary axis, the splittings (1)

and (2) reduce to the trivial case

ti

C=S, U={0}, C=S, IIS=idC,IIS=idC. Hence xt = xt for all t > 0; so all conclusions of the corollary follow immediately from the theorem.

Corollary (4.2.2):

o

Suppose all the conditions of Corollary (4.2.1) hold and

let g be constant i.e. g(t) = G E L(Rm,Rn) for all t E R.

Then the stochastic

FDE

dx(t) = H(xt)dt + Gdw(t)

t > 0

(XVI)

is (globally) asymptotically stochastically stable i.e. for every n E C lim

Po(nxt)-1 exists in Mp(C) and is an invariant Gaussian measure for the

t->W

stochastic FDE (XVI), (§(III.3)).

Proof:

As °°x is stationary, let Po("xt)-1 =

show that the transition probabilities

of (XVI) converge to y0 in Mp(C) as t

110 =

POO-1 for all t > 0.

We

Po(nxt)-1:t > 0, n E C} CD for every n E C.

Thus it is

sufficient to prove that l

J

mm

J

EEC

CEC

for every bounded uniformly continuous function 0 on C. But by Corollary 2 = 0 for every ri E C; so it follows from (4.2.1) above, lim EI1TI xt - xt1l

t-

C

217

the proof of Lemma (LLL.3.1)(i.i) that li.m [J

4(Xt(w))dP(w)] = 0,

q(nxt(w))dP(w) - J

in

t-

0 E Cb*

0

i.e.

(E)p(O,n,t,dE) -

lim CJ CEC

0(&)d j(&)] = 0, 0 E Cb. J&EC

Thus lim p(O,n,t,.) = u0, a Gaussian measure on C.

This implies that

t-,

Pt(u0) = vo for all t > 0, where

{Pt}.b0 is the adjoint semigroup on M(C)

associated with the stochastic FDE (XVL) (§(ILL.3)).

a

Remark (4.3)

Note that the invariant measure u0 of Corollary (4.2.2) is uniquely determined independently of all initial conditions n E C (or in 92(c,C)). In contrast with the asymptotically stable case, we finally look at the situation when a(AH) has some elements with positive real parts.

In this

case it turns out that the variance of every one-dimensional projection of nxt in U diverges to + - exponentially as t

for each n E C.

Indeed we

have

Theorem (4.3): (4.1.1).

Define T c U*, B E L(Rd), z:sl x e _ Rd as in Corollary (Rm,Rd).

For the stochastic FDE (XVI) (constant G), let C = '(0)G EL

Suppose {Ai = aj + ib:1 < j < p} is the set of all eigenvalues of B where a. > 0 and bj E R, j = 1,2,...,p.

Denote by

the (Euclidean) inner

product on Rd i.e. for u = (ul,...,ud), v = (v1,...,vd) E Rd, =

E uivi. i=1

(i)

Assume that the pair (B,C) is controllable viz.

(18)

rank (C, BC, B2C,...,Bd-1C) = d.

Then for every v E Rd, there exists 1 < j < p and t0 > 0 such that given e > 0 we can find D1, D2, D3 > 0 with the property

(2a+e)t

2a.t D1e

J

for all t > t0. 218

- D2 < EkI2 < D3e

i

(19)

( i i )

If the rank condition (18) is not satisfied, then either the inequal-

ities (19) hold or else EIl2 = 0 for all t > 0 and any v E Rd.

Proof:

The following argument is taken directly from (Mohammed, Scheutzow,

WeizsScker [61]). Fix t > 0.

Since z satisfies the stochastic ODE

dz.(t) = Bz(t)dt + Cdw(t)

it follows that z(t) =

etBz.(0)

+

t

(t-u)BC fe

dw(u)

0

and

Ez(t) = etBz(O) = etB(T,n),

t > 0.

Let v = (v1,v2,...,vd) E Rd, w = (w1,w2,.... wm).

Then

jt

e(t-u)B Cdw(u)>12

EI12 = EI t0.

The molecule is of mass m and moves under no external forces; a represents a frictional (viscosity) coefficient function having compact support, and Y:R3 x 1R3 , 1R is a function giving the random gas forces on the molecule.

The position and velocity of the molecule at time t > t0 are denoted by E(t), v(t) E I R 3 respectively.

Without loss of generality we may assume that

t0 = 0, supp a c [O,r] for some r > 0 and m _ 1.

It is clear that one needs

to specify vIJ, J = [-r,0], for equations (I) to make sense. stands for 3-dimensional Brownian motion.

As usual w

Now (I) is a stochastic FDE.

To

see this note that

223

t

to S(t-t')v(t')dt' =

Jo

J-t

S(-s)vt(s)ds,

t > 0,

ti and define the mappings H:J° x C(J,R3) . R3, H:R> x C(J,R6) --> p0 : C (J ,R3)

i R3 by

R6,

0

- f-t 0(-s)n(s)ds

0 < t < r

0

H(t.n) = I

s(-s)n(s)ds

t > r

pO(n) = n(0)

0 o

H

p0 H

J

(0) for n E C(J,R3).

Setting x(t) _ `v(t)) E R6, t > -r,

w(t) _

E

R6

w(t) it is easily seen that equation (I) is equivalent to the stochastic FDE

dx(t) = H(t,xt)dt + Y(x(t))dw(t), in R6.

t > 0

(II)

Note that this stochastic FDE is time-dependent for 0 < t < r but

becomes autonomous for all t > r.

If U E r1([0,r],R), it follows that H and

ti

H are continuous and Lipschitz in the second variable uniformly with respect ti

to t E R>O.

In fact H(t,.), H(t,.) are continuous linear maps with norms ti

uniformly bounded in t E R>O.

Now xJJ is specified by

v.1J and so from Theorem (11.2.1), (111.1.1) the stochastic FDE (II) has a

unique Markov trajectory {xt}b0 in C(J,R6) with given v!J. cess {xt}b0 is time-homogeneous for t > r.

The Markov pro-

In contrast to the classical

Ornstein-Uh.lenbeck process, observe here that the pair {(fi(t), v(t)):t > -r}

does not correspond to a Markov process on R6, yet the trajectory {(Et,vt): t > 0} in C(J,R6) does have the Markov property. We would like to consider the velocity process {v(t):t > -r} in the simple

case when the noise coefficient Y is identically constant i.e. let Y(x,y) _ mYO E R for all x,y E R3.

Then v satisfies the autonomous stochastic FDE

dv(t) = H0(vt)dt + YO dw(t)

224

(III)

for t > r, where HO:C(J,R3) 0 H0(n) = - f-

.

R3 is given by

S(-s)n(s)ds,

n E C(J,R3).

r

By Theorem (VI..4.1), write t

vt = Tt-r(vr) + J

a.s., where {Tt)

r

Tt-u Dy0 dw(u),

t > 2r

is the semigroup of the deterministic linear drift FDE

t>O

t > r.

dy(t) = H0(yt)dt,

HO),

O

Now suppose

S(-s)ds < 7/2r.

f-r

We show that if A E a(A

the spectrum HO

of the generator of {Tt}, then Re A < 0.

Write A = Al + iA2 E a(A

Suppose if possible that Al > 0.

some A1,A2 E R.

), for

Using Hale [26] (Lemma

(2.1), p. 168), A satisfies the characteristic equation A + JO

S(-s)eAsds = 0

(1)

Hence

A1s

fo

+

S(-s)e

-

cos A2s ds = 0

2)

sin A s ds = 0

3)

r

and a1s 2 +

S(-s)e

fo

r

But from (3),

IA2sI < IA2Ir < r 1-r

Is

Isin A1sIds

O 0

< r J-

B(-s)ds < ir/2 r

for all s E J.

Therefore cos a2s > 0 for all s E J and is positive on some

open subinterval of J.

So from (2), Al = - t0r B(-s)exis cos A2s ds < 0

where S is assumed positive on a set of positive Lebesgue measure in J. is a contradiction and Rea

This

must be less than zero for all A E a(AHO). There-

fore, according to Corollary (VI.4.2.1), we obtain

225

In the system (I) assume that y is constant (-myO),S has compact

Theorem (2.1):

support in [O,r], a E f1([O,r],RO) and 0 < fr s(u)du < n/2r.

Then there

is a sample continuous Gaussian solution {( &(t),°°v(t)):t > r} of (I) and

positive real numbers K,a

such that

rt

v(u)du

0%(t) = fi(r) +

(i)

J

t

00

=

vt

J

00

r

Tt-u Ay0

dw(u)

t > r,

a.s.,

for every solution (E,v) of (1),

(ii)

E JI&t -t,2
2r, -v is stationary and E has a.a. sample paths C1.

(iii)

Remark

Physically speaking, the above theorem implies that the 'heat bath' will always eventually stabilize itself into a stationary Gaussian distribution for the velocity of the molecule.

U.

Stochastic FDE's with Discontinuous Initial Data

This is a class of stochastic FDE's with initial process having a.a. sample paths of type t2 allowing for a possible finite jump discontinuity at 0. These equations were studied by T.A. Ahmed, S. Elsanousi and S.E.A. Mohammed and can be formulated thus: dx(t) = H(t,x(t),xt)dt + G(t,x(t),xt)dz(t), t > 0 x(0) = v E r2(s,Rn) s

x(s) = 0(s)

for all s E [-r,0).

In (IV) the initial condition is a pair (v,0) where v E C2(S2,Rn) and 226

9 E C2(c,r2(J,Rn)).

Note that here we confuse t2 with L2, the Hilbert space

of all equivalence classes of (Lebesgue)-square integrable maps J -> Rn.

trajectory of (IV) is then defined as pairs {(x(t),xt):t > 01 in

The

Rnx.C2(J ,Rn).

We assume that the coefficients

H:R>0

x Rn x C2(J,Rn) -+ Rn, G:R0 x Rn x t2(J,Rn) - L(Rm,Rn)

are measurable with the maps H(t,.,.), G(t,.,.) globally Lipschitz on

Rn

x

92(J,Rn) having their Lipschitz constants independent of t E Rte.

The

is a sample continuous martingale on the filtered

noise process z:R x S2

probability space (SZ,F,(Ft)t>o,P) with z(t,.) E E2(S2,Rm;Ft) for all t E R>0

and satisfying McShane's Condition II(E)(i).

Using the method of successive

approximations (cf. Theorem (11.2.1)), it can be shown that there is a unique measurable solution x:[-r,o') x 0 - Rn through

(v,o) E C2(S2

n;F0) x C2(S2,L2(J,Rn);FO)

with a continuous trajectory {(x(t),xt):t >01 adapted to (Ft)t>O (Ahmed [1]). From the point of view of approximation theory, a Cauchy-Maruyama scheme

can be constructed for the stochastic FDE (IV) in the spirit of McShane ([53], Chapter V, §43,4, pp. 165-179).

For more details on this matter see [1].

In

addition we would like to suggest the following conjectures:

Conjectures (i)

In the stochastic FDE (IV), suppose the coefficients H,G satisfy the

conditions of existence mentioned above. motion adapted to (Ft)t>O.

Let z = w, m-dimensional Brownian

Then the trajectory {(x(t),xt):t > 01 corresponds

to a Feller process on Rn x C2(J,Rn).

If H, G are autonomous viz

dx(t) = H(x(t),xt)dt + G(x(t),xt)dw(t), t > 0, then the above process is time-homogeneous.

(V)

The transition probabilities

{p(t1,(v,n),t2,.):0 < t1 < t2, v E Rn, n E t2(J,Rn)} are given by

p(t1,(v,n),t2,B) = P{w :w E S2,

where B E Borel {Rn x

c2(J,Rn)}

((v,n)x(.)(t2), (v'n)xt2(w)) E B}

and (v.n)x is the unique solution of (IV)

through (v,n) E Rn x L2(J,Rn) at t = t1.

227

Let Cb = Cb(Rn x t2(J,Rn),R) be the Banach space of all uniformly

(ii)

continuous and bounded real functions on Rn x c2(J,Rn).

Define the semi-

group {Pt}y for the stochastic FDE (V) by

t > 0,

Pt(4)(v.n) = E4((v,n)x(t), (v,n)xt).

0 E C.

Define the shift semigroup St:Cb

Cb, t > 0, by setting

ti

St(4)(v,n) = 4(v,nt), for each 4 E Cb.

t > 0,

The semigroups {Pt}

t>O

and {St }will have the same domain

of strong continuity Cb c Cb (cf. Theorem (IV.2.1)), but it is not clear if Cb # Cb in this case (cf. Theorem (IV.2.2)).

However it is easily shown

that both semigroups are weakly continuous.

Let A,S be their respective

weak infinitesimal generators (cf. IV §3); then we conjecture the following

analogue of Theorem (IV.3.2): Suppose 0 E V(S),$ C2, D@ globally bounded and D20 globally bounded and globally Lipschitz.

Then 0 E V(A) and

A(4)(v,n) = S(4)(v,n) + D14(v,n)(H(v,n)) n +

2

D, 4(v,n)(G(v,n)(eG(v,n)(ej))

E

j=1

1 where D0, D20 denote the partial derivatives of 0 with respect to the first

variable and

is any basis for Rn.

Remark

In contrast with the non-Hilbertable Banach space C(J,Rn), the state space

Rn x r2(J,Rn) carries a natural real Hilbert space structure and so Cb(Rn x L2(J,Rn),R) contains a large class of smooth (non-zero) functions with bounded supports.

By a result of E. Nelson (Bonic and Frampton [6]), a

differentiable function on C(J,Rn) with bounded support must be identically zero.

§4.

Stochastic Integro-Differential Equations

In the stochastic integro-differential equation (SIDE)

228

0

r0

dx(t) = {I

h(s,x(t+r(s)))ds}dt + {J-

r

r

g(s,x(t+d(s)))ds}dz(t),t > 0 (Vt )

x(t) =

t E J = [-r,O]

z:52- C(I O,Rm) is a. continuous Rm-valued martingale on a filtered probability

space (S2,F,(Ft)t>,O,P), satisfying the usual conditions of McShane (Conditions The coefficients h:J x Rn -> R'

E(i) of Chapter II).

g:J x Rn

L(Rm,Rn) are

continuous maps which are globally Lipschi.tz in the second variable uniformly

with respect to the first.

Denote their common Lipschitz constant by L

0.

The delay processes r,d:J x S2-> J are assumed to be (Borel J ® F0, Borel J)-

measurable and the initial condition e E 92(S2,C(J,Rn);FO).

To establish the

existence of a unique solution we shall first cast the stochastic IDE (VI) into the general format of Chapter II. §1.

h .j 2(S2,C)

L2(S2,Rn), g:r2(S2,C)

h(W)(w) = J

Indeed, let us define the maps

C2(S2,L(Rm,Rn))

as follows:

h(s,W(w)(r(s,w)))ds Or 0

ti

9(s,W(w)(d(s,w)))ds

9(W)(w) = J -r

for all T E 92(S2,C), a.a. W E Q.

Observe now that (VI) becomes the stochastic

FDE

dx(t) = h(xt)dt + g(xt)dz(t),

t > 0

x0 = 0.

Note also that the coefficients h, g are globally Lipschitz because if

y,1,W2 E

t2(S2,C), then IIh(T1)-h(W2)

112

2 n t (S2,R )

(0 f

= 1 /-r

{h(s,,U1(w)(r(s,w))) - h(s,W2(w)(r(s,w)))}dsl2dP

r




h(s,W(w)(r(s,w)), (s,w) -a g(s,W(w)(d(s,w))) are(Borel J ® Ft)-measurable Thus by Theorem (11.2.1) the stochastic

whenever W E t2(St,C;Ft), for t > 0.

IDE (VI.) has a unique sample continuous trajectory {xt:t > 0} in C(J,Rn) through 0.

The trajectory field of (VI.) describes a time-homogeneous Feller process

on C if z = w, m-dimensional Browni.an motion and the delay processes r, d are

According to Theorem

just (deterministic) continuous functions r,d:J -> J.

(IV.3.2), the weak generator A of the associated semigroup {Pt}

t>O

is given

by the formula 0

A(O)(n) = S(O)(n) + DO(n)

h(s,n(r(s)))ds X{0}) J-r

1

+ -Z

n -2-

0

0

D 4(n)([

E

g(s,n(d(s)))(e.)dsX{0}, f-r g( s,n(d(s)))(ej)dsX{0})

J-r

j=t

where ¢ E Cb = Cb(C,R) satisfies the conditions of Theorem (IV.3.2), and the notation is the same as that of IN §3. If the delay processes r, d:J x Q - J are assumed to be independent of the

Brownian motion w, we believe that the trajectory field of the stochastic IDE (VI) corresponds to a random family of tiarkov processes on C(J,Rn) in very much the same spirit as that of Chapter VI. §3 (Theorem (VI.3.1) for stochastic DDS's). For deterministic delays r, d:J - J and Brownian noise z = w the coeffic-

ients H:C - Rn, G:C H(n) = J O

r

L(Rm)Rn), h(s,n(r(s)))ds, G(n) =

O f-r

g(s,n(d(s)))ds, n E C,

a re clearly globally Lipschitz and so all the regularity properties of

Section V §4 hold in this case viz. Theorems (V.4.2), (V.4.3), (V.4.4), (V.4.6) and Corollaries (V.4.4.1), (V.4.7.1).

It is not clear, however, if

the trajectory field admits versions with continuous or locally bounded sample functions.

§5.

Infinite Delays

The problem of determining sufficient conditions for the existence (and 230

uniqueness) of stationary solutions to stochastic FDE's was first considered by K. Ito and M. Nisio ([41], 1964) in the context of an infinite retardation time (r = °°).

We quote here some of their results wi.thout proofs.

For

further details the reader should consult [41]. For simplicity, we only consider the one-dimensional case.

The state

space is the Polish space C = C((-,O],R) of all real-valued continuous functions (-°°,0] -> R furnished with the metri.c

E 2'" n=1

Iln- Iln

1 + Iln-On

where IIn -EIIn = - supO In(s) - Vs)I,

n,

E C.

Consider the stochastic

FDE

dx(t) = H(xt)dt + G(xt)dw(t),

t > 0.

(VLI)

for a one-dimensional Brownian motion w on a filtered probability space

(S2,F,(Ft)t,.0,P) and continuous coefficients H,G:C -+R. Theorem (5.1) (Ito-Nisio):

Assume the following:

Equation (VLL) has a solution 0x:R x S2-> R such that °x(s) = 0 a.s. for

(i)

all s < 0 and there Is a constant a > 0 with EI°x(t)I4 < a for all t > 0;

(ii) There is a number M > 0 and a finite positive measure p on (--,0] so that J0

IH(n) I + IG(n) I < M +

- In(s) Idu(s)

for all n E C. Then the stochastic FDE (VII) has a stationary solution. In [41] several conditions on the coefficients H, G are given in order to guarantee the existence of a stationary solution e.g.

Theorem (5.2) (Ito-Nisio): (i)

Under the assumptions

H is of the form H(n) = -H0(n)n(0) + H1(n),

n E C,

231

for some continuous HO,H1:C + R; (ii)

G is continuous;

(iii) There are constants m, M, M1, M2 > 0 and finite positive measures 111, on (-oo,0] such that for all n E C 112

m < HO(n) < M 0

1H1(n)14 < Ml + J

In(s)14 duff (s) 00

j0

IG(n)14 < M2 +

1n(s)14 du2(s) Go

11111111/4 +3y

1111211 1/2 < m

where 11111 II

=

111(-°°,0],

1111211

= 112(--,oJ,

the stochastic FDE (VII) has a stationary solution.

Theorem (5.3) (Ito-Nisio):

Suppose H, G can be written in the form

H(n) = -H0(n)n(O) + H1(n), G(n) = -G0(n)n(0) + G1(n), Ti E C,

with HO, H1,GO,G1:C - R all continuous and bounded on C.

If there is a

positive constant m (> 0) such that 2H0(n) + IG0(n)12 < -m for all n E C, then the stochastic FDE (VII) has a stationary solution. Now in (VII) assume that the coefficients H, G are linear or rather affine of the form f0

H(n) = M1 + I-

n(s)d111(s)

1

G(n ) = M2 + fo

n(s)du2(s),

n E C,

for Ml, M2 E R, and 111,11 2 finite signed measures on (-°°,0].

denote the total variation measures of u,, 112.

232

Let 11111,

11121

Define the constants c E R,

:1,c2 > 0 by

c = - jump of

at 0, 111

c1 = 11111(-°°,0), C 2

= 11121(--,0].

Then the stochastic FDE (VII) will have a stationary solution in each of the following cases: (i)

c1 +

2 < c, provided 111,

have compact supports; 112

c2)1/2 < C. 2 + 1(c2 + 8c1 In this case the stationary solution tc2 IT is unique among those with sup EIx(t)12 < . ([41] §11 pp. 47-56). tEE (ii) c1 +

Taking the further special case G(n) = 1 for all n E C i.e. M2 =

1 and

112 = 0, Ito and Nisio ([41], §12, pp. 51-56) also proved the existence of

a unique stationary solution to dx(t) = H(xt)dt + dw(t),

t > 0, 0

under the conditions 11 1(-°°,0] < 0 and Jisi dl1111(s) < 1.

Note here that

in the case when 111 has compact support (i.e. a finite retardation time) the

last two conditions imply that the characteristic equation 0

n(A) = A -

eXs d111(s) = 0

J

-00

of H has all its roots to the left of the imaginary axis (Lemma (12.1), p. 54, in [41]).

Therefore this last result of [41] is indeed a special case

of our results in VI §4 viz.

Corollaries (VI.4.2.1), (VI.4.2.2).

Finally, in view of our analysis in Chapter III and the fact that C is a

Polish space, it is tempting to believe that for the general non-linear stochastic FDE (VII) the trajectory field {nxt:t > 0, n E C} also describes a time-homogeneous Markov process on C.

233

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239

Index

Absorbing state

Adapted

Cauchy-Maruyama scheme

111

14, 30

Cp-bounded function

Adjoint semi-group

69

a-Hdlder continuous

norm

98

Characteristic equation

(iii)

function

(i)

49

Chapman-Kolmogorov identity

1.14

Algebraic tensor product

8

Almost sure convergence linearity

Chebyshev's inequality

6

Compact Hausdorff space

190

map

Asymptotic behaviour

1

121

Compactifying version

172

113

Compactness in probability

165

19

(iii), 31

Coefficient process

5

Approximation of stochastic DDE's

227

155

fixed point theorems

236

Complete filtered probability space

stochastic stability

184

Completion of a probability space Condition (D)

Backward stochastic DE's

237

(DA)

(ii)

Banach space

independence

Bilinear pairing

integrable random variable Borel-Cantelli. lemma

m-dimensional

Brownian a-algebra

240

(i), 22

bilinear maps

dependence

99

148

12

(iii)

Feller process function

122

40

Continuous bilinear form

46

169

(ii)

Continuity of the semi-group in probability

22

31

Continuation property of trajectories

(iii)

(iv), 2

motion, one-dimensional

time

4

6

Borel measurable version

Browni.an filtration

7

Configuration space

4

7

7

probability

2

Bochner Integration

Bump function

32

Conditional expectation

4

a-algebra

94

Condition (E)

Banach-space valued random variable

41

(ii)

(ii)

linear functional

2

12

71

28 1

Continuous path

Drift coefficient

(iii)

20, 67

Contraction semi-group 218 rk

Controllable Convergence in

in probability Convolution

RFDE

46

Dunford-Schwartz (D-S)

8, 10

integral

6

(D-S)-u-integrable

5

21

Eigenfunctions

function

92

Eigenvalues

Cylinder sets

92

92

Eigenvalue problem

23

Erratic behaviour Decomposition of solutions

moments

processes

229

a-algebra

170

norm

150

(i)

(i)

Evaluation map Existence

31

Deviating argument

235

30

(iii), 33

Expectation

4

(iii), 117

Differential system

Feller process

(i)

234

difference equations

Filtration

(ii), 20

14

Filtered probability space (iii), 14

70

Differentiation of stochastic

Finite-dimensional distributions

integral with respect to

jump discontinuity

parameters

memory (v)

196

Diffusion coefficient RFDE

First hitting time

(iv)

Dirac measure

151

Forced linear system

77

Formal adjoint

79

236

191

206

Discontinuous initial data (v), 226

Frdchet differentiability

Discrete-time Gaussian system

Frictional coefficient function

Distribution

21

Frobenius condition (Fr)

59

Distributional regularity Domain of strong continuity Dominated convergence Doob's inequality

17

6

(iv)

15

79

Fluctuation-dissipation

46

Direct sum

218

23

space

(iii)

Deterministic delay equation

Diffeomorphism

144

Euclidean inner product

Delayed diffusions

memory map

13

Estimates on higher order

236

(i)

notation

9

99

Covariance

Delay

(iv)

Functional calculus

41

223

(iv), 114

185

(iv), 76

Gateaux differentiable Gaussian distribution

42 21

241

Gaussian process random field

(1)

Its formula

21

random variable system

Ito calculus

21

Ito integral

21

126

27

21

Generalized. eigenspace

Kolmogorov-Bochner theorem

191

223

Ornstei.n-Uhlenbeck process

Generic conditions

Kronecker delta

stochastically stable Globally Lipschitz

146

24

Laplacian

46

Law of the iterated logarithm 117

t2-continuous process

119

Linear drift

Growth of solutions

238

Hausdorff topological space

222

123

144

Linearity in probability

58

Lipschitz coefficients

46

Local uniqueness

Independence of events random variables sub-6-algebras

23

6

(v), 239 24

I.nfi.ni.tesimal generator, strong

r1-semi-norm

5

r2-semi-norm

(iii)

Markov behaviour

31

property

I.nitial path

(ii)

trajectories Martingale

(i)

I.nvariant Di.rac measures

242

69

type noise

Mean

15

26

(ii), 19, 51

46

16

35

McShane belated integral

165

Invariant probability measure

Ito'belated integral

inequality

112

(iii)

19

Initial data

Isonomous

(iv)

p-almost everywhere convergence

process

77

Gaussian measure

143

49

Infinite memory (or delay)

process

(iv)

(iii)

compactifying version

6

6

Indicator function

weak

Locally bounded

28, 190

150

Localization technique Increments of Brownian motion

191

(iv)

stochastic DDE

Hypotheses (A) (M)

maps

223

Hyperbolic case

34

FDE's with white noise

2

23

165

growth condition

Heat bath

147

216

165

Group of diffeomorphisms property

Kolmogorov-Totoki theorem

69

Globally asymptotically stable

147

21

Measure

1

(iii)

31

9

Measure, vector-valued finite

Physical Brownian motion

1

positive signed

path

1

(v), 223

15, 148

Mercer's theorem

2

223

Position

Probability measure

13

Probability space

24

p-essentially bounded Minimal representation

1

(i),

1

Product probability measure

10

98

Progressively measurable Prohorov's theorem

9

Noise

148

16

155

Projective tensor product Neutral FDE's

187

100

Polish space

1

Measurable version

u-measurable

209

Piece-wise linear approximations

1

Measurable space

Mesh

Periodicity in distribution

1

8

235

Pseudo-metric

224

Quasi-tame function

(iv), 105

Random coefficients

33

5

(ii.i)

coefficient Noise process

31

Non-existence of continuous versions

delays

144

locally bounded versions

family of Feller processes 169,

145

measurable linear versions

148

Non-negative definite kernel

167

92

171

field

15

gas forces One-dimensional polynomial

delay equation

infinitesimal generators

189

RFDE

quadratic delay equation One-parameter semi-group

semi-groups

(iv) (iv)

237 185

transition probabilities Regular measure

235

22

in probability

Partition

24

Resolvent

belated

24

Retardation

Cauchy of unity

53

Periodic family of Gaussian measures

165

113

113, 149

78 (v)

Retarded FDE's

24

(v)

Riemann-Stieltjes integral sums

171

2

Regularity properties Parameter space

185

122

retardations

186

Ordinary diffusion coefficient Oscillations

223

35

35

Riemann sum

24

243

Riesz. representation theorem.

Right continuous

It

square-integrable vector-valued

16

14

14

with continuous sample functions. a-additive a-algebra

Stochastic indefinite integral

t

Sample function regularity

(iv)

Semi-martingale

228

68

32 163

71

15

(iv), 20

Strong continuity dual

2

infinitesimal generator

semi-group

78

Submartingale

Signed bimeasure Simple delay

Symmetric kernel

8

continuous

(ii)

Solution process Splitting of C

31,

208

Stationary transition

(1), 167

dynamical system

(i)

(v), 49, 165

Stochastic process a.s. continuous continuous

rk 14

183

Total variation measure norm

3

77 (ii)

Trajectory field

(ii), 111

Transition probabilities

19

14

Uniform Lipschitz constant

14

locally bounded of class

(i)

(ii), 20, 64

235

Tonelli's theorem

Trajectories

3.1

95

2

Time-homogeneous -lags

23

probabilities

Stationary solutions (v), 231

Stochastic DDE

98

Tame function

Tight measure

15

memory map

13 13

Taylor's theorem

165

State space

13

191

Stable projection subspace

positive

36

(ii)

Supremum norm

169

166

17

Successive approximation

12

Simple function

244

194

151

Stricker-Yor lemma

9

measurable version

ODE's

(i), 31

Stopping time

Separable martingale

Slice

RFDE

variation of parameters

143

(ii.)

Semi-variation

Shift

integro-differential equations

(iii)

Semi-group property Semi-flow

24

integral (McShane)

1

14

Uniformly continuous Lipschitz

32

14

26

(iv)

33

(v),

Uniformly tight Uniqueness

155

Wiener process

(iii.), 33

measure

Unstable stochastic ODE subspace

23

206

165

Vague topology

3

Vector fields

118

Velocity process Version

22

224

(iii)

Borel measurable discontinuous smooth

15

145

(111)

non-linear measurable

148

Weak continuity of linear operators

192

property (w1) property (w2)

79

82

Weak convergence in Banach space in Cb

11

77

Weak Derivative

generator

78

(iv)

sequential compactness

Weak * topology

11

3

Weak (or narrow) topology

3

Weak topology on Banach space

11

on Cb

77

Weakly closed

78

Weakly compact map complete

11

12

continuous extensions

79

continuous semi-group

20

dense

(iv.)

measurable White noise

4 (v)

245