Stochastic Analysis and Related Topics II: Proceedings of a Second Workshop held in Silivri, Turkey, July 18-30, 1988 (Lecture Notes in Mathematics, 1444) 9783540530640, 3540530649

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Lecture Notes in Mathematics Edited by A. Dold, B. Eckmann and F. Takens

1444 H. Korezlioglu

A.S. Ustunel (Eds.)

Stochastic Analysis and Related Topics II Proceedings of a Second Workshop held in Silivri, Turkey, July 18-30, 1988.

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FOREWORD

This volume contains the contributions of the participants to the second meeting on Stochastic Analysis and Related Topics, held in Silivri from July 18 to July 30, 1988, at the Nazim Terzioglu Graduate Research Center of University of Istanbul. The first week of the meeting was devoted to the following lectures: - Short Time Asymptotic Problems in Wiener Functional Integration Theory. Applications to Heat Kernels and Index Theorems, by S. Watanabe (Kyoto, Japan). - Applications of Anticipating Stochastic Calculus to Stochastic Differential Equations, by E. Pardoux (Marseille, France). - Wave Propagation in Random Media, by G. Papanicolaou (Courant Institute. New York, USA). The lecture notes are presented at the beginning of the volume. We regret the absence of the lecture notes by G. Papanicolaou, who was too overloaded at the time. The presentation of the papers contributed to the volume ranges from the construction of new distribution spaces on the Wiener space to large deviations and random fields. We would herewith like to thank the Scientific Direction of the ENST for its support in the preparation ofthe meeting and the present volume. During the year of this meeting we lost our dear friend and colleague Michel METIVIER; we are dedicating this volume to his memory.

H. KOREZLlOGLU

A.S. USTUNEL

CONTENTS S.WATANABE Short time asymptotic problems in Wiener functional integration theory. Applications to heat kernels and index theorems E.PARDOUX Applications of anticipating stochastic calculus to stochastic differential equations H.KOREZlIOGLU and A.S.USTUNEL A new classof distributions on Wiener spaces D.NUALART. A.S.USTUNEL and M.ZAKAI Some remarks on independence and conditioning on Wiener space

63

106

122

N.BOULEAU and F.HIRSCH Some results on Lipschitzian stochastic differential equations by Dirichlet forms methods

128

M.JOLIS and M.SANZ On generalized multiple stochastic integrals and multiparameter anticipative calculus A.GRORUD Un crochet non-symetrique en calcul stochastique anticipatif

141

183

P.BALDI Large deviations and functional Levy's modulus for invariant diffusions

193

M.CHALEYAT-MAUREL and J.F.LE GALL On polar sets for hypoelliptic diffusion processes

204

Ph.BLANCHARD and Z.MA New results on the Schrodinqer semigroups with potentials given by signed smooth measures

213

F.RUSSO linear extrapolation concerning Hilbert valued planar functions

244

SHORT TIME ASYMPTOTIC PROBLEMS IN WIENER FUNCTIONAL INTEGRATION THEORY. APPLICATIONS TO HEAT KERNELS AND INDEX THEOREMS. Shinzo Watanabe

Department of mathematics, Kyoto University

INTRODUCTION. Since the Wiener measure space was introduced by N. Wiener in 1923, a rigorous theory of path space integrals has been developed with many interesting applications to mathematics and mathematical physics.

Especially, the Feynman-Kac formula was established by M.

Kac and it was applied to several problems in the spectral theory of Schrodinger operators and potential theory.

If we want to extend

Kac's theory to curved Riemannian spaces, we need to make use of an important stochastic calculus on the Wiener space, that is, Ito's stochastic calculus.

Indeed, such important notions as Brownian

motions and stochastic moving frames on Riemannian manifolds can be constructed by solving Ito's stochastic differential equations. The main purpose of my lecture is to discuss this probabilistic approach by the Wiener functional integration to obtain short time asymptotics of traces (supertraces) of heat kernels.

It is well-

known that many important problems in analysis, geometry and mathematical physics, such as asymptotics of eigenvalues of the Laplacian, index theorems, fixed point formulas, Morse inequalities for Morse functions, Poincare-Hopf index theorem for vector fields and so on are essentially related to this problem of estimating traces of heat kernels.

The method of Wiener functional integration

consists of first representing the heat kernels by integrals of

2

certain Wiener functionals and then study the asymptotics of There is however a

functionals by probabilistic techniques. crucial difficulty in this approach.

Heat kernels, i e . fundamental v

solutions of heat equations, can not be represented by an ordinary expectation of Wiener functionals but by a conditional expectation, which, in a standard theory of probabiLity,

everyuhere sense.

is defined in

Thus an disintegration theory,

i.e. a refinement

of condi t i o na l expectations, is needed in this approach.

One

approach given so far is to use pinned diffusion. or tied-doun

diffusion prooesses.

But the very definition of tied-down diffusion

involves fundamental solution of heat equation

in essential way so

a kind of tautology occurs and some analytical knowledge of heat kernels is inevitable in this approach.

Here we appeal to the

HaLLiavin oaLouLus for this disintegration problem.

Namely, we

Introduce a family of Sobolev spacec of Wiener functionals by refining the usual family of Lp-spaces.

Among these Wiener

functinals are generaLized Wi.ener funotionaLs,

an analogue of Similarly as in the

Schwartz distributions over the Wiener space.

Schwartz distribution theory, we can generalize the notion of Using these

expectations to these generalized Wiener functional.

notions. the disintegration problem for Wiener functional Very roughly. our approach

integration can be well discussed.

systematically used in this lecture may be described as follows. (typically a heat kernel or the

We represent the quantity

trace of a heat kernel) for which we would estimate the asymptotic with respect to the parameter

S

as

by a generalized

Wiener functional expectation of a generalized Wiener functional We decompose that for

as

$(6) = $} (6)

+

$2(6)

in such a way

we can estimate its Soholev norm and thereby show

that it is negligible. secondly

is a generalized Wiener

functional having a rather simple structure so that we can manage to

3

compute the generalized expectation

explicitly.

Finally, we explain the content of this lecture.

In §l, we

review the fundamental concepts and results in the Malliavin calculus; the Sobolev spaces of Wiener functionals and differential calculus defined on them, pUll-back of Schwartz distributions by finite dimensional non-degenerate Wiener maps, dependence on parameters, especially, the asymptotic expansion of Wiener In §2, we discuss the application of the

functionals and so on.

Malliavin calculus to Ito functionals, an important case of Wiener functionals defined by solutions of stochastic differential equations To illustrate our method, we reproduce in §3 some results of McKeanSinger [30] for heat kernels on a compact Riemannian manifold with and without boundary.

In §4, we give a proof of index theorems by

our probabilistic method.

§1. 1.1

A survey of the Malliavin calculus SoboLev spaces of Wiener functionaLs and generaLized Wiener

functionaLs. Let

r (WO,P)

W0r =

(

be the r-dimensional Wiener space: wEC( [0, 1 J-+R r); w(O)=O

endowed with the supremum norm and P is the standard Wiener measure on Wr is denoted simply by W in the sequel. We restrict 0 ourselves to the Wiener space wi th the time interval [0, 1 J jus t because of simplicity and that it is sufficient in the problems discussed here. By a Wiener functional, we mean a more precisely, an equivalence class of coinciding with each other usual

P-measurable function on

W,

P-measurable functions

P-almost surely.

LP-space of real-valued Wiener functionals with the

L -norm p

4

If we set (1. 1)

then

L.,,_

is a Frechet space and it is an algebra, i.e. if

L..._. then

f·1S' E L..._

generally, i f Lp (E)

f. g E

Its dual is clearly! L I+ e l':t p('" Lp' More is a real separable Hilbert space, we denote by

E

the real

Lp -space of

" "P.E'

or simply by

Lp(E).

Thus

E-valued Wiener functionals and by

I Up

when there is no confusion, the norm of

Lp(R):: Lp'

L..._(E)

and

LI+(E)

are defined

similarly. Let all

H

be the Cameron-Martin Hilbert subspace of

w E W which are absolutely continuous in

W formed of

tE[O,I]

with square

integrable derivatives and endowed with the norm (l .

Z)

We identify the dual then For

H'

-+

h E H, define [h](w)E L

(l.

Then

3)

F(w)

Z

with

c

-+

H

by the Riesz theorem and

denotes the continuous inclusion.

by the usual Wiener integral

cx dh dh (s)dw CX (s) [h](w) :: JI dt(s)'dw(s) :: r JI dt cx= 1 0

°

{[h](w)j hEH}

covariance

H

where

W'c H':: H c W -+

of

L ) is a mean zero Gaussian system with the Z E([h](w)'[h'](w» = (h,h')W A Wiener functional (c

is called a polynomial functional if there exist a real

polynomial

P(tl •...• t

n)

and

hI , ... ,h

n

E H

such that

The totality of polynomial functlonals is denoted by P c L..._

and this inclusion is dense.

separable Hilbert space

P.

Clearly

More generally, given a

E, an E-valued polynomial functional F(w)

is any E-valued Wiener functional expressible in the form of a finite sum

F(w)=

F. (w)e .• 1

1

valued polynomial functionals is denoted by

The totality of EP(E).

Then

5

P(E)CL00- (E)

and the inclusion is dense,

Hilbert space and is decomposed into a direct sum of mutually orthogonal subspaces of Wiener's homogeneous chaos: (1,4)

The projection of FeP(E)

L

then

L

is denoted by

n

F

and

n

Therefore, operators

C (E)

onto

2(E) J FeP(E)

=2

J, If n is actually a finite sum.

J F n

= e

f

0' i (X ( t

for every

pE(J,a»

be the one-dimensionai Wiener measure on

E (SYI 1

we mean the measure

x exp{-(yl-X

2 2 1'(YI-x l ) / 8 0 (dl{t) l) 128 }P O,

on

and WI

s>O.

and by

(2n8 2)-1/2

WI'

where

1 a P O: O' aER,

is the pinned Wiener measure: POl'oa ,

=

Define 8 M

-

ds] where measure space

is understood as Ito's stochastic integral on the (W1,E 1 (Sy

This is well-defined I

because, on this measure space,

22

Ct)

where

BCt)

is an adapted Wiener process and by definition

It is easy to see by Fernique's fameous theorem that, if

0 < y < 2

then (2.23) Using this fact, we can conclude that, for fixed

and, for every

pEC1,m)

and

£>0,

and

x

s>O, the Sobolev norm

has all moments as a function of

i.e., it belongs to

with respect to the measure

in the sense of subsection 1.4.

Hence the generalized expectation -8

E[M

-

is well defined

-8

a.a.t (mCdt»

-

and belongs to

is well-defined which we write symbollically as

E[M 8

-

£

Y1

E = E1 X E. Similarly we can define where

Y

(l,x,w;t»)

Thus

23

THEOREM 2.4

For

8 E (0,1)

and

x,y E D, (denoting

y = (Y1'Y»

(2.24)

§3

Probabilistic proofs of McKean-Singer estimates for heat kernels To illustrate applications of our methods to short time asymp-

toties of heat kernels, we will reproduce by them some results of McKean-Singer (30). of dimension

Let Let

d.

M be a compact smooth Riemannian manifold

p(t,x,y)

be the fundamental solution with

respect to the Riemannian volume of the heat equation

2Y = l 6 u at

2

+

M

h.au

6

being the Laplace-Beltrami operator and M field. It is proved in [30) that (3.1)

(2nt)d/2 p(t,x,x) = 1 +

where

R(x)

o(t)

as

uniformly

is the scalar curvature, div h(x)

thereby deduce that

Y n

r

Ih(x)1

xEM. the We can

(dV(x)= the Riemannian volume)

(2nt)d/2 Z(t) =

Z(t)=

and

in

h(x), respectively.

rMdV(x)

+

t·r

R(x) - } div hex) M

- }lh(X)1 2 )dV(x)

where

a smooth vector

R(x)- } div h(x)- }lh(X)1 2 )t

+

divergence and the Riemannian norm of

(3.2)

h·a

p(t,x,x)dV(x).

Note that

M

are the eigenvalues of

L:=

t 6M

+

h·e

+

Z(t)=

o(t)

2n

as exp(ynt)

in the case

L

where is

24 symmetric with respect to some volume.

The proof of (3.1) given

in [30l is based on the method of parametrix and the LevI sum.

Here

we give a proof of (3.1) by our method of generalIzed Wiener functional integrations, cf.[39l for related topics. As in [30l, the problem can be localIzed and so we choose a coordinate neighborhood U of x, view U as a part of Rd and extend components of the metric tensor

gij(Y)!U

and

Rd

h·al u

= hi (y)alaY i to the whole of such that gij= 0ij and hi=O near Then, omitting the summation sign for repeated indices,

1 i 2 g

j

e2

+

where

and

g(y)=(gij(Y»' g(y)-l= (gij(y».

symbols.

is the Christoffel

Furthermore, we may assume that the local coordinate in

U chosen above is a normal coordinate around 13 Rimnj(O)y my n

+ O(

I y 13 )

as

x

so that

Iv l

x = 0 ,

J. 0

and as

lr l

J.

o.

Hence hI (0) + ajhI(O)yj

b i (y)

1 Rij(O)yj 3

and Rij(y) = Rmi jm (y) a i = .Q-. aX i be the square root of gij(y).

where Let

oi

k

Let XS(O)= 0

+

XS (t)

1 R. (O)ymyn 6 lmnk (XS ( t , 0 , w) )

(hence r e d ) ,

+

o(lyI3)

+

o(lyI2)

as

lr l

0

is the Ricci curvature. Then as

lr l

J. O.

be the solution of SDE (2.9) wIth

Then, by (2.12),(2.13) and (2.14), we have

25

where h(O) and

=

i

Rimnj(O)j:wm(S)Wn(S)dwj(S) +

Rij(O)j:Wj(S)dS

ajhi(O)jlWj(S)dS ,

l, ... ,d

o

Hence by (2.15),

as

s.j. 0

where and

(3.3)

c

o

= (27t)-d/2

and

because the generalized Wiener functional under integration is odd (i.e.

it satisfies

F(w)= -F(-w».

Now

(3.4)

E[

2

fi·fLaiaj,s ( f ) 2 2 0 1

]

= .i2

hiCO)hj(O)

x aiaj[(27t)-d/2exp(-lx/2/2)]lx=o

Also,

26

and

dw(t) = dB(t)+ x-w(t)dt where Bt t ) is an adapted dI-t dimensional Brownian motion under the pinned Wiener measure Since

P(

IW(l)=x) cr .

= ErJ

we have

[14].

1

o

m n J' w (s)w (s)(x -wJ(s» 1 - s

dslw(l)=x]

=fo and if

m. nand

are different from each other. then this is

equal to

' j . x Er(x j-wJ(s»lw (1)=x J]ds = J1-l.- (sx m'sx n '(I-s)x J )ds 01-S

Similarly. if

m= n

;o!

if

m=

;o!

j

n.

Hence

1 R. 6

+ (-

i

i

j(O)Er Jl wm(s)w n (s)dw j (s)lw(I)=x] mn 0

g Rij(O)+ aJhi(O»E[Jlwj(S)dSlw(l)=X] }(2n)

o

2

e

27

and therefore, (3.5)

I i E[fs·a .sO(f l)] = (21t)-df2 +

(-

-.l R (0) + -.l R'imlm(O) 18 36 l mml

16 Rli(O) _ 1 alh i (0» 2

-.l = (21t)-d/2 (- -.l 36 RII(O) - 18 Ril(O)

= (21t)-df2 where By

R(O)

-.-l.

12 R(O)

= Ri l (0) = Rl mmi (0)

- 12

alhl(O)

+

1 RII(O) 6

-

laih i (O)} 2

}

Is the scalar curvature at

x

= o.

(3.3), (3.4) and (3.5), we can conclude that c

2

= (21t)-df2

RCO) - } div h(O) - } Ih(0)1 2

and this completes the proof of (3.1). Next, we consider the case of manifold with boundary. result of McKean-Singer In §5 of [30] is as follows: a compact smooth Riemannian manifold with the boundary p%(t,X,y)

with the Neumann boundary condition n:

the uni t outer normal

and with the Dirichlet boundary condition ulaN = 0

(3.6)

aM

M be and

are the fundamental solutions with respect to the

Riemannian volume of the heat equation

respectively.

Let

The

Then

J

1 dV(x)

M

28

± 8 ±

where

/ 2 T1:

4

.1 2

!h(x)!

I dS(x) +

and

dS(x)

t

M and

aM

8 J. 0

are the Riemannian volume and the surface respectively. R(x)

is the scalar curvature.

is the RiemannIan norm of the vector field

flux hex) = (h(x).n). xeaM xeaM

R(x)dV(x) -

M

JaM flux h(x)dS(x) - f2 JaM J(x)dS(x)]

dV(x)

element of

IaM

given by

If the positive

and

J(x)

h

at

x.

is the mean curvature at

2xthe trace of the 2nd fundamental form ( = xl-direction Is perpendicular to

aM).

(x).

By the

Gauss theorem. J M dlv h(x)dV(x) = JaM flux h(x)dS(x) and hence (3.6) can also be written as (2T1:8 2 ) d/2

(3.6) •

Ip M

± (8 2 .x.x)dV(x)

=

J

1 dV(x) ± 8L1E M 4

JaM 1 dS(x)

IMR(x)dV(x) - t JMdlv h(x)dV(x) - t J Ih(x)1 2 dV(x)

+

({ ±

+

{)JaMflux

IaM J(x)

hex) dS(x) -

M

dS(x)]

+ 0(8

2

)

To obtain (3.6)'. we first note that at each boundary point. we can choose a coordinate nlghborhood

U

of

M such that x = 0 ) .1

and 2 •...• d.

cf.C32].also [6].[13].

Hence

(3.7)

I

2"

6 M + h·a

1 2

{(

L

aX d

+

)2 I

+

2:

d

l.j=2

bi(x) 2aX I 1=1

2:

2 gi j (x) a ax. ax. 1

J

29

where (3.8)

... ,d

and

(3.9)

x E aM

(3.10) and (3.11)

J(x) = (det g(x»-l x

a1 (det

a1 ( de t

a) (x )

g)(x) = - (det a(x»-l

= a1 ( l o g t d e t

g)} (x).

Note that by the same localization argument as in the case of the manifold without boundary, we may assume that

x E Rd

globally defined on D = ( a l j (x) :: cS l j and hi(x) _ 0 §2 are clearlY satisfied.

near

aij(x), hl(x)

are

and satisfy that so that all the conditions in

(3.7) has the same form as (2.16) and

(3.12)

(3.6)' follows from the following integrated local formula;

(3.13)

(2ns 2)d/2

=

J

J

p±(S2,x,X)(det g(x»1/2 d X UnM

(det g(X»1/2 d x ± unM

J

(det g(O,X»1/2 dx unaM

30 + s2r-lJ R(x)(det g(x»1/2 d X - lJ Ih(x)1 2(det g(X»1/2 d x 12 unM 2 unM if dlv h(x)(det g(X»1/2 d x -(i±t)f h 1(0,x)(det g(O,x )1/2 dx unM unSM

If

M'SM,

Is of the form In

Rd- 1•

So we may assume that

(3.13) follows from (3.U.

U where U Is some bounded Borel set 1 1 Is fixed but It can be made arbitrarily small. So

no

k

we Introduce the following notation: A(S,Y)= 1 1m

1 1m sup

I A(S , Y) lIS k

If

} = O.

It Is now sufficient to show that

(3.14)

(2ns 2 ) d / 2 f

= fYdX1J

o

p±(S2,x,x)(det g(X»1/2 dx

(det g(X U

1,X»1/2

dx

1

as

By (3.12),

U

the left-hand side of (3.14) Is equal

(2ns 2)d/2 J 1

x

to

S

O.

31 by for simp I i cit Y,

Now we shall estimate this integral by the Malliavin calculus depending on the parameter

,

in the measure space

W1,m(d,»

where

by applying Th.l.4 and its obvious modification.

namely"

By Th.l.4 deduce that

with (2.21),(2.22) and (2.23), we can easily

(denoting by

g-I(X)=(glj(x»

1

d

U(X)=(U k(X»i,k=2

the square root of

d

i,j=2

+

M6(1) = 1 +

Io

I -i b

+ 0(6

o

-

2)

as

s.\.O

6.\.0

In

in L

... - (0"')

and

b

O(

XSi1)-X )

+ s[

x

= b

O

o

1

1

o

0

+ 0(6 2 )

Also

II

Is as

L

... - dreo) .

32

Hence,(denoting by

E

the integration on the

d-1 dim. Wiener space

(W,P»,

as

8,1. O.

The forth term in the integrand of the right-hand side vanishes because the generalized functional under the integration is odd in

w E W.

Clearly,

and hence (3.15)

=

I

{O:S:X

We show that (3.16)

Id"eT" g(x) 1:S: Y}XU ll

dx

33

Iot 1

x 2

1 {X +s t {S ) , X) dt ( s ) ] m{dt) 1

IU t 1 ( O+ , x ) / d e t

g(O,x) dX

1

For this, it is sufficient to show that (3.17) / det g(O,x) = - .6. 2 b 1 (O+,x)' +

uniformly

in

xErr.

Since the left-hand side of (3.17) is equal to

it is not difficult to show (3.17) by a direct calculation. Here we are contented with showing it only in the follwing simple case

the general case being similarly proved with a slight more careful estimate.

Then the above integral is equal to

I /27ts (1-s)

= 2b 1 (O,x)

GO

-x Is -

1

f_: -

1 IS

)

2S(1-8) Y d y

e

IYo

/det g(x

II

rics --"'-"1,X)dX 1 0 l-s

I-

Xl / S

_GO

1 e 2s{1 - s) y dy /27ts(1-s)

34

= -2eb 1(O,x)

I

1

o

I

sds

a2 b 1(O,x) "'det

x 2

v te 0

1

1

/det g(ex 1,x) e /27(s(1-s)

g(O,x) +

0

yJ. 0 (e) OJ

2s(1-s)

dX

1

•

Finally, we show

(3.18)

(27t)d/2

Io Y

dx

1I

dX U

1

I

WI

0

-

.a:

= 24

J

U

1

1 0 (det g) (0+ ,x) '/det g(O,x) dX + ( 2) det g °yJ.O s

.

For this, we note that 1

E[

-

o

Ioe II (0 du

=

1 i j

g

)(x

0

+

0(e 2)

1

+ut ( S ) , x ) t ( S ) d S

2 0(e ) ,

being in the sense of

Hence, by the same proof as in (3.17),

= -

2

/det g(O,x)

d

'Z

_

1 i"

-

gij(O,x)'(O g J)(O,x) i,j=2

Io 1

1 uduI s(1-s)ds 0

+ °yJ.o(S

2

)

35

By (3.15),(3.16) and (3.18),

This completes the proof of (3.18). we can conclude that

(3.19)

x jdet g(O,x)dx

Next, we estimate

P

By the same way as above,

2(S,Y).

1

+

+ E[

+

o 1

-

o

+

Fi rs t ,

Iovte

(3.20)

(2n)-1/2 e

-2x 2 IdX1I dX U 1

l,-2x 1 Po, 0

= S(2n)d/2

Y/S

Io

(2n)

-1/2

e

dX

1

36

g)'(detg)

x Idet g(O,x) dx

S

J

-1

-

](O,X)

+

Idet g(O,x) dx

VI

x Idet g(O,x) dx

+

Next, we show that

J dx J

(3.21>

{E[&o(O(SX 1 , X) WI

VI

x

1,-2x

1

W(I»].J t 1 ( s x 1 +S",, ( S ) , X) d",, ( S ) o

J

IP O,

b1(O,x) Idet g(O,x) dX

VI

The left-hand side of (3.21> Is equal to

S2

Io

YIS

e

x d'/!(s)] dx

0

1

(d,/!)

37

dX

J1o 4JL Ja> l-s -a>

1 +

By a direct calculation, I ds

Io l-s

J""e o

(Y+2x

1s)2 2s(1-s)}

1

/27'(s(1-s) exp{-

1

2

and therefore (3.21) is established. Finally, we show that

(3.22)

By the same argument as for (3.18), the left-hand side of (3.22) is

1,-2x 1 x EO,O [

II ds II du o

]

0

+

and (3.22) is obtained by verifying through a direct computation that

a>

Jo =

2

e -2x 1

1,-2x 1 EO, 0 r

II ds II du t(s)'sgn(x 1+t(S)U) 0

0

I ds II du Ia> e -2x 21 dX I 1 -a> /2ns(l-s) Io OX>

0

0

I

exp{-

] dX 1 (y+2X 1S)2 2s(1-s)

38

=

1

1

t Jo ds J0d u

s czu-t i j 1 - 4su(1-u)

O.

By (3.20), (3.21) and (3.22), we can conclude that

s

(3.23)

JU jdet

g(O,x) dX

I

+

Combining this with (3.19) and noting (3.9), we can now conclude the proof of (3.14).

§4.

A probabilistic approach to index theorems. Let

M be a smooth, orientable closed surface and K(x) R I 2 12(x) The classical be the Gauss total curvature. det g(x) Gauss-Bonnet theorem asserts that -l

27t

where X(M)

JM K(x) dS(x)

dS(x)

=

X(M)

= 2(1-g)

is the surface element,

g

is the Euler number of the surface

is the genus and hence M.

Chern [7l

extended

this formula to any smooth, orientable, compact Riemannian manifold of even dimension and expressed its Euler number as the integral of a certain explicit polynomial of curvature tensors.

A proof of this

Gauss-Bonnet-Chern theorem through the asymptotic study of heat kernels was suggested by McKean-Singer [30l and accomplished by Patodi [33l.

Here we will simplify the proof of Patodl by our

method discussed above.

The Hlrzebruch signature theorem ([Ill,

[I2]) can be obtained at the same time and, indeed, the Atiyah-Singer Index theorem for classical elliptic complexes can be obtained in

39

essentially the same way by our method (cf.{16l,[36l,[38l). Before proceeding. we prepare some notions and notations in linear algebra. Rd is the d-dimensional Euclidean space with the 1

canonical base

2

, ...

d

i - th ...• 0.1.0 •.•• 0)

i

and

is .as usual, its exterior algebra or the which is the 2d-dimensional Euclidean space with i i i i the canonical base l A & 2A .•. p. •••

Grassmann algebra

p =O.I •.•.• d.

For

E A Rd.

d

the exterior product

E A R

Let is defined in the usual way and satisfies = d) End(A R be the algebra formed of all linear transformations on d) by For each = 1.2 •••.• d. define E End(A R (4.1>

and

a

E End(A Rd)

i

define

by

D ( a ) e End(A Rd) 1

(he dual of

For

by

(4.2)

(the summation sign is omitted)

(4.3)

Let d

M

= 2L

be a compact oriented smooth manifold of even dimension Let

Ap(M) • p=O.I •..• ,d

be the space of differential

p-forms and d

A(M)

$

p=O

A (M). p

Consider the heat equation on

=

(4.4)

t

A(M):

nu

u ] t=O = a where d

0

=

is the Laplacian of

being the exteior derivative and

Hodge-de Rham-Kodaira, cf.[34l.

Let

OeM)

be

40

the orthonormal frame bundle over

M;

of

e = tel' ...• e d]. eiETxCM). of For-given r = Cx.e) E OCM). there

x E M and an orthonormal basis

the tangent space

TxCM)

at

x.

r =Cx.e) e OCM)

is any pair

is a oanonioaL isomorphism for each

r

and hence an isomorphism between

where

[fl. f2 •...• f d ]

p=O.I •..•• d. A Rd

and

uCt.x)

C4.5) where

T

x

CM).

r = Cx.e)

is the stoohastio moving frame C[28].[14])

as defined by the solution of the following SDE on d-dimensional Wiener space

defined by

OCM) fCr

lim

t)

tJ.O

the Stratonovich differential)

Is the system of oanonioaL horizontaL

= 1.2 ••••• d.

veotor fieLds on LafCr)

C •

=r

rr ) ,

OeM) over the

CW.P):

d

drCt) = rCO)

-

fCr)

f E C

let

A

and let

= k = even, 1"1" k I -Cv-l) r

and hence

= O.

TrCi'A) TrCYA)

=

=Hence

TrCi'A)

If

#A

= odd,

choose

1\

)

E A and write

=-1

)

=-

TrCYA)

= O.

It Is easy to see that the system all the subsets of

{1,2, .... 2d},

{y } , where

K

forms a basis of

Indeed, Independence of this system is clear from

Also.

}.

A"

1f

K if. K'

If

K

K'.

K ranges over End CAe Rd ) :

51

and the assertion follows.

Thus every

A e End(A Rd )

is expressed

uniquely as (4.37) and (4.38) (4.38) is known as the Berezin formuLa. (4.39)

(-1)

F

= (-1)

d

Next we claim that

l'(1.2 ••..• 2dl'

Indeed. if we denote the right-hand side by {ai' * al = 0.

and hence conclude =

C.

a

=

(_l)F

a. then

{1'jJ..

al

=0

From thiS it is easy to

= 1 •.•.• d.

for

if we can show that

(I)

o

d

e A (R )

C

But

and hence if

(I)

e C.

Thus

Combining (4.39) with (4.38). we have (4.40)

Now the proof of (I) is easy. (4.37). then. in each term. Hence if

m

+

1"S

If we express

appear at most

2n < d •

For the proof of (ii). we first note that

2m

A +

in the form 4n

times.

52

(4.41)

if c

b 1j km satisfies (4.31). 1,

c

where

Here

bi j

are some universal constants.

2

11

= b i mmj,

b

= bt t

and

Indeed, we have

or

Y2 1and the exponent S depends only on the 1 way of this choice. Noticing a well-known property of b i j km satisfying (4.31) that the alternation over any three indices Y2i

vanishes (cf.C2]), it is easy to deduce that

+

a polynomial in

Y's

of degree

2.

It is easy to deduce from (4.31) that the first terms in are equal and

the remaining four terms cancel.

Also, the remaining

polynomial of degree 2 is easily seen of the form

This completes the proof of (4.41).

From this and (4.40), we can

easily conclude (4.32).

Next we take, instead of (4.39), another involution (a Fermion number operator)

(_l)F E End (A

Rd )

defined by

(4.42) where In

d

= 2L

as before.

End (A T:(M»

(_l)F

and hence in

is similarly defined as an element End (A(M».

It is easy to verify

53 that

(_l)F

sends

Ap(M)

into

Ad_P(M)

and is given by

=

where

is the adjoint form of (cf.[34]). Define another supertrace of A e End(A Rd ) (similarly for A e End (A T:(M» or

e End (A(M»

by

(4.43) Then by the Berezin formula (4.38), (4.44) Instead of (4.9), of

t)O

J

M

Str(2)[e(t,x,x)]m(dx)

by a similar proof as above)

invariant of the manifold [12]).

(which is independent

gives another topological

M, called the signature of

M ([2],[11],

Also, we can compute the d-form

as a polynomial of curvature forms, indeed as an explicit polynomial of Pontrjagin forms of

M as given by (4.50) and (4.51) below,

thereby obtain the Hirzebruch signature theorem: . (4.45)

Signature of

M:

J H(x). M

Before proceeding, we first remark the following: be defined by (4.23).

LEMMA 4.2

On the set

{w;

w(1)

:

O},

(4.45) Proof.

and hence

On the set

{w; w{O): I}, we have

54

I

II0 w (s)'dw k

= i Rijkm(O)

m

(s)

Hence by (4.23),

1

II

1

6 Rijkm(O) - 3 Rijmk(O»

= 12

Rijkm(O)

II w (s)'dw 0

k

m

0

(s).

We can use this lemma in computing a generalized expectation of form E[

cS 0 (w (1 )

) • ill (w)

= (21t ) -

1

E[

(jI;)

=0

(w) I w(1)

l.

ill

e D"".

Next we remark the following:

lEMMA 4.3 Let d. e RdeRdeRdeR Let

A

e

be a product of

D2[b I l , D2[b 2l, ••• , D 2[b nl (4.47) and if

if

+

(4.31).

DI[ail. D1[a 2l •...• D1[aml .

in some order.

Str (2) (A) = 0 m

satisfy

Suppose further that all

Rd )

End(A

and

d =

m

Then

+

n
,786806. [24]

S.Kusuoka:

The generalized Malliavin calculus based on

Brownian sheet and Bismut's expansion for large deviation, in Stochastic Processes - Hathematics and Physics, Proc.l-st

61 Bibos Symp. LNM.1158,(1985),141-157. (25J

S.Kusuoka and D.W.Stroock:

Applications of the Malliavin

calculus, Part I, in Stochastic AnaLysis,Proc. Taniguchi Symp.(ed. by K.Ito), Kinokuniya/North-Holland,(1984),271306. [26]

S.Kusuoka and D.W.Stroock:

Applications of the Malliavin

calculus, Part II, J.Fac.Sci.Univ.Tokyo, Sect.IA,Math.32, (1985), 1-76. [27]

P.Malliavin:

Stochastic calculus of variation and hypo-

elliptic operators, in Stochastis DifferentiaL Equations, Proc.Intern.Symp.SDE Kyoto 1976,(ed. by K.Ito),Kinokuniya (1978),155-263. [28]

Geometrie differentieLLe stochastique,

P.Malliavin:

Presse de Universite [29]

P.Mallaivln:

de Montreal, 1978.

Analyse differentielle sur I 'espace de

Wiener, Proc.ICM.Warszawa,PWN, (1984),1089-1096. [30]

H.P.McKean and I.M.Singer:

Curvature and eigenvalues of

of the Laplacian, J.Diff.Geom.l,(1967),43-69. [31]

P.A.Meyer:

Quelques resultats analytiques sur Ie semi-

groupe d'Ornstein-Uhlenbeck en dimension infinie, in Theory

and appLioations of random fieLds, Proc.IFIP-WG 7/1 Working Conference,(ed.by G.Kallianpur), LNCI.49,(1983),201-214. [32]

S.A.Molchanov:

Diffusion processes and Riemannian

geometry, Russ.Math.Survey 30,(1975),1-63. [33]

V.K.Patodi:

Curvature and eigenforms of the Laplace

operator, J.Diff.Geom.5,(1971),233-249. (34]

G.de Rham:

(35]

I.Shigekawa:

DifferentiabLe manifoLds, Springer, 1984 Derivatives of Wiener functionals and

absolute continuity of induced measures, J.Math. Kyoto Univ.20,(1980),263-289. [36]

I.Shigekawa and N.Uekl:

A stochastic approach for the

Riemann-Roch theorem, to appear in Osaka J.Math.

62

[37]

H.Sugita: Sobolev spaces of .Wiener functionals and Malliavin

calculus, J.Math.Kyoto Univ. 25,(1985),

31-48. [38]

N.Ueki:

Proof of index theorems through stochastic

analysis, Master thesis, Osaka Univ. [39]

H.Uemura:

On a short time expansion of the fundamental

solution of heat equations by the method of Wiener functionals, J.Math.Kyoto Univ.27,(1987),417-431. [40]

S.Watanabe:

Leotures on stochastic differentiaL equations

and HaLLiavin caLouLus, Tata Institute of Fundamental Research/Springer,1984. [41]

S.Watanabe:

Analysis of Wiener functionals (Malliavin

calculus) and its applications to heat kernels, Ann.Probab. 15,(1987),1-39.

Applications of Anticipating Stochastic Calculus to Stochastic Differential Equations Etienne Pardoux Mathematiques, URA 225 Universite de Provence F 13 331 Marseille Cedex 3

Introduction There has been recently a very significant progress in stochastic calculus where part of the usual theory is generalized so as to allow anticipating integrands, see in particular Skorohod [20], Nualart-Zakai [13], Ustunel [22], Nualart-Pardoux [11]. For an exposition of these results and a more complete bibliography, we refer the reader to Nualart [10] . This new theory makes it possible to study various classes of equations where the coefficients and/or solution are non adapted processes. The simplest such equation is an "ordinary stochastic differential equation"

where the given initial condition X o at time zero is not independent of the driving Wiener process {Wt } . A second type of equation of interest is a stochastic differential equation with a "boundary condition" of the type h(X 0, Xl) = h, instead of an initial condition at time zero. A third example of stochastic differential equation with anticipating coefficients is given by a stochastic Volterra equation where the coefficients anticipate the driving Wiener process (in the situation which we have in mind, the solution itself is an adapted process, but the notion of anticipating stochastic integral is needed to study the equation). Let us indicate moreover that there is a serious difference between an equation interpreted in the "Ito-Skorohod" sense and in the "extended Stratonovich" sense. Our goal in these notes is to review most of the results known to date by the author. The notes are organized as follows. Chapter I reviews the anticipating stochastic calculus, introducing the precise notions and notations which will be used later. Chapter II studies two kinds of stochastic differential equations in the sense of Skorohod: a class of linear equation with anticipating initial condition, following Buckdahn [4], [5], and a class of Volterra equations, following Pardoux-Protter [17]. The main difficulty here is that the usual estimation techniques do not work as in the adapted case. Chapter III studies stochastic differential equations in the sense of Stratonovich with anticipating initial condition and drift, following Ocone-Pardoux [15]. The basic technique there is to represent the solution by means of the flow associated with the equation, and use a generalized "It6-Ventzell formula". Finally, Chapter IV studies two classes of stochastic differential equations with boundary conditions,

64

following Ocone-Pardoux [16] and Nualart-Pardoux [12]. For that class of problems, two questions are of interest: existence and uniqueness of a solution to the equation, and the possible Markov properties of the solution. A striking result is that for certain classes of equations, the solution possesses a Markov property iff the coefficients are linear. It is my pleasure to thank H. Korezlioglu and A.S Ustunel for having invited me to present this series of lectures to the second Silivri Conference on Stochastic Analysis, and for having created such a nice atmosphere which made the Conference an unforgettable experience.

65

Chapter I : Anticipating stochastic calculus. In this chapter, we recall the basic notions which we shall use in the sequel. Most of the results indicated here can be found in [11] ; see also [10} for a more extensive introduction, with proofs. 1 - The derivation operator on Wiener space.

Let 0 = C(JR+; JRk), equipped with the topology of uniform convergence on compact subsets of 1E4, T be the Borel field over 0, and P denote the standard Wiener measure on (O,T) i.e. {Wt(w) = w(t),t O} is a standard (EWtW: = tI) Wiener process under P. If hE H = L2(JR+;JRk), we denote by W(h) the Wiener integral:

1

00

W(h) =

< h(t),dWt >.

Let S denote the subset of L 2(0) consisting of those random variables F which take the form :

(1.1) where n E IN;h1, ... ,h n E Hi! E C'b(JR n ) . If F has the form (1.1), we define its derivative (or "gradient") as the kdimensional process {DtF; t O} defined as :

{DiF; t 01 will denote the j­th component of {DtF}. One can define more generally the p­th order derivatives:

Proposition 1.1. Di,j = 1, ... , k (resp. D) is a closable unbounded operator from L 2(0 ) into L 2(0 x JR+) (resp. L 2(0 x JR+; JRk)). We identify Di (resp. D) with its closed extension, and denote its domain by JD},2 (resp. JD1 ,2). o! and D are local operators, in the sense that if FE JD}'2 (resp. JD1,2), then D!F = 0 (resp. DtF = 0) dt x dP a.e. on {F = O} X JR+. 0 Note that JD 1,2 respectively :

= nJ=l JD}'2, JD},2, JD1,2 are the closures of S

IlFlli,1,2 = 11F112 + IIDi FII£2(nxR+) 11F1h,2 = 11F112 + IIDFII£2(nxR+iRk)

with respect to

66

More generally

and lDl,P(p ;::: 2) are the closures of S with respect to :

1IFlli,l,p = 1IFIlp + 1IIIDiFII£2(R+) lip 11F1h,p = 1IFIlp + 1I1I DFII£2(R+;Rk)llr Finally, we shall also use the spaces lDJ'p and lD 2,p, which are the closures of S with respect to respectively :

IlFlh,p = IIFllp + II

k

L

IID iDiFII£2(Rpllp,

i,i=l

1.2. Definition of t/», We now define some classes of processes. For j = 1, ... , k; .e = 1 or 2 and p ;::: 2, we define: = U(JR+, dt; lDJ'P)

ILl,p

= LP(JR+, dt; lDl,P).

We shall mainly use the space ILl ,2 . Note that U E JL l ,2 iffut E Il tll i,2dt < 00.

Jo= u

1.3. Definition of which are such that :

(i) (ii)

For any T

> 0,

uv>, t

a.e., and

will denote the set of those elements U E the set offunctions {s

--t

D!u s ; s E [0, T] - t}tE[O,T]

is equicontinuous with values in LP(fl)

< 00, VT >

esssup

(s,t)E[O,T]2

l,p-eIL e

°

nk

eIL'2P =

i=l ILl,p i,e n 1L2 ,p

1.4. Definition of the operators D +, D _, '\7. Let U E ILs, (DiU)t

= LP(fl)

lime!oDtut+e

= LP(fl)

lime!oDtUt-e

('\7iU)t = (Diu)t

+

If now U E lLi/, we define (D+u)t,(D_u)t,('\7u)t as the processes whose j component is given as above.

th

The reason for taking into account the possibility of a jump of Dtu s accross the diagonal of is related to the :

67

Proposition 1.5. Let

U

be mean-square continuous. Then

E

'L)UCi+ 1)z-nl\t - Uiz-nl\t)(WCi+l)z-nvs - Wiz-nvs)

--t

iEN

in £2(Q), as n

--t

t[(D+u)r - (D_u)r]dr

is

00.0

2. Definition of the Skorohod integral. Definition 2.1. Let bj be the adjoint of t», i.e, it is the closed unbounded linear operator from LZ(Q x lR+) into LZ(Q) which is defined as follows: • Dom bj is the set of those U E LZ(Q x lR+) to which we can associate a constant c such that:

IE 1'XJ D:Fut dtl cllFllz, VF

E

S

• Ifu E Dom bj, bj(u) is the unique element of LZ(Q) which is such that:

Proposition 2.2. C Dome, and the restriction of bj to is characterized as being the unique continuous linear mapping from 1L? into LZ(Q) with the properties that: Ebj(u) = 0

E[bj(u)Z] = E

1'XJ

dt + E

1'XJ 1'XJ

dsdt.

Moreover, bj is local on 1L?, in the sense that bj(U) = 0 on the set {Wi Ut(w) = o dt a. e.}, if U ElL?

O} x

Proof of the local properties of t» and bj In order to simplify the notations, we restrict ourselves to the case k = l. a) We want first to show that if F E DJ1 ,z, then DtF 0 a.e. on {F

u.:

Let 'PI!, 'l/JI! : JR. --t lR be defined by :

'Pe(x) =

l + X/C:, {

1

0,

x]«,

-c:

o

0,

x

x

e,

otherwise.

=

68

IE

f Dt?/;t;(F)ut dtl JR+

=

IE[?/;t;(F)8( u)]I c:EI8(u)1

Then from Lebesgue's dominated convergence theorem,

Since 1£1,2 is dense in L 2(Q x JR+), the result follows. b) For U E 1£1,\ define f2- n ) :

8(u) in L2(Q). But:

= 0 on {w;

Using the first part of the proof, we get that 8( un) Proposition 2.3. • Let h E L2(JR+), FE

8j(hF) = F8 j(h) • Letu E Dom8j,F E and

lD?

-1

= 0 dt

a.e.}.D

Then hF E 1£Y and

00

h(t)DfFdt.

ThenuF E Dom S] iffF8 j(u)- Jo UtDfFdt E L 2 (Q), OO

Y is that

U

lDY_

Ut(w)

An important property of 1£ E 1£},2, we define:

o

U

Y:: :} ul[o,t) E 1£?, Vt > O. For

E 1£

Using LP estimates for the Skorohod integral, one can establish:

69

Theorem 2.4. Each of the following conditions implies that possesses an a.s. continuous modification:

(i) u E

and SUPtElO,T] E

T>O. (ii) U ElL? and E

JoT (JoT

[(J:

ds

ds

Y

dt
2 and all T > O.

One can compute the quadratic variation of the process

0

{J; US dWl}:

Proposition 2.5. Let u E 1L 1 ,2 . Then

in probability, as n

- t 00.

2.6. Extensions of the Skorohod integral (k = 1). a. It is clear from Definition 2.1 that S extends to a continuous linear mapping from L 2 (fl x JR+) into (BJ1,2)' = BJ- 1,2. This means that S(u) is well defined for any u E L 2 (fl x JR+), but need not be a random variable: BJ-l,2 is a space of distributions over Wiener space. Let us consider a particular example. If f E Cl(JR) and h E L 2(JR+), then from Proposition 2.3 f(Wt}h E Dome and:

Let now = rn=E(GjW1 = 0). y27l"

b. We now point out the fact that some properties which we have for u E ]£1,2 are no longer true for u E Dome. First of all, we have no expression for E(S(u)2)

70

in general. Second, u E Dome does not imply that u1[o,t) E Dome for all t Indeed, if we choose Ut = 1R+(W2)h(t), where 1,

h(t) = {

1,

0,

0 t 1 1n

x

t) exp(Wt - t/2). Again: X: = fn(WI

)

+ i t x; dWs '

For any t 0, X n1[o,tJ ---t X1[o,tl in £2(0, x JR+) and Xf' sign(Wt} in £2(0,), where sign(x)

-1, 0, { 1,

x x

---t

Xt, fn(WI )

---t

0

Since 8 is closed, it follows from the above identity that X1[o,t) E Dome, t and:

0,

x, = sign(WI ) + i t x, dWs But X, = sign(WI - t)exp(Wt - t/2), and the process t on the set {WI> O}.

---t

J; X

S

dWs has a jump

71

2.7. Interpretation of D and S in terms of It 0- Wiener chaos expansions. For simplicity, we restrict ourselves in this subsection to the case k = 1. Let F E £2(0). It is well-known that F can be expanded in an £2(0)converging series of the form : 00

where Io(fo)

= 10 = E(F), In(fn)

and for n =

f

JR'+

1,

In(tl, ... ,tn)dWt 1 · · .dWt n

with 1 E j,2(.JRi-), the set of symmetric square-integrable real valued functions defined on .JRi-. These multiple Ito-Wiener integrals are characterized by the fact that In is linear, and

EIn(fn)Im(gm)

=

o { n! < In,gn > £2 (R,+)

ifn#m if n = m

(fn E j,2(.JRi-), gm E j,2(.JR+)). Now F E ]Dl,2 iff the series nIn-1(fn(" t)) converges in £2(0 x .JR+), and in the latter case DtF is given by that series. Let 0, Ut E £2(0), so that it can be represented in the now U E Dome. For any t form: n=O

where Vn E IN, t

E

0, gn(., t) E j,2(.JRi-). Now:

j,= D,Fu,dt j,= E

n

+ 1)InUn+' (., t))]

t))] dt

+ I)! < In+l(" t), gn(., t) > £2(Rn) Jo(>0 dt f(n + o oo Suppose that we can interchange Jo and Then: =

00

= L(n + I)! < In+l, gn > L2(R++ 1 ) o

=E L In(fn)In(gn-l) 1

72

where gn(tl ... , tn, t) is the symmetrization of gn, or in other words the orthogonal 1 projection of gn onto j}(lR+.+I). But U E Dome implies that E fo DtFut dt = E[F8(u)]; it then follows that the above interchange is justified and that:

Lln+1(gn). 00

8(u) =

o

3. 'The extended Ito formula. Let us formulate a first version of the extented Ito formula. Note that we shall use below the convention of summation over repeated indices. Theorem 3.1. Let E Ct(lR d ) and X o be a d-dimensional random vector, {At, BI, ... , Bf; t 2: a} be d-dimensional random processes such that: (i) X o E (JDl,4)d

(ii) A E (JL 1 ,4 ) d (iii) B i E (JL 2,p)d, i = 1, ... , k;

Let:

x,

for some p

>4

= X o + it As ds + it Bt dWj; t 2:

We then have :

(Xt) = (Xo)+

I

t

(I(X s ), As) dS +

I

a

t (I(X s ), B ;) dW j

t(II(X s)(VjX)s,Bt)ds 2 io where:

t (vjX)t =2D{Xo +21 D{Asds +21

+ D{X t - ,;) Sketch of the Proof:

where we have used :

t +Bl

73

(d. Proposition 2.4). 0

Note that part of the hypotheses made on {A, B 1 , ••• ,B k } are used to establish some properties of {Xt } . One can instead formulate some conditions directly on X. This gives another version of the above result : Proposition 3.2. Let ep E Cl(JR d), A E (LFoc(JR+»d a.s., Bi E 1, ... , k, Suppose that the process:

=

has an a.s. continuous version and that:

Then the conclusion of Theorem 3.1 still holds.O As we will see in the next section, Proposition 3.2 still makes sense and is correct for ep E C 2(JRd). We have in particular the following. Let {Xt} and {Yf} be scalar processes ot the form :

Yf = Yo

+ it c.e, + it

0

s», ...

and suppose that the k + 2­tuples (X, A, ,B k ) and (Y, C, t», ... ,D k ) both satisfy the assumptions of Proposition 3.2. One then has:

XtYf =XoYo + it(YsA s + x.c., ds (3.3)

+

r

Jo

+

dwl

+

2

r

Jo

+ (V'i

ds.

74

4. The localization procedure. All the processes which we have integrated so far satisfied moment conditions, which one would like to remove, as well as the boundedness condition imposed upon q, and its derivatives in Theorem 3.1 and Proposition 3.2. Let us now indicate how we can localize the processes which we want to integrate. Definition 4.1. For E = 1,2; p 2, we define F which are such that there exists a sequence

as the set of random variables

{(Sln, F n), n E IN} C F

X

IDl,p

with the following two properties: (i) Sln i Sl a.s., as n ---+ 00 (ii) F = F n a.s. on nn, n E IN. and DtF is defined We say that the sequence {(nn,FnH localizes F in without ambiguity (thanks to the last statement of Proposition 1.1) by:

rr»,

DtF = DtFn on nn x JR+, n E IN.

IDJ;foc is defined analogously. Definition 4.2. Again for.e = 1,2; p 2, we define lLf:c as the set of measurable processes u which are such that for any T > 0 there exists a sequence {(n;,u;), n E IN}

cF

X

lLl,p

such that: (i) n; t n a.s. (ii) u u; dP x dt a.e, on x [0, T], n E IN. In that case, n E IN} will be said to localize u in lLl,p on the interval [0, T). and are defined analogously. 0 Let u E

We define its Skorohod integral with respect to {Wi} by :

iot

Us

. iot

dWI =

T un,s

.

dWI on

si;T x [0, T]

This definition is not ambiguous, thanks to the last statement of Proposition 2.2 . Note that the generalized Ito formula could be reformulated with localized hypotheses on the data. In particular, it makes sense and is true with q, E C 2(JRd). We shall need a more restrictive localization procedure in chapter II.

Definition 4.3. Let us define lL1,loc as the set of those measurable processes u such that for any T > 0 there exists a sequence {,8;;, n E IN} c np>2ID1,p satisfy-

(i) {,8;; = I} i Sl a.s. (ii) 1[0,71,8;;u E Vn E IN (iii) ,8;; Du E L 2([0, Tj2)), Vn E IN The set of sequences {,8;;, n E IN}r>o will be called a localizer.

analogously lL"2 loc with l ,p lLl,loc C n lL1,p 0 n lL loc» e e,loc'

replaced by

We define Note that lL1,loc C

75

5. The extended Stratonovich integral. For any n E IN, let = {O < t; < ... < < ...} be an unbounded sequence. We shall assume that the sequence n E IN} satisfies - t 0, as n - t 00. Definition 5.1. A measurable process {Ut, t O} whose trajectories are locally dt-integrable a.s. is said to be Stratonovich integrable with respect to {WI} if the sequence.

e, = fi=o (ti+n 11

ti n

lit

lit Us dS) (WA+l

converges in probability as n - t 00 to a random variable of the sequence for any t > O. We then write: = i t Us

0

which is independent

dwl

Proposition 5.2. Let Then U is Stratonovich integrable, with respect to {WI} and its Stratonovich integral is given by :

t i0 Proof: Let

Us dwl

lit j + -20 (\7 u )s ds.

tl+ 1 lit u; = 2:(2 [l n i=O Jt',.lIt 00

n

From Proposition 2.3,

But un - t ul[o,t] in lLY, so from Proposition 2.2, 8j(u n) - t 8j(u) in L 2 (!1 ). It follows easily from the definition of lL that the last term of the above right hand side converges a.s, towards

Jot (\7i u) s ds. The result follows.D

Theorem 5.3. Let!P E Ct(JRd),{Xt,At,BL ... ,Bf;t random processes s. t. : (i) X E and is continuous (ii) A E (L70c(JR+))d a.s. (iii) Bi E j = 1, ... ,k and

x,

= X o + Jot As ds

+ Jot Bt 0 dWj.

Then

O} be d-dimensional

76

(Note that iP'(X)Bi is Stratonovich integrable with respect to Wi, 1

j

d).D

Remark 5.4. Comparison between Skorohod's and Stratonovich's integrals (in case k = 1). a. First note that, unlike in the adapted case, we can produce many processes which are Stratonovich integrable, with a square integrable Stratonovich integral, but do not belong to Dom8. Let u = Fh, where F E h E L 2(lR+). Then

But Fh needs not belong to Dotub (see section 2.6.a above). b. If {Ut} is continuous and of bounded variation a.s., then u1[O,71 is Stratonovich

integrable VT > 0, and JoT Ut 0 dWt is the Riemann-Stieltjes integral. c. If {Ut(x); 0 t 1, x E lR P } is a random field, which is P ® B p measurable (P is the a-elgebt« over lR+ x n of progressively measurable sets with respect to the natural filtration of {Wt } ) . Suppose that • x -t u(t,x) is C 1 V(t,w) 1 • E Jo sUPxERP 1\7 xu(t, x Wdt < 00 1 • 3q > pV 2 s.t. E JRP Jo (lu(t,x)lq + l\7xu(t,x)IQ) dxdt < 00 • t -t (u(t, x),\7xu(t, x)) is continuous with values in U(n), uniformly with respect to x. • 3 a measurable function a: [0,1] x n x lRP - t lR s.t. 2"'_1

:L [u( t7 + a( t7+1 ­ £=0

1 1

tn, x) ­ u(t7, x

-t

a

a(t, x )dt

in probability, uniformly with respect to x, as n - t 00, 0 a 1. Let finally I} is Stratonowich F be a p dimensional random vector. Then {u( t, x)j 0 t 1 integrable Vx E lRP and {Jo u(t,x) 0 dW t, x E lRP } possesses an a.s. continuous modification. Moreover {u( t, F); 0 t I} is Stratonovich integrable, and

This property of the extended Stratonovich integral will be crucial in Chapters III and IV. On the other hand, if we suppose moreover that FE (BJ 1,4)p, then {u(t,F), 0 t I} E Dom8 and

77

Chapter II : Stochastic Differential equations in the sense of Skorohod

1. Introduction - The basic difficulty (k = 1). Let us formulate the following "simple" problem : X o being a given function of {Wt, t 2:: O}, and a being possibly a smooth coefficient, can one prove existence and' uniqueness of a solution to the equation:

The difficulty is the following: X t being anticipating like X o, we can only estimate the mean square of the above stochastic integral as follows :

E[(l

t

a(Xs)dWs )2]

=El l

t

a t

a

2(X 2(X

s)ds

+

El l t

t

a'(Xs)a'(Xr)DsXrDrXsdsdr

ll t

s) ds

+

t

(D sXr )2 ds dr

Consequently, estimating the L 2 (fl ) norm of X t brings in the derivative of {Xt } . Estimating the derivative of X; would bring in the second derivative of X t , etc ... It is therefore not easy to adapt the traditional approach to study existence and uniqueness of a solution to the above equation. We shall present two results: one due to Buckdahn [4] applies to the above equation with a linear coefficient a and improves over earlier results of Shiota [19] and Ustunel [21]. The second, due to Pardoux-Protter [17] applies to a class of Volterra equations.

78

2. A linear stochastic differential equation with anticipating initial conditi In this section, we suppose for simplicity that k = 1, and all processes will be one dimensional. We consider the equation : (2.1) where: X o E LP(f2), for some p > 2, and b,(J' E x (0, t» where the subscript "prog" stands for "Ft-progressively measurable". We assume moreover that VN > 0, 3KN > 0 s.t.:

s

°

sups O. Moreover, it is easily seen that X E n t>oL2(f2 x (O,t») solves equation (2.1) iff for any F E S,

Vw,w' s.t.

J;

Before constructing the solution X, let us introduce some new processes. It follows from the local Lipschitz property and the boundedness of (J' that the equation

pathwise solution. Consequently, for any t has a unique non following defines a map 'P(t) : f2 f2:

r: (J's( ds +2i < + it < > ds.

IIXt l12 = IlXo ll 2 +

Proof: Let {el'

.e E

t

t

>

IN} be an orthonormal basis of K. Apply the extended

Ito formula to < X t , el >2, sum from .e = 1 to.e = N, and let N Let now {Xtl be a d-dimensional process of the form:

-> 00.0

(2.2.i) and {Ft(x); x E lRd ,

t'2. O} take the form:

(2.2.ii) We assume:

1-

X

and is a.s. continuous

A E(Lloc(lR+ ))d a.s.

Bi

1

i

k

II- For a certain measure J.l on lR d with = q(x), q E COO(lR d ) , q(x) > 0 \:Ix, {Ft , Gt , Hl,..., s», t '2. O} are processes with values in K = L 2 (lRd , J.l) satisfying:

86

We interpret (2.2.ii) as an equality between K-valued processes, and assume moreover:

Vt

> 0, VC

compact subset of lR d , the following quantities are finite:

E t (t

io io

ds, E xEC

rr

io io

xEC

Theorem 2.3. Under the above conditions, the processes {F;(Xt)B;; t . 1 2 {H;(Xt ) ; t O} belong to lLiioc' 1 i k, and: ,

Ft(Xt) = Fo(Xo) + 1

+

+

t

< F;(X s), As > ds +

II(Xs)(V

1\F t

io

iX)s,B;)dS

+

2

+1

1 t

< F;(Xs), B; > dW;

t Gs(Xs)ds

io

+

0

t

Proof. Let cP E CC(lRdj lR). Then, from Proposition 1.3.2 by q-l(x):

+ multiplication

t q-l(X)cpe:(Xt - x) = q-l(X)cpe:(Xo - x) + 1 q-l(x)(cp'(X s - x),As)ds

+

r

io

q-l(X)(cp'(Xs -

O} and

+

2

r

io

q-l(X)(cp"(Xs -

87

This can be viewed as an equality in K = L 2(JRd j p). From Proposition 2.1 + integration by parts in all dx integrals containing derivatives of 0,

np ?: 2LP(Sl ),

i :::; d. (ii) b : f'lxJR d --+ JRd is a measurable mapping s.t. s « G2(JRd; JRd); b, ]£1,2(L2(JRd; J.l)d) where J.l = N(O, I), and Dtb, E G(JRd; JR d), (t, w) a.e, and moreover VT

n E lN, 1

Ve > O,:JGe s.t. Ib(x)1

Ge (l + lxiI-e)

+

Gp,T(l

and 3p, Gp,T s.t.

IDtb(x)1

+

(iii) a E GOO(JRd, JRd); 1 of: i

+ i

+ IxnVt E

[O,T], x E JRd

k, all its derivatives are bounded, as well as those

m(x) =

"2 L 1

k

aa i . ax (x)at(x)

i=l

We associate to (3.1) the equation:

(3.2)

d:: = (l,keN} equipped

with the projective limit topology is denoted by D(X) and called the space of X-valued test functions (or functionals). Its continuous dual is denoted by D'(X') and called the space of X'-valued distributions. Let us recall the following fundamental identity

L=oV' and the fact that the norm defined on the X-valued polynomials by k

L i=O

IIV'

LP(Il;XI8lHl8li)

is equivalent to the one that we have defined above using the Ornstein-Uhlenbeck operator thanks to the inequalities of P.A.Meyer (cf.[6]). For typographical reasons, in the case of X=R, we shall omit to write the inside of the parantheses in Dp.k(X), D(X), etc., and note simply Dp.k, D, etc. Let us finally recall that D is an algebra as one can see by the Meyer inequalities.

If.Spaces of test functions and distributions Let A be a self-adjoint, positive operator with dom(A) in H and bounded inverse on H. We suppose that the spectrum of A is in the interval (1,00), hence IIA-UII O , H 00 = n dom(AI) is dense in Hand a-+(A uh,h) is increasing. We denote by H a the completion of H 00 under the following norm

a

2

(A h,h)=lhl , ue R. u

109

The dual of H that H

a

a

coincides with H

In this way we obtain a scale of Hilbert spaces (H

-a

a

; ae R ) such

is continuously injected into Hg for a>(3. We provide Hoo with the obvious projective limit

topology. Let he Roo and define. for ae R, the operator I'(A a) by f(Aa) [exp(Oh-(l/2)lhI1J=exp(OA ah-(l/2)IAahI 2). Another way of describing it is the following. decomposition :

Let be a nice random variable having the Wiener chaos

= E{1 + n=l where ne H®n is a symmetric tensor and In is the n-th order multiple Wiener integral which is defined in the abstract Wiener space setting as the n-th order divergence that we denote by o(n)=(V define I'(A a) as r(Aa) = E{} +

I

We then

In«Aa )i&ln n)

n=l where (Aa )®n is the n-th order tensor product of the operator Aa (cf.[12]). This construction of the operator T(A a ) is well known and called the second quantization of the operator Aa is a contraction operator on each

for p> I,

. Let us note that

(cf.[12]).

Definition II.I For p> I, ke Z, ae R, we denote by D i.k the completion of real-valued smooth polynomials defined on W

p.

with respect to the following norm 1IIID a = II(I+L) k/2r (A p.k

,

where is of the following form =p(Oh l ,....oh n)' hie H oo• i=I,... .n, and p is a polynomial on R

If X is a separable Hilbert space we denote by D ak(X) the completion of Xp. valued smooth polynomials under the following norm:

IISIID a

p.k

(X)= 1I(1+L)

k/2

f(A

al2

where the operators are applied componentwise. Theorem II.I For any aeR, denote by (W a, H , Jl ) the abstract Wiener space whose Cameron-Martin space is H . a a a Let us also denote by D(a)(X) the Sobolev space of X-valued random variables defined on (Wa,H ,Jl ) p.k a a as in the preceeding section. Then D ak(X) and D(ak)(X) are isomorphic.

p.

p.

Proof: The proof is a direct consequence of the construction of D

(X). I!QED

110

Remark: One has to pay attention that the above isomorphisme is not an algebraic one, i.e, it does not commute with the pointwise multiplication.

Theorem

n.2

The norm of D Uk(X) is equivalent to the following one: p. k

LII(A

all)@if(ACJ12yV

i=O The equivalence of the norms follows from the Meyer inequalities and from the following commutation relation which is easily verifiable on the smooth cylindrical functions:

Definition

n.2

Let X be a separable Hilbert space. We denote by i) DU(X) the intersection of (D Uk(X); ue R, po l , ke Z) equipped with the projective limit topology, p. ii) (X) and p.k(X) are the projective limits of (Du(X); Ue R) and of (D ue R), respectively.

Remark ILl Note that

if and only if

Hence, if l3 is an element of DU(H

@k k ) for any u. Hence (Aal2)@kV q> belongs to

DU(H@k) for any u, As an application, let us take a particular case: H= L Sobolev injection theorem

A=I-d. If e , by the dk ... ,xk) becomes a Coo-function with respect to its arguments in R

after a modification on a set of zero measure with respect to the measure dxjx...xdxkXdjl . The following result shows that, as for the Sobolev spaces on a finite dimensional Euclidean space, the integrability of p-th order (p>2) can be replaced with the integrability of second order by increasing the Aregularity of the trajectories:

Theorem 11.3 Let be a scalar test function.i.e.ioe . Then for any p>2, Ue R, ke Z,there exists some 13 >CJ12, such that

Consequently we have n

u.k

DU=n D U 2.k p.u.k p.k

111

Proof: Using the Wiener chaos decomposition, we can write a = 1I(I+L)k/2r(Aa/2)ql1I LP p.k

L 00

= II

"LP

n=O

From the hypercontractivity of the Omstein-Uhlenbeck semigroup, there exists some t>O, such that 00

" P L

II

So

n=O 00

So

II I e

nt(l+n)k/2

n=O

In)«(Aa/2)®nn) "L2

where H o n denotes the n-fold symmetric tensor product of H. By the hypothesis we have made on A (recall that IIA-CIlI 2, ke Z, there exists some

such that 1I!;11 D

a (X) s 'p 1I!;11D

p.k

2.k

(X)"

Proof: Let (e] ; ie N) be a complete orthonormal basis in X. We have

R

112

where

is the component of

in the direction of ei. If we denote by (Yi ; ie N) an independent sequence of

coin tossing with possible values I and -1, the Khintchine inequalities imply OQ

E [ ( 2:1(I+LF2r(A

2 1

l /2]i

i=o OQ

IP

:So c p E'I LYi (I+L)kI2r(A i=o

where E' is the expectation with respect to the product measure dvxdu and v denotes the law of the coin tossing process. We can now apply the preceding theorem. IIQ.E.D.

Theorem II.S For any e , for any p>2, ue R, ke Z, there exists some P>u such that

Similarly, if

is in (X), where X is a separable Hilbert space, we have

Proof: In the proof of Theorem IT.3 we have majorated the square of 1III f,(I+n)k nl

,

D

u

by

p.k

n=O for some pe R, now let us choose p>o such that the norm of (AP)®n compensates the growth of the factor (1+n)\or any n. For this, it is sufficient to choose P such that

IIA- lll nP:So (l+n)-k for any ne N. With such a choice we obtain OQ

2

2:(l+n)knl II(AP)®nnIlHon i

n=O

The vector valued case goes exactly as in the proof of Theorem ITA. IIQ.E.D.

113

Corollary 11.1 We have the following identities: (X) = 11 k D a (X)= Ilk D a (X)= .a 2.k p.. a p.k

D a (X) a 2.0

11

algebraically and topologically. We have also the following important result:

Theorem 11.6 is an algebra. Proof: If , let us denote by

the following functional:

trace

, where the trace is taken in H®H.

Theorem 111.2 We have the following a priori majoration: 2

42

McpliD a sc IIA-a/2I1H.s.llcpIiDa , 2.0 2.2 where c is a constant independent of cp. Consequently, if A-a is a Hilbert-Schmidt operator on H, for some

ce-O, then

is a linear continuous mapping from 4> into 4>.

Proof: If (ei; ie N) is a complete, orthonormal basis in H, we have, from the commutation relations of Theorem II.7,

119

Consequently IIVT(A a / 2)ql1I

E[ Ir(Aa/2)Mll i

2 0 2.2

i

2

s

2 H®2]

a

2.2 where the constant c comes from the Meyer inequalities. The rest of the theorem is now obvious from Corollary 11.1. IIQ.E.D.

Remark If A-a is a Hilbert-Schmidt operator for some a>O, then, for any injection from

there exists some

such that the

into F(lb) is Hilbert-Schmidt, where F(H ) (respectively F(lb» represents the

W

symmetric Fock space over H (respectively over lb). Since, each F%) is isomorhic to L where isomorphic to

W

is the Abstract Wiener space corresponding to lb and since

is

Ddb (cf.the Theorem 11.1) for any B, we see that the Frechet space is nuclear, hence its dual

is also a nuclear space under the strong topology. Similarly, the couple ((Hoo),'(H_oo) ) is also a nuclear couple in duality. Under the hypothesis of nuclearity, since to. is continuous on , its adjoint to.* is a continuous operator on '. Using a complete, orthonormal basis of H it can be represented as to.*4>=-L4>+

i:

[-V ($ Oei i=O ei

.

Since to.*$ and L$ are basis independent, the sum is also independent of the basis that we have used. In case of nuclearity we can say much more about and ' as shown by the following theorem(cf. also [3] for a different proof).

Theorem 111.3 There exists a negligeable set N in W such that, for any roe W-N, the Dirac measure £co is a distribution. Proof: Since the divergence 15 maps (Hoo) continuously into nuclear map. Consequently it can be represented as

1511 =.L A.i 1=0

;11> 0i'

and since (Hoo) is nuclear it is a

120

where (A.i )e 11,

) and (Gi ) are bounded sequences respectively in '(H_ oo ) and Let be a ,fixed once and for all. They are defined up to a random variable from the equivalence class denoted by

q

negligeable set N. If roe Q-N, then define

It is easy to see that TJ-t[OTJ](ro) is continuous on (Hoo ) ' Now any uniquely as

with $e, uekero(cf.[l3],[14]). If

(Hoo ) can be represented

goes to zero in then oV4h=L; hence so does 4h (since we suppose E[ cr{ I l(h).

h

E

H }

which proves part (a). Part (b) follows from proposition 3. IV. CONDITIONAL INDEPENDENCE Let F

= I p (f) and let SF denotethe subspace of L 2(T) induced by F:

_ L 2(T)

.

f f (t. t l ' ... , t n - 1'ltl-'t" TP-'

dt p _ 1• i

E

IN}

where g,i is a base in L 2(T P- l ). Let 't(F) denote the subsigma field where {hi.

i

E

SF = Span

L

{

... , tp_,dt 1

...

(12)

IN } is a base on

2(T)

't=o { 2 k' ij 1 k L (T ) (roughly speaking. 't

=0

E

{F, DF , ...• Dp- 1F }).

IN.j

•

(13)

127 Lemma 5: If F I p if), then for every h E SF, I 1(h) is 't measurable and SF is the tangent space of 'to Otherwise stated, in this case 't satisfies the assumptions of proposition 4 with K = SF. Proof: Note that, for every h e: SF, I 1(h) can be expressed as the L 2(O) limit of finite linear combinations of

therefore

h e SF, I 1(h)

for

is

't(F)

measurable.

If

hE

S/

then

< D k I p (n, h 0 hi 0 ... 0 h ik_ >L 2(Tk ) = 0 a.s, and consequently SF is the tangent space 1 associated with 'to Proposition 6:

Let F

=I p (n, G =I q (g)

and let 't(F), 't(G) be as defined above, then 't(F) and

't(G) are conditionally independent given a {l1(h), h E (SG /\ SF)} where SG /\ SF denotes the largest subspace of L 2(T) contained in both SG and SF. (Therefore, I p and I q (g) are conditionally

(n

independent given (SG /\ SF»' Proof: By lemma 5, both 't(F) and 't(G) satisfy the conditions of proposition 4 and the result of proposition 6 follows from well known properties of sigma fields generated by random variables which are jointly Gaussian. References

[1]

Nualart, D., Noncausal stochastic integrals and calculus. In Proceedings of the Silivri Workshop. Lecture Notes in Math. 1316, pp. 80-129,1988.

[2]

Nualart, D., Ustunel, A.S. and Zakai, M.: On the moments of a multiple Wiener-Ito integral and the space induced by polynomials of the integral. Stochastics, Vol. 25, pp. 233-240, 1988.

[3)

Shigakawa, I.: Derivatives of Wiener functionals and absolute continuity of.induced measures. J. Math. Kyoto Univ., 20-2, 263-289, 1980.

[4)

Ustunel, A.S. and Zakai, M.: On independence and conditioning on Wiener space, Annals of Probability, Vol. 17, pp. 1441-1453, 1989.

[5)

Ustunel, A.S. and Zakai, M.: On the structure of independence on Wiener space, J. of Functional Analysis, 1990. D. Nualart: Facultat de Matematiques, Universitat Barcelona, 08071 , Barcelona, Spain.

A.S. Ustunel: Ecole Nationals Superieture des Telecommunications

46 Rue Barrault, 75634, Paris Cedex 13, France M. Zakai: Department of Electrical Engineering Technion ­ Israel Institute of Technology Haifa 32000, Israel

Some Results on Lipschitzian Stochastic Differential Equations by Dirichlet Forms Methods. Nicolas BOULEAU ENPC La Courtine 93167 Noisy-le-Grand cedex France

Francis HIRSCH ENS Cachan 61 Av. President Wilson 94230 Cachan France

-=-

Since the impulse given by P. Malliavin, the stochastic calculus of variations has been mainly applied to stochastic differential equations with Coo coefficients, see Ocone [01] for a comprehensive exposition. But it is also important for applications to get regularity results for solutions of SDE with less smooth coefficients and in particular under Lipschitz hypotheses which are, in dimension greater than one, the most natural hypotheses of existence and uniqueness of solutions. The celebrated integration by parts method cannot apparently be extended beyond the case of functionals in the domain 1)£ of the Ornstein-Uhlenbeck operator (1D 2 ,2 with the notations of Watanabe {WI]), so that the regularity of solutions of Lipschitzian SDE must come from specific technics. Especially well adapted are Dirichlet forms methods which allow to exploit intensively the fact that Lipschitz functions operate on 1D 2 •1 = 1)...;:::£. We give here an account of results already obtained in this direction by Dirichlet forms methods and we present in details a new example which gives rise to an extension of the stochastic calculus. The first part introduces the framework of the Dirichlet space related to the Ornstein-Uhlenbeck semigroup on the Wiener space and recalls the absolute continuity and some consequences on Lipschitz criterion (d [B-HIJ [B-H2J) for functionals in 1D2 •1 or SDE. The second part is devoted to the regularity of solutions of Lipschitz SDE with respect to initial data. It is shown that the solution is differentiable in a slightly weakened sense. That gives for example the following simple result: under these hypotheses, if the initial variable X o has a density, then X; has a density for all t. After recalling the definition of the capacity associated with the Ornstein- Uhlenbeck Dirichlet form, it is shown in the third part, that the solutions of Lipschitz SDE can be refined, by taking quasi-continuous versions for each t, into processes with continuous paths outside a polar set and unique up to a quasi-evanescent set. The main tool here is an extension of the Kolmogorov theorem on existence of continuous versions to the case where the measure is changed to a capacity. This allows to study the solutions of Lipschitz SDE under measures which do not charge polar sets. In the last part, using Wiener chaos decompositions of positive distributions, we show that this property allows an extension of the stochastic calculus by constructing a finite

129

energy measure singular with respect to the Wiener measure and for which the coordinates do not build a semimartingale. This answers a conjecture formulated in [B-H4].

I

The structure of Dirichlet space on the Wiener space associated with the Ornstein - Uhlenbeck semigroup.

The Wiener space

11 = {w E C(lRt,lRd);w(O) = O}

is equipped with the topology of uniform convergence on compact sets, with its Borelian aalgebra and with the Wiener measure m which makes the coordinates a standard Brownian motion. :F denotes the m-completed o-algebra of u(B t ; t E lR+), and :Ft the :F-m-completed o-algebra of u(B6 ; S t). We consider on L2(m) the Ornstein-Uhlenbeck semigroup Pt, a strongly continuous symmetric Markovian semigroup characterized by

Pt[exp{j h(s).dB6

= exp{e- t / 2 j h(s).dB6

-

-

Vh E H = L 2 (lR+, JRd). The self-adjoint operator generator of Pt is denoted by L. There corresponds to Pt (cf [B-HI]) a Dirichlet form with domain

ID =V(yCL) given by

«u,u»

= IlyCLulli.(m)'

That means that the space lD with the norm (I!ulli.(m) + «u,u» )1/2 is complete and that normal contractions operate: For all u E ID, for all measurable v such that Vwlv(w)1 lu(w)1 and Vw,w'lv(w) - v(w')1 :::; lu(w) - u(w')I, one has v E ID and «v, v» :::; «u, u». This Dirichlet form is local ([B-HI] p239) and possesses a carre-du-champ operator, i.e. a symmetric bilinear continuous map r from lD x lD into L 1(m) such that Vu, vElD n LOO(m),

2«uv,u» - «v,u 2»

«., .»,

= j vr(u,u)dm.

This Dirichlet structure (11,:F, m; lD) is related to the Sobolev spaces which are classically defined on the Wiener space in the following way: Let lDp ,6' p E (1, (0), s E JR be the closure of the linear space generated by polynomials in continuous linear forms on 11 for the norm IIFllp,6

(cf [WI]). Then lD = lD 2,l and IlFlli. Let

= 11(1

+ «F,F»

L t/ 2 Flip IlFllt1'

en = 10r·en(S) ds

where enis a complete orthonormal system of H limit exists in probability (cf [B-H2])

L 2(JR+, JRd), then for all u E lD the following

130

and one has

f(u,u)

= ECV'enu?

The derivation operator D (cf [WI)) which can be defined by

o« = E(V'enu)en n

and which is continuous from 10 I' by

= 102 ,} into L 2 (fl , H) is related to the carre-du-champ operator

f(u,u) =< Du.Du >H, "I'll E 102 ,} . This relation between the carre-du-champ operator and the derivations in the directions of Cameron-Martin vectors E fl s.t. E H) allows, by an extension of the co-area formula of Federer [FI], to obtain the following absolute continuity criterion:

e

Proposition 1 . Let u

= (u}, ... , un)

E (102 ,})n. Then the image by u of the measure

det[f(u, u*)].m

is absolutely continuous with respect to the Lebesgue measure on IRn • When n = 1, this result is true for any local Dirichlet space (cf [B-HI)) and also for the local energy part in any Dirichlet space on a locally compact space (cf [BI)). In fact proposition 1 remains valid for u in defined by

= {u : fl ---+ IR; 3fl n E F,

fl n Tfl, Vn3u n E 102 ,} ,

U

= Un on fl n}

and for u E I'( u, u) depends only on u. An important application of the extension of Dirichlet forms methods to the case of the Wiener space is the study of stochastic differential equations. Let us specify the Lipschitz hypotheses which will be in force in the sequel: Two Borelian functions (7', b are given (7'

:

IR+ x IRn

b : IR+ x IRn and there exists K : IR+ ---+

---+ ---+

IRn x d IRn

such that "IT E IR+, "It E [O,T], Vx,y E IRn

1(7'(t,x)1 V Ib(t,x)1 :s; K(T)(I

+ Ixl)

1(7'(t, x) - (7'(t,y)1 V Ib(t,x) - b(t,y)l:s; K(T)(lx - yl). We are interested in the equation

(1) From the fact that contractions hence Lipschitz functions operate on the Dirichlet space, it follows (cf [B-HI] [B-H2)) that the solution of (1) is such that the map t ---+ X t is continuous from IR+ into (102 ,})n and by writing down a stochastic differential equation satisfied by the matrix f(X t , Xn it is possible to bring out conditions under which X, has a density by application of proposition 1. For example if A k = ((t,y) : cr(t,y) is of rank k} and if Tk is the essential beginning of A k for one gets the fact that, for t such that m( {t > Tk } ) > 0 and for almost all subspace V of IRn of dimension k, the projection of X t on V, knowing {t > n}, has a density with respect to the Lebesgue measure on V.

131

II

Regularity of solutions of Lipschitz SDE with respect to the initial data

Under these Lipschitz hypotheses, it is known (d [K2]) that there exists a version (Xf(w))(t,x}ER+XR" of the solution of (1) starting at x, such that for almost all w the map (t, x) - t Xf (w) is continuous and for all t 2 0 X - t Xf (w) is an onto homeomorphism of IR". If it is supposed further that a and b are aI,,,, with respect to x then x - t Xf(w) is an onto I _ diffeomorphism. Under the only Lipschitz hypotheses, Xf is of course not l with respect to x in general, but it is possible to show that the Jacobian :x(Xf(w)) exists in a weak sense and satisfies a SDE which can be written explicitly. For this, consider the space = IR" x n equipped with the probability m = hex) dx x m where m is the Wiener measure on nand h a strictly positive continuous function such that J h(x)dx = 1, JlxI 2h(x)dx < +00. The cr-algebrasgenerated by applications B., s :5 t and completed for m are denoted by i; (n, m) gets a natural Dirichlet form associated with the derivations in directions given by the canonical basis of IR". In other words the tool is here the form

a

a

n

a/ax,

suitably defined. with domain and operators We denote by (Xt)t>o [resp. (11t)t>o] the class of the process (Xnt>o [resp. of the Brownian motion enlarged up to m-ev';:nescent sets. Proposition 2 . Under the Lipschitz hypotheses, a) for m-almost all w, Vi 2 0, X:(w) E

b) there exists a process (Mt(x,w))t>o, (J:t)-adapted, with continuous paths and values in GL,,(IR), such that for m-almost all w, Vi 2 0, [:x(Xf(w)) = Mt(x,w) dx - a.e.] c) let a' and b' be fixed Bore/ian versions of the derivatives /;cr(t, x) and :x b(t, x), then M is the unique (J:t)-adapted continuous solution, defined up to an m-evanescent set, of the SDE [cr'(t,Xt).Mt] dB t + [b'(t,Xt).Mt] dt

I

It follows from this proposition and from a variant of proposition 1 applied to the Dirichlet structure on explained above, that the equation (1) with initial value a random variable independent of (B t ) possessing a density, has a solution which admits a density for all t 2 O. This was known, apparently, in greater than one, only under aI,,,, hypotheses. In dimension 1, there is an explicit solution: if we write as before cr;, b' for fixed Borelian versions of the derivatives of cr and b with respect to x, the process

n

r;x = exp

crHs, Xn dB; -

i

fot[crHs,X:W dS)

is such that for m-almost all w,

Va,.8 E IR, Vt 2 0,

Xf(w) - Xf(w)

=

J:

+

l

b'(S,Xn dS}

r;X(w) d».

132

III

Regularity, up to a polar set, of the solutions and their flows.

The Dirichlet form on the Wiener space associated with the Ornstein- Uhlenbeck operator makes it possible to look at properties of the Brownian motion satisfied up to a zero capacity set (cf

[F3], [KI],[S2D. We study here, from this point of view, properties of solutions of Lipschitz SDE. A work in the same spirit was done independently by J. Ren (ef [RID for equations with Coo-coefficients and with thin sets associated with C".s-capacities (ef [MID. We denote by C the capacity associated with the Ornstein-Uhlenbeck Dirichlet form. It is defined by C(G) = u E ID, u 1 m - a.e. on G} if G is an open set, and by C(G)

inf{C(G)j G open and G :J A}

if A E:F. If C(A) = 0, A is said to be a polar set. f: n -+ lR is said to be quasi-continuous with respect to the capacity C if V > 0,3n. open with C(n.) < e such that f restricted to the complement of n. is continuous. Two processes (U).heA, (V>')).EA defined on n are said to be C-indistinguishable if there exists a polar set A such that Vw r/:. A, V>. E A, u>.(w) = v>.(w). Under the Lipschitz hypotheses, we know that the solution X;(w) of equation (1) starting at x is such that for fixed t, x, Xf E It follows that this random variable admits a quasi-continuous version defined up to a polar set . The following extension of the Kolmogorov theorem gives conditions under which it is possible to put these quasi-continuous versions together to get a continuous process outside a polar set. Proposition 3 . Let (u",)",eRr be a family of elements of ID and p, al, ... , a r stricly positive real numbers. Suppose the following conditions hold

t

o c

o

Ixl VIyl

Ei=l < 1 Vx, Y E lRr lux - u ll !" E ID 3L: lR+ -+ lR+, VR> 0, Vx,y E lRr :s: lR = } II lux :s: L(R) Ei=l Ix; - y;i'"

Then there exists a family (V",)",eRr such that i) x -+ v",(w) is continuous ii) for all x V x is a quasi-continuous version of U x ' The family (v x) is unique up to C -indistinguishability and the following uniformity properties hold: There exist open sets (n.).>o with compact complement such that -+ vx(w) E lR is continuous a) V > 0, C(n.) < e and·the map (x,w) E lRr x b) V(3;, 0 < (3; < a;(l - Ej=l l/a;)/2p i = 1, ... , r, 3K

> 0, V > 0, VR> 0, 31] > 0,

IxIVlyl:S:R, Ix

yl:S:77) =>

Ivx(w)

vlI(w)I:S:KEi=llx;

y;I!3,·

133

This criterion allows to show that under the Lipschitz hypotheses and for a given fixed initial condition x E lRn , the solution Xf of equation (1) can be made more accurate into a process (Xt)t>o unique up to C-indistinguishability such that i) t - t x, is continuous, ii) for all t X t is quasi-continuous and X t = Xf m - a.s .. This result has been extended, by using a Banach valued space ID 2,1 by D. Feyel and A. de la Pradelle [F2] to the case of Ito processes of the form

x, = l

a•.dB.

+ ll3·· ds

with a, 13 E L 2(lR+, ID) and adapted. The previous criterion of Kolmogorov type, allows also to obtain a quasi-continuous version X; of Xf which is for w outside a polar set, continuous in (t, x) and an onto homeomorphism with respect to x; but for this C 1,a-hypotheses in x are needed for a and b (cf [B-H4]). With C 2 ,a-hypot heses, the differentiability with respect to x of the flow is obtained with a quasi-continuous regular Jacobian matrix :xX;(w) continuously depending on (t,x) for w outside a polar set (see [B-H4] theorems V.l and V.2 for more precise results).

IV

Stochastic calculus under a probability which does not charge polar sets

We keep in what follows the preceding globally Lipschitz hypotheses and look at the solution of

Xt=X+ la(s,X.).dB.+ lb(s,X.)ds

(2)

which is continuous in t, quasi-continuous in wand unique up to C-indistinguishability. This process is well defined under any probability measure on the Wiener space which does not charge polar sets.

A. The first case is when the right hand side of (2) also makes sense under such a measure u, To be precise with the changes of measure we introduce the a-fields ;F? = a(B., s S; t) without any completion. It can be shown (cf [B-H4]) that there exists an (;F?)-adapted solution, Xt , of (2) such that, for fixed t, X; is quasi-continuous in w, and for quasi every w, t - t Xt(w) is continuous. Then if v is a probability measure on n which does not charge polar sets and such that the process (B t ) is an (;F?)-semimartingale under u, the process s; is the solution of the same SDE under u , that is to say Xt satisfies v-a.e.

where f a(s, X.) dB. denotes the stochastic integral under v, For a one dimensional Brownian motion (d = 1), the law of the Brownian bridge 1E[. IB1 = a] is an example of such a measure v which is singular with respect to the Wiener measure (cf [B-H4]). For d > 1 the same result is obtained by taking the conditional law of the Brownian motion given that B 1 belongs to an (n 1)-dimensional hyperplan with the Gauss measure on it.

134

B. The case which gives rise to a true extension of the classical stochastic calculus is when, under t/, (B t ) fails to be a semimartingale so that the right hand side has no direct meaning by itself. We construct now a family of such measures on the Wiener space in the case d = 1 for simplicity. The idea is to consider a conditional law of the form lE[ . If; ho(s)dB6 = 0] for ho E L2([0, 1]), f; = 1. For using computations by decomposition on the Wiener chaos, we define this object as the positive measure which coincides on ID n C(O) with the distribution on the Wiener space [1

-

ho = Jo ho(s)dB

(3)

6

in the sense of Meyer-Yan [M2]. The characteristic functional of v is

(4)

1])

E

so that its decomposition on the chaos is written

with

p if n = 2p f n -- (2p)!(-W p!2 h@2 0

{

=0

fn

P

if n

(5)

= 2p + 1

where

In(f) = n! [

JO2.

A similar computation as the previous one gives

The family (Vt) is a distribution martingale in the sense ofYan [Y1] and the vt's are probabilities on I]. Following the notations of [Y1J we compute now the adapted projection of the distribution Dt/ where D is the gradient operator. We have 1 I (h 1 )02q-l (2q)!( -l)q h ( )) ( D v )ad() t = L.J (2 _ 1)1 2q-l 0 [O,t] '2q 0 t q=l

.

q

q.

and by using the formulae of Shigekawa [Sl]

hln(h0n) = In+l(h0(n+l») + nllhIl 2In_1(h0(n-l») we obtain

ad

(Dv) (t) where

-ho(t)-

= l{t 0, then

f:' dulE[IJ"I{Rt>u}] = f:' dulE[(IJ" -1{Rt$u})+] f:' du(lP(Jn) - v([O, u]))+

and by the hypothesis 2) lim,._o v([O, u]) integration

= 0,

hence the last integral is equal to an

which gives ii) by contraposition. It follows by taking Rt

=

> 0. By

o that 1

= +00 as soon

[00 U(8)JS d io (8+1)3/2 S

as

There are several functions satifying (11) and (15), for example u( 8) gives 1

ho(t) =

(15)

+00

= 1/(

+ log(8 + 1)), which

1

e:r;;-2(4-t)

1[O.a)(t). a-t Let us summarize the preceding discussion. Let ho associated with u by (12) and (13) and let v be the distribution on the Wiener space associated with ho by (3) and (4). v is a distribution of Watanabe in ID2.-1 and is also a positive measure which does not charge polar sets. For t < a, on the a-field :F'; the measures m and v are equivalent, (B.).9 is an (:F';)Brownian motion under m and an (:F';)-semimartingale under t/, For t a, the measures m and v are mutually singular on the a-field :F';, (B.). ... ,Sq)dUl'" dUt;

149

The symbol" • " means that the corresponding variable has been removed, and "--+" means replacement. Notice that

Using (3.7) we get

E

hi

00

(m-i)A(k-i) 1

,

11l:.(mr:: i )![ t=o l: E.{" ... l: m=1

,;,)C{1 •...

Im+k-2i-U(c (i l

it,

, ... ,

,m-i}

.. ,j,)C(1 •... ,k-;}

(j"

il,"" it) fm(.,:2.) 0 h(.,:2.»)]

We will check that for each £ = 0, ... , k - i, 00

,

Ill:. (mr:: i)!. m=.

A

.

l:

.

{'l .... ,Il}C{ll···,m-l} {j ,.' .. ,j C {', ... , k- i}

ImH-2i-U(c (i l , ... ,it,il,· ..




E T k } we define

where ..'.,ik=1 means that the sum is extended over all indices i 1 , .•• , i k = 1, ... , km such that no two of them are equal.

Proposition 3.5. (3.9) converges in

If

the sequence {u(m), m 2: I} of processes defined by

u E

as m -+

CXJ

to u.

151

Proof. Let us introduce a new sequence -(m) ( Zl, ••• ,Zk ) U

{u(m),

m

I} of processes defined by

=

Notice that, by Propositon 3.4, u(m) and

u(m)

belong to Li· 1 , for any m

1.

As in the case k = 1 (see Lemma 4.2 [7]) it is immediate to check that

Li·

1

,

as m

where

-4

00.

u(m) -

u

in

On the other hand

(resp. U:',... i_) denotes the sum (resp. the union) over all Zl>"" Zk = such that there exists j, j' E {I, ... , k} with i j = i i': Since the Lebesgue

II

L:i , ...;.

1, ... , km

U;',... i_

measure of the set

tends to zero as m

X .•. X

0, as m

lIu(m) - u(m)II£2(T-xo) -

-4

00,

we conclude that

00.

By the same argument we also have that IID i m -4 00, for any i == 1, .... k, Consequenly lIu(m) the Proposition is proved.

-4

u(m) -

D iu(m)lI£2(T_+iXO) -

u(m)1I _

0 in Li,l, as m

-4

00,

0 as and

_

It has been established in [7] that the operators D and 8 are local in the following sense: (a) If FE Dom D, then l{F=o}D t F == 0, dp,(t) x dP - a.e on T x (b) If u E Li· a.s on A.

1

,

n.

and A E.r is such that Ut(w) = 0, dp,(t) x dP a.e on T x A, then 8(u) = 0

This property can be also extended to Proposition 3.6.

Let

F E Dom

Proof. We use induction on k. For

parameter space

T = [0,1]

Assume that Wehave Equivalenty, the set

Dk

tr, Then k= 1

and 8k respectively. 1 {F=o}

Dr F = 0, dtx dP

on

T k x n.

we refer the reader to Lemma 2.6 in [7]. The

can be replaced without any trouble by

l{F=o} D;-l

a,e.

F = 0,

dt x dP

a.e,

on

T

k

-

1

x

[o,l]n.

n.

Fix

t

E

Tk-

1

•

a.e.on Txn. -

152

has ds x dP measure 1 for any

t E T k- 1•

Df-l

Let 'R= {(t,W):1{F=O} F=O}.The dtxdP measureof 1. Define A = {(s,t,w): (t,w) E 'R, (s,w) E S1} = (T

The set A has ds

dt

X

X

X

'R)

'R

isequalto

n {(s,t,w): (s,w) E Sil.

dP measure equal to 1, and, on the other hand

{(s,t,w), 1{F=O} Ds(Df-l F)

O}. This completes the proof of the Proposition.

A C

-

The local property for the operator Sk can be now established using the approximation by Riemann sums and the preceding proposition.

Li,l

Proposition 3.7. Let U E and A E F. Suppose that u!Jw) = 0, dt X A. Then Sk(u) = 0, a.s. on A.

Tk

XdP

a.e, on

Proof. By virtue of Proposition 3.5 it suffices to prove that sk(u(m») = 0, a.s. on A, for any m 2:: 1. According to Proposition 3.4,

L

I

+ (ii 1 we define

1£=(Zl, ... ,Zk)ET

are unordered}.

k

.

... ,k,

Zi

and

Zl

V ... V Zi-l V Zi+l V ... V Zk

159

For

s, s' E T, s

we define

s',

Proposition 4.7. Let

:F.'.'

FE Dam D";

as the

e--field

s,s' E T, s

and

k\ f

F=E(F/:F6,.,)+'t

.11.,.'] 4 such that such that (ii)

Ut

sUPzET

Eluzl P

2+

sup {EIDz, uz,I

t E T,

2

t-f EID;,zs u ,I z

00,

D 5u E L2(T6

X

f

dZ3 +

iT2

{!ro,Z] U dW

Then, the Ramer-Skorohod integral version.

Proof. If {X z , z E T} condition:

Eluzl P dz
2

< 00.

E Dam D5 for ahnost every

zt,z2ET

and to remark that the

r ,

r

n),

and

EID;,zsz4uzl12dz3 dz4 } < +00.

z E

T}

has an a.s, continuous

is a process continuous on the axes and satisfying the following

There exist positive constants

p

C, a,

t3 and ..\0 such that

{IX(z,z'])1

l(z,z'Ji1+P,

(4.12)

for any ..\ "\0, then, {X z , z E T} possess a version with a.s. continuous paths (see [2]). We will use this fact in order to get our result. Fix

z, z' E T, z < z',

1

Ur

dWr =

1

Following [7] we consider the decomposition

E(ur/:Fz,Z')dWr +

1

[U r

E(ur/:FZ'Z')] dWr

= A 1 +A2 •

E(uri:Fz,z') is independent of the increments of Won [z, z'], A 1 is an Ito integral. Therefore, using Burkholder and Holder inequalities, we obtain, for p > 2 Since the process

0,

and the proof is finished.

_

5. An Ito's formula in the two-parameter case Along this section T would be [0,1]2. As in sections 3 and 4, {Pm, m ;::: I} is a sequence of partitions of [0,1]2 such that IPml -+ 0, as m -+ 00. The points of Pm will be specified as follows:

Pm = {(sr, tj), i

o=

O, ... ,Pm, j

t;{' < tr' < ...
2 such that

E{ {

h

+ IT'

Assume that

U6

E Dom Da

df3 + { IDa uplP da df3 + {

h2

dp d, do: df3}
'(U)(D.;j)lAii(,8)-O in

in

Li,l

L2(n) as

as

up Li,j '(U)(D.;j) lAii(,8) -

Hence

-+

00,

we will check that

m-+oo.

It is clear, by dominated convergence, that

L4(T x n).

m

Li,j '(U)(D.;j) lAii(,8) 0 in L

2(T

-+

0 in

x n).

On the other hand

D 2: i,j

a

['(U)(D.;j)] lAii(,8)

Indeed

E

/']'2

{D a['(U)(D.7j)] I,)

r

m

-=:

00

lAii(,8) da d,8

0

III

L 4(T2 x n).

(5.11)

167

=E

iT2 L.. {

{//(U. ifj+l )DOI U. itj+l - //(U. it;) DOl U. ifj }

',1

:5 C E

£2 { £2

"(U.itj+l) - //(U.itj)}

4

IDOl

o.; 14 l.£l;j

4

l.£lij (.8) do: df3

(f3) da df3

',1

W'(U'itj+l)14ID" U' itj+l -DOl U.itjI 4 ) l.£lij(f3) da df3.

+C E

(5.12)

',1

The first term of the right hand side of inequality (5.12) tends to zero, as m 00, by dominated convergence. The conclusion (a) of Lemma 5.1 ensures the same property for the second one. Hence (5.11) is established, and this finishes the proof of the convergence

= O.

L 2 -limm -.oo Fm

We now take care of

£

E IDp['(U)

l.£lij(f3) df3

',1

E {W'(U.itj+.) - //(U.itjW IDp U' it;+t 14 } l.£li; (f3) df3

:5 C

+

G m . We have

£

',1 4

E {W'(U'i tj+1W ID p U' itj+l -Dp U.it;1 } l.£lij (f3) df3}. ',1

By the same arguments as in the proof of (5.12) it is easily seen that the right hand side of this last inequality tends to zero as m 00. This suffices to prove that L 2 _ lim m -. oo a; O. Step 3.

Due to the continuity of U and " we can replace

The two-dimensional continuous process

X.

(U.,1' U.,1)

1 from Lemma C2 of [7] with a. = f0 U;t dt. Hence

in probability, as Consequently

m

satisfies hypotheses (C iv)

00.

P -limm -. oo D m

exists, and the formula (5.5) is proved.

-

168

Our next purpose is to give an explicit expression for the process

O.

for some T/ Then

P - lim

m-+cx:>

L

..

'(U) (6.;j) U(6.L)

= J 1 + h + la,

I,J

where

and

= {

h

+

Jrr

2

(2 /V(U.vt) (rJR.

JT

/I'(U.vd

v,

D.U r dWr)

(rJRoY< o, Ur dWr) U.Ut ds dt

(rJR.v, D;t Ur dWr) U.Ut ds dt

+ (

JT2 "(U. vt) D.Ut o,«, ds dt

+

JrT

"1 (U.vt) (D .Ut) U.Ut 1v( 8, t) ds dt + 2

(l.v, o,«; +

(2 11I(U.vt) (rJR,v, o,«; dW r)

JT

UtDtu. ds dt,

{(S,t), 8 = (Sl, 82), t = (t 1 , t2): 81 :;::: t 1 , t2 :;::: tIl.

V

Proof.

Without loss of generality we can assume that

Uz

'(U)(6.;j) U(6.: j) = I,J

r=4

L

i,j

'

= xk,

k :;::: 2,

and that the

we obtain

"(U'i tj) U(6.;j) U(6.: j) I

k

(x)

is bounded.

Using the Taylor expansion of

+L

/V(U. vt)

2

dWr) U.U; 1v( s, t) ds dt}

where process

JrT

1

J

(r II)! (r) (U'itj) U(6.;jr- 1 U(6.:j).

+

11I(U.itj) U(6.;j)2 U(6.:j) t,J

171

Step 1.

We show that

P - lim

m ......oo

+ +

{T ( (

J7

JT

h(h

+

T

+ (

JT2

+

2: ¢>"(U'it) ..

U(6.}j)

1

',}

= 2 Jt (

¢"(UsVt) U.Ut

2

¢/"(U. vt) ({

JR,v,

o,«; dWr) U.Ut

lfw. t

11'2 (S, t) dW.) dt

¢1I(U.vt) Ut o,«, 11'2 (S,t) dW.) dt ¢'V(U.vt) ( (

JR,v, D.UrdWr) ({ JR,v, DtUrdWr)U.Utdsdt

¢'II(U.Vt) ( (

JR,v,

D;tUrdWr)U.Utdsdt

¢"(U.Vt) D.Ut tu«, ds dt} + ¢"'(U.vt) ({ Jt2 Jt2 JR,v,

Let Ri,;/ "(U. vt) D.Ut Dtu. ds dt r/>"(Usvt)

(third term of J3 ) . Property (5.15) follows from the following majoration

IL4>"(Us;t;) 161;(8) 16?/t)-r/>"(U.vt) 11'(a,t)1 i,j

k(k - 1) IL

16 1; (8) 16 ?; (t)1

(U:;"'i;2 -

i,i

+ elL v.i

16 1; (a) 16 ?; (t) - I'll (a, t)l,

and dominated convergence. For the proof of (5.16) we write

I'" L-t DOl r/>"(U.;t;) 16,,{a) 16?,(t) - DOl 4>"(U.vt) 11'(a,t)1 t.i

s

4

I).)

IDOl 4>"(Us;t;) - DOl r/>"(U.vt)1416i;(a) 16?;(t)

L i,)

+ I DOl

r/>"(U.vt)1

4

I'" Z:: i.i

16!,{a) 16dt) -11' (a, t)1 I)

4.

IJ

Then,

f ('" iTa 4-! ',J =

4 E IDO/4>"(U.;tj) - DOir/>"(U.vt)1 16!,(s) 16?(t)) da ds dt .) I)

f '" (f iT' 4-! iT ',J

f

sup

EID OIr/>"(U.;t;)-DOIr/>"(Usvt)1

(f

iT2 Iz-z/1:5I'Pml l-

4da)

16!,(s) 16?,(t)dsdt 'J

4 E IDOIr/>"(Uz) - D Oir/>"(Uzl)1 da) L

i,i

.J

16!,(a) 16?,(t) ds dt, 'J

'J

173

and this expression tends to zero as

m

-+ 00,

as has been established in part (a) of

Lemma 5.7. This is enough to prove (5.16). The proof of (5.17) follows the same lines. It uses part (b) of Lemma 5.7. We now study the second and third terms of the right hand side of (5.14). We prove that

and

11) (s,t) Do: [4>JII(Usvt)

(

i.:

o,«; dWr]. (5.19)

Then, using Lemma 5.5 (b), we obtain that the sum of the second and third terms in the right hand side of (5.14) tends to

Notice that I)

It: :S

o,

L:ij

4>"(USitj)

(s)

i J (t ' 'j tu«, dWr) i C {;;; I4>JII(Usitj) (t'i'j o,«; dWr) -

(rl(USitj) JR"'j

4>11I(USYt)

t

+ 14>III(USyt) h.v, ti.«; dWr l

4.

t:

14>III(Usitj)

:S C{L: i,j

+ L:

4>1II(USyt)

o.«, dWr).

(t.v, o,«; dWr)

(h.v, o.«; dWr)

IL:

11) (s, t)1

4 1

4}. 11)(s,t)1

',J

It is clear that this last term tends to zero in

iti

L:i,j

Then

IJ

4>111(Us

(t)

L 1 (T 2 x n).

(t'i'j o.«; dWr)

On ther other hand

4>1II(USyt)

(t.v, tu«, dWr)

4 ( D tu rdWr l JR.v , \R'i'j 4 14>III(Us,tJ-4>III(USyt)1 o». dWrI4}.

11

1 4

(5.20)

R. v t

The first term in the right hand side of (5.20) tends to zero in

L 1(T2 x n), (see

Lemma 5.1 (a) for an analogous result). The same property is true for the second one, by dominated convergence. Consequently (5.18) is proved.

4

174

Let us now check (5.19).

o;

=

[/I/(U8 itj)

the following facts

L.}.

/V(U8it; )

(JR' i' j tu«;

(h... ,.

¢/'(U8i t;) 1Llr.ds) 1Llr.? (t)

We notice that

"

(h..,

o;»; dWr)

• J

I

tu«, dWr)

1Llr.?/t)

J

/V (U.vt) (h..v, ti;«; dWr) . : o.«, dWr) 11>( s, t),

7: /11(U8it;)

(

1(R.

)

dWr 1Llr.i/s) 1Llr.?/t)

i,;

/11(U8Vt) ({

(

/V(U8Vt) Ua 1R.v,(a)

(5.21)

£4(T'xfl)

(5.22)

dWr) 11>(s, t),

i.:

7: /V(U8itj) Ua 1R'i'j(a)

IJ

Hence (5.19) will follow from

dWr)]

(

lR'i'j o,«; dWr

)

1Llr.?/t)

(ri.: o,«; dWr)

l1>(s,t),

£4(T'xfl}

(5.23)

(5.24)

/I/(U8 V t} 1R.v,(a) Dtu a l1>(s, t). In order to check (5.21) we write

If.;: /V(U8it;} (h.'i'j o;»; dWr) 'l.; o,«, dWr) - /V(U8Vt) (h..v,

s

n;«,

dWr) (h..v,

o.«;

C {f.;:I/V(U8it;) (l'i'j ti»; dWr)

_/V(U8Vt)

+ 1/V(U8Vt}

dWr ) 11>(s,t)1

(h..

i

4

,; o.«; dWr)

(rt.: DaurdWr) (ri.:

(l.v, o,«: dWr) (l.v. o.«, dWr)

4

1

I f.;:lLlr.i;(S)

-11>(s,t)14}. (5.25)

L 1(T3 X S1), by dominated convergence. Furthermore, the first term in the right hand side of inequality (5.25) can be bounded by 41(j C 1/v(U8 it;)-IV(U8Vt)j Dour dWr)(j o.«;

This last term tends to zero in

{L i,j

'l.;

+ f.;:1/V(U8 itj t I

- (rlR.

Dour dWr) VI

R,vt

R. v t

Dour dWr) (l'i'i o,«; dWr)

(ri.: tu«; dWr)

4

1

175

It is clear that the first of these tenus tends to zero in

lTd

E

{f11

-ui.:

L 1 (T3 x 0).

Moreover

(l'itj o,»; dWr) (l'itj tu«, dWr)

o;«, dWr)

.ti.:

4

o,«; dWr) 1 1a 1j (s) la?/t) } da ds dt

< C{Fm+G m}, with Fm

= f a E {2:1f JT

i,j

JR.Vt\R'itj

r; s

(£a f1

D au rdWr

l4 !

f

I«;

DtUrdWrI41a1j(s)la?j(t)}dadsdt,

and

Then,

(ira

E I,}

E (L.vt\R.itj Dau r dWrf 1ai/s) 1a?/t) da ds dtr/2.

(L

o,«; dWrf 1a;j(s) 1a?j(t) da

and, due to Lenuna 5.1 (a),

L1

lim G m = 0,

ds dtr/2,

.Vt

Fm

-+

0

in

L1(T 3 x 0),

as

m

-+

O.

Analogously

and consequently (5.21) is proved.

The proof of (5.22) is every closed to that of (5.18) (using Lenuna 5.1 part (h), instead of part (a)), and therefore it will be omitted. Similar arguments apply to (5.23). Finally (5.24) is inunediate. This ends the proof of (5.19). We will now deal with the terms

(and the analogue with

lads) 1a d t)). Their sum converges to 'J .)

as a consequence of (5.18). To finish this step it remains to study the convergence of

176

D;t ¢"(U••t; ) 1A !/ t) 1A?/s) = (¢IV(U••t;) (JR•• ,; Dvu; dWr)

We first remark that

(JR•••; b.«, dWr) + ¢"'(U••t; ) JR•• ,; D;tUr dWr)l A!;(t) 1A?;(s). Consequently

t.

+ f. JT

as

m -->

(l.v, o,«, dWr) -i. o,«, dWr) U.Ut 1v (s,t) ds dt

IV(U8vt)

00.

(f

¢/II(Usvt)

JR m

D;tur dWr) U.Ut 1v (s, t) ds dt,

Indeed, this convergences are similar to (5.21) and (5.18).

Step 2.

L

p -

¢1I1(Us.t;) U(e:.;j? U(e:.L) =

i,j

t (t

¢"'(U.vt) usu; 1v (s, t) dW.) dt

+ f. ¢/'/(UsVt)(Dsut) UsUt 1v(s,t)ds dt + f q,1v(U.vt)( f o,«; dWr)lv(s,t)u.u; ds dt. JT JT' JR.v, Denote by

H,j

the random variable

and Proposition 2.7 with

F

••t;)U(e:.;j)2u(e:.L) = I,}

+ +

t. (2.::

H"j1 A!;(S)l A?/t)) U.Ut d2Wst

1)

f .. iT2 1,1 f JT2

+ lT2 f +

q,1II(Us; t; ) U(e:.;j)' We use Proposition 2.8

== H'i> then

I,}

c: ',J

f z: iT2 t,}

+ iT2 [

',J

'J

t,

•. 'J

",}

t)

f c: JT2

'J

zw, dt

utDtu. dW. dt

.)

.J

t,)

.) (t)lM

UsUt

U.Ut ds dt

•.(t))UtDtu. ds dt +

.)

'J'J

",}

f iT2

dW. dt +

'J

t)

f lT2

dW. dt+

'J.)

t)

o.o,«, ds dt

(s)) D.Ut o,«, ds dt.

We first check that

L..." i,i

s., 1

A •tJ

(s) 1A • (t) 'J

---+

0

in

L 4(T2

X

[l),

(5.26)

177

L o; n., It>.l.(s) It>.?(t)

--0

in

L 4(T 3 x Sl),

(5.27)

s) It>.?(t) --0 D;aHij It>.d '1 IJ

in

L 4(T4 x Sl).

(5.28)

i,i

and

'1

IJ

L

i,i

These convergences imply that

IT' (EiJ· Hijlt>.l(S)lt>.dt)) ,

IJ

I)

U.Ut

.l.(t)lt>.?(s)) D.utDtu.dsdt, ,J IJ 'J 00, due to Lemma 5.5. Property (5.26) follows from the majoration

L

It>.;j(s)

L 2(Sl) as

m-+

L

C

i,j

i,j

L

tendtozeroin

In order to prove (5.27) we notice that

ID",HijI4 It>.;/s)

C

{L

Da(U.,tj

It>.;j(s)

i,i

i,i

+L

o;

It>.;j(s)

(5.29)

i,j

IT sUPz E (D a Uz )8 da < + 00. Hence, by dominated convergence, the first term in the right hand side of (5.29) tends to 0 in L 1(T3 x Sl). Moreover, It is easy to verify that

L. 2::

E IDa

It>.;j(s)

ds dt da

',J

:::;

{

i-

sup ({ E I[D",Uz Iz-z'IIII(Us.t

The same arguments apply to

t

4

J

1

t

1

)(Do1II(Usvt) usu; l1)(s, t) dWs) dt,

For the proof of (5.31) we write

t/>'V(Us;tJ(l

=

+

DtUrdWr) I,)

ij

DtUrdWr)U(.6.;j)lai/s) 6i t }

1,J

(5.33) 1 ,}

179

By Schwarz inequality and dominated convergence, the first term in the right hand side of (5.33) tends to zero in L 4 (T 2 x 0). The second one can be treated as (5.18), and tends also to zero in L 4 (T 2 x 0). On the other hand, it is obvious that E;,)' 4/"(U.;tj )Ut l,().dt) l,().!.(s) 'J '1 (5.31) is proved.

(s, t)

tends to

L 4 (T 2 x O).

in

Hence

We now prove (5.32). Applying the differentiation rules we have

+ O. (x, y) E T,

it holds that

ifJ(Ux) with

= ifJ(O) + I1(z) + I 2(z) + I 3(z) + I 4(z),

JR, xR. ifJ"(Usvt)usUt

ifJ'(Us)usdWs, Iz(z) = { {{

JR, JR. + (

JR.

+

i.:

DtUrdWr)usUt 11"2 (s,t)dWs

ifJlII(UsVt}UsU; Iv(s, t)dWs} dt,

and

JR.

JR,

+ { {

D;tUrdWr)usUt

+ ifJ'V(UsVt) (

DsUrdWr)( {

i.:

(

Jo

JR,v,

+ifJlII(Usvt)( (

JR,XR.

r U;tdt) ds

ifJl/(U. y)(

Jo

{ifJ"(U.vt)D.utDtu s + 2ifJlII(U.vt) ( (

JR,XR.

+

I"

ifJl/(U.)( ( DsutdWt) u.ds +

I 4 (z ) = (

(

i.:

11"'(s, t)d2Wst,

ifJ"(Usvt)utDtus 11"' (s, t)dWs

ifJlII(Usvt)( {

{ JR,

(5.38)

JR.vt

{ifJ'V(U. vt)( {

JR,v,

DsurdWr)UtDtU.

DturdWr)u.Ut} 11".(s,t) ds dt

DsUrdWr)U•. U;

+ ifJlII(U.vt)(D.Ut)U.Ut}

Proof By a localization argument we can assume that positive constant k.

sUPzET IUxl

Iv(s, t) ds dt.

k, for some

Consider a sequence ifJn of polynomials such that limn..... oo sUPlxl9 lifJ!;)(x)ifJ( r) (x) I = 0, for r 0, ... ,4. Then, by Theorems 5.4 and 5.8, we know that

ifJn(Uz)

= ifJn(O) + If(z) + I;(z) + If(z) + If(z),

If(z), i=1,"',4 are the analogue of Ii(z), i=1,"',4 with ifJ replaced ifJn. Thus it suffices to show that If(z) -+ Ii(Z) in LZ(n), for i = 1,'" ,4.

where by

It is obvious, using dominated convergence, that ---+

and

Do.

ifJ'(Us)u.

in

L 2(T x n),

+

=

---+ D cr [ifJ'(U.)u.J

= q/'(U.)(DcrUs)us + ifJ'(U.)Do.u., in

L

2(T2

x n), as

n

If(z)

-+

Consequently

-+ 00.

I1(z)

in

L2(n),

as

n

-+ 00.

On the other hand, Lemma 5.5 ensures the convergences If(z) -+ I;(z) in as n -+ 00 for i 2,3. Finally, If(z) -+ I4(z), as follows easily by dominated convergence. Therefore, the extended Ito's formula is proved. _

182

REFERENCES 1. BERGER, M., MIZEL, V. An extension of the stochastic integral.

Ann. Probab. 10,

435-450, (1982). 2. CHENTSOV, N. N. Wiener random fields depending on several parameters. Dokl. Akad. Nauk. SSSR 106, 607, 609 (19M).

3. FOLLMER, H. Calcul d'ItO sans probabilites, Seminaire de Probabilites XV. Lecture Notes in Math. 850, 143-150. Springer Verlag (1981). 4. MEYER, P. A. Transformations de Riesz pour les lois Gaussiennes. Seminaire de Probabilites XVIII. Lecture Notes in Math 1059,179-193. Springer Verlag (1984). 5. NUALART, D., ZAKAI, M. Generalized stochastic integrals and the Malliavin Calculus. Probab. Theory and Related Field» 73, 255-280 (1986). 6. NUALART, D., ZAKAI, M. Generalized Multiple stochastic integrals and the representation of Wiener functionals. Stochasiics 23, 311-330 (1988). 7. NUALART, D., PARDOUX, E. Stochastic Calculus with Anticipating integrands. Probab. Theory and Related Fields 78, £)35-581 (1988). 8. OCONE,D. Malliavin Calculus and stochastic integral representation of functionals of diffusion processes. Stochastic» 12, 161-185 (1984). 9. OGAWA, D. Quelques propietes de I'integrale stochastique du type noncausal. Japan Journal of Appl. Math. 1, 405-416 (1988).

10. RAMER, R. On non-linear transformations of Gaussian measures. J. Funct. Anal. 15,

166-187 (1974). 11. SHIGEKAWA, I. Derivatives of Wiener functionals and absolute continuity of induced measures. J. Math. Kyoto tu:« 20-2, 263-289 (1980). 12. SEKIGUCHI, T., SHIOTA, Y. L2-theory of noncausal stochastic integrals. Math. Rep. Toyama u­s« 8, 119-195 (19815). 13. SKOROHOD, A. V. On a generalization of a stochastic integral. Theory of Probab. and Appl. 20, 219-233 (19875). 14. USTUNEL, A. S. Representation of the Distributions on Wiener Space and Stochastic Calculus of Variations. J. of Functional Analysis, 70, 126-139 (1987). 15. USTUNEL, A. S. The Ito Formula for Anticipative Processes with Nonmonotonous Time Scale via the Malliavin Calculus. Probab. Theory and Related Fields 79, 249-269

(1988). 16. WATANABE, S. Stochastic differential equations and Malliavin Calculus. Tata Inst. of Fundamental Research. Springer Verlag (1984). 17. WONG, E., ZAKAI, M. Differentiation formulas for stochastic integrals in the plane. Stochastic Proc. and their AppJ. 6 339-349 (1978).

Maria Jolis Departament de Matematiques Facultat de Ciencies Universitat Autonoma de Barcelona. 08193 BELLATERRA (Barcelona) Spain.

Marta Sanz Facultat de Matematiques Universitat de Barcelona Gran Via, 585 08007 BARCELONA Spain.

Un crochet non - symetrique en calcul stochastique anticipatif AxelGRORUD Universite de PROVENCE 3 Place Victor HUGO 13331 MARSEILLE

Resume, Nous definissons un crochet qui generalise Ie crochet de variation quadratique de deux semi-martingales au cas de certains processus qui ne sont pas necessairement adaptes au processus de Wiener integrateur, Ce crochet permet d'ecrire de maniere plus concise la fonnule d'Ito et aussi de retrouver une fonnule de correction Ito-Stratonovirch.

Abstract :We define in this paper a non-symmetric bracket of two processes for a class of anticipative stochastic processes. This bracket is the quadratic variation bracket if these processes are semi-martingales. We can then write an Ito-formula for the anticipative stochastic calculus in a way which is very close to the adapted case, We also prove an Ito-Stratonovitch correction formula,

I . Introductjoo Le calcul stochastique anticipatif a ete inaugure par les travaux de Skorohod [9J, il s'est developpe ensuite grace, en particulier a Nualart-Zakai [6J, Nualart-Pardoux[7J, Ustunel [1OJ, qui ont permis de rapprocher les formalismes du calcul stochastique adapte et du calcul stochastique anticipatif , de demontrer une formule de changement de variable de type Ito et de definir une integrale de Stratonovitch de processus non adaptes (Cf.Nualart [4J ). Dans cet article nous definissons Ie crochet non-symetrique de processus non-adaptes integrables au sens de Stratonovitch ..Ce crochet pennet de retrouver une ecriture de la formule d'Ito tres proche de la formulation analogue en calcul adapte et aussi de demontrer une formule de correction Ito-Srratonovirch . Apres des rappels sur l'Integrale de Skorohod donnes dans la section II,nous considerons dans les sections III et IV Ie cas oil Ie processus de Wiener W par rapport auquel on integre est a valeurs dans Rd ; puis nous donnons rapidement dans la section VIes resultats lorsque Ie processus de Wiener est cylindrique sur un espace de Hilbert H . L'integrale de Skorohod a ete definie pour ce dernier cas dans Grorud-Pardoux [2] , la seule difficulte est la definition rneme du crochet non-symetrique lorsque West un processus de Wiener cylindrique .

184

II· Rapoels sur I'jntW3le de Skorobod Soit IE un espace vectoriel reel de dimension finie, soit (0, r, P) un espace de probabilite et W un processus de Wiener standard sur IE .On note r t = o ( W s ' O$;s$;t ) la filtration de r associee aW . :S(IE) designe l'espace des fonctions simples it valeurs dans IE ( Cf, p.e. Nualart [ 4 ] , Pardoux [8] ); pour p 2,on definit sur :S(IE) la norme : II F " I P

,

2 dt) p/2]) lip . =( E IIF IIp + E[(J0II DtF 11 HS la fermeture de s (IE) par rapport it II. 111 p . D) 1,2(IE)

D) 1,P(IE) sera est alors le sous-espace de L2(0; IE), domaine de l'operateur de derivation stodlastique D par rapport au processus de Wiener standard sur IE (Cf.Nualart-Pardoux [5] ). Le dual de D est l'operateur de divergence 0 appele aussi integrale de Skorohod.

Lorsque IE =.f(lR d; lRk), on note D...1,2(lR d;lRk ) l'espace L2 (lR+,dt; D)1,2(IE». D...1,2(lR d ; lRk) est indus dans Dom (0) (Cf. Nualart-Pardoux[5] ; Nualart-Zakai [6] ) qui est lui-meme un sous-ensemble de L2(OxlR+; .f(1R d; IRk» ) . En designant 111.111 (er-e .,. »respectivement) la norme (et Ie produit scalaire) dans .f (lR d; .f(lRd;IRK», on ala formule d'isometrie :

II O(u) II 2

E(

IRk

)=E

r 0

2 fX'OO IlusllHS ds+ ELJ «Dsut,Dtus »ds dt,

0 0

pou: u dans D...1,2(lRd . lRk) ; ou (Dtus (ej»(ej) = (Dsut(ej»(ei), pour une base orthonormee {ei,l = 1,..,d} de lRd. Pour u E n.),2(lRd; lRk) et pourtout t de lR+, u l[O,t] E Dom(o); t

on note alors

Us dWs l'integrale 0 ( u I[O,t] ).

Citons enfin deux resultats importants pour Ie calcul anticipatif (Cf.Nualart-Pardoux [5] ) Proposition 2,1. Soit u E D...1 ,2(lRd ; IR k) tel que : 'V t E lR + , le processus { D t us' S E IR+} soit dans Dom (0) , et tel que Ie processus { rD t Us dW s, t E IR+ }

aw,

o

ait une version dans L2(Q x lR+; .f(lRd; IRk» alors Jt Us E D)1,2(lR k) et t t 0 Da JOusdWs= JOD a usdWs+ua1[O,t](a) 'Va E lR+.

• Proposition 2,2. Soit FE D) 1,2(.f(lRk ; IRQ» , u E Dom (0) Fe! Us dW s) on a :

if

o

Fu

E

Jt

0

o,

alors, pourtoutt E IR+, si

F(u s) ds E L2(0; lR Q); et si F u E L2(0 x IR+; lR Q)

Dom (0)

iii O(F u) = F

s> -

tDsF(uS>dS

•

185

Nualart-Pardoux [5] definissent l'integrale de Stratonovitch de u :

a

Dffinition 2,3 ; Un processus mesurable u

valeurs dans

.:E(lRd ; lR k) tel que:

pour toute suite {An= (0 = tn 0 < tn 1 ds . 1=1

0

La definition 3.4 du crochet non symetrique et la proposition 3.2 permettent facilement de conclure. • Remaroue3J. 1- Nous noterons dans la suite dX t = Vt dt + Ut dW t lorsque X est un processus comme ci-dessus. La formule d'Ito peut alors s'ecrire , de maniere plus usuelle : cp(X t)

t

= cp(Xo) + I

1

< cp' (X s) , dX s > ds + - (cp '( X), X

o

2

It .

2- On peut Iocaliser I'integrale de Skorohod (Cf. p.e, Nualart-Pardoux [5]) afm d'etendre la proposition 2.6 ades fonctions cp de C2(IRk), ce qui permet de deduire le

Corollaire 3.8: Soit X,Y deux processus verifiant les hypotheses de la proposition 3.6, en utilisant la notation de la remarque 3.7, on a : t t i/ «x, Yt>=+I +I dX s ' on a :

0

It +

(It y s ®

0

ex, + 2 ((Y,X) It)* . .•

la formule classique

II x, 11 2

II Xo 112

+ 2

t

I

< x;

°

ax, >

+ (X,X

It

188

IV • IDteerale de StratoDoyitcb t t Soit { x, = Xo + I Vsds + I Us 0 dW s, t e IR+ } un processus tel que v e L4 (Q x IR+; IRk)

o

0

et u e o..l,4(IRd; IRk).

C

PrQPosjtjoD 4.1. Soit X comme ci-dessus , on suppose que Ye 1. 1,4 (IRk,lR l!) et C {.1n =(0:::; tn, 1 < tn,2 O, notons (Y+y)s l'element (V+u)s (Id) de L2(Q,IRk)

ou Id est

l' identite dans IRd .selon la notation de la definition 3.4 . 1 f tn k+ 1l\t Notons Yn,k ' Ys ds; nous obtenons : tn,k+ 1 - tn,k tn,kl\t tn,k+ 1 tn,k+ 1 1 tn,k+ 1 X X = vsds + usdWs+(Y+Ws ds . tn,k+ 1 tn,k tn,k tn,k 2 tn,k , 11=.1 n.:J tn,k+ 1 1 pUIS: 2. Yn,k(X X ) =.2: Yn,k (vs + - (Y+Y>s)ds k=O tn,k+ 1 tn,k tn,k 2 1 n-I _ n,k+ l +L Ynk usdW s' Notonsas=vs+-(Y+y)gVselR+, , tn,k 2 1 (s ) . et 1n k Ie processus s [tn,k; t n,k+l [ ,

J

f

J

f

f

1 t-J 00

Le premier terme du second membre s'ecrit alors Y n k as 1n,k (s) ds , n-I 0 k,;n, Mais, a e L4 (Q x IR+; IRk) et Y. Yn k In k e L4 (Q x IR+ ;.f(IRk;IRl!» ; done 00 n-I k,;{) " n.:J (Y. Yn k as I n,k (s» ds e L2 (Q; IRk). Par ailleurs Yn,k In k 01(=0' k=O ' converge vers Y l[O,t] dans L4(Q x IR+; .f(lRk;IR l!» ; done,

J

fo(

Yn k as k=O" dans L2(Q; IRk),

1n k (s)

:fa Y

n.k

ds converge vers

1n'f

t.-o

00

converge merne dans o..l,4(IRk,IR l!) vers Yl[O,t]

et, comme u e 1.24 (lRd;1R k) , Yn.k u 1n,k 1 1. 1,2(lR d ,IR l! ) , alors Yn k u In ,k ) k=O'

s( i

f 0 Ys (vs + 2 (V+u )s ) ds

converge vers Yu 1[O,t]

f 0ooYsus

dW s·

dans

189

D'apres la proposition 2.2 : tn,k+l Q;I {l;l 2.. Yn k Us dWs = 2.. k=

+

t

0

Io< $ '(Xs),

1 0

(V+u)s > ds + - { $ '(X), X t 2

I.

Le corollaire precedent nous permet de conclure.

•

V• 1.& cas bjlbertjen Nous avons decrit le calcul stochastique non adapte dans le cas hilbertien dans Grorud-Pardoux [2,3]. Nous don nons ci-dessous les notations essentielles et Ies problernes specifiques au cas hilbertien . H et K sont deux espaces de Hilbert separables reprenant les roles de IR d est IR k des paragraphes precedents. Leurs produits scalaires respectifs seront notes < , >H et < , >K .W est un processus de Wiener cylindrique; c'est a dire une famille {Wt, t E IR+ I de fonctionnelles aleatoires lineaires sur H (cf.par exemple Bensoussan [1]) qui verifient : i I vt , w.. H LZ (0; F ,P) est lineaire et continue, ii/ 'V hn ,..., h n E H: { W t (h n), ..., W t (h n t ;::: O} est un processus de Wiener (non neeessairement standard) a valeurs dans IR n . • iii/ 'V t;:::0, Vh , k E H; E(WtH .

»,

Lorsque la matrice de covariance de Wtest nucleaire , il resulte du theoreme de Minlos qu'il existe un processus it valeurs dans H, Yi.. tel que Wt(h) = < Wr-h >H p.p.t. . On dit alors

190

que West decomposable. En particulier un processus de Wiener avaleurs dans un sous-espace de dimension finie de H est decomposable. La difference, pOUT ce qui conceme le crochet non-symetrique, entre la dimension finie et la dimension infinie est que ici W n'est plus dans !L 1,2(H); en particulier l'ecriture {u,Wh =

f

t

(V+ u)s

o

«V_ W)s) ds

C

n'a plus de sens .

Notons !L 1,2(K) l'espace L2(1R +j ; ID 1,2(K» ; ou ID 1,2(K); pour un espace de Hilbert K est construit de facon anqlogue aID, (IRk) (Cf. Grorud-Pardoux [2] ); notons !L1,2(H ;K) l'espace L2(1R+, dt ; £2 (H; ID 1,2(K»). On definit de maniere similaire !L I,P(H ;K) .

Definition 5.1 !L 1,2( H;K) designera le sous-espace de !L1,2( H; K) des u tels que: C 2 i!

Du E L2( Q x IR+ ; £1 (£C( H; H) ; K».

iii V T > 0, les ensembles de fonctions {s ----7 D t Us ' S E [O,t [ hE fC1,Tl et (s ----7 D t Us , S E) t,T] hE [O,T] sont equicontinus a valeurs dans L (n ;£1 (.tc( H; H) ; K» . T

f (

iii! V T > 0 , 3 P > 0:

E 1 D s Ut 1 2 £1 (.tC(H;H);K»

sup

o t-p s s s r-p

) dt < 00

•

ou :f.C( H; H) designe l'ensemble des operateurs compacts de H dans H. RemaIlJ.Ye :

La condition

if peut sembler obscure ; si K = IR elle devient

2

Du E L2( Q x IR+ ; £1 (H; H», i.e. Du de donner un sens a f Pour u E

9...il2( H; K)

•

est un " operateur a trace" ; ce qui permettra

(Dus) ds . , on definit (D+ u)t, (D-uh, (V + u)t (V_ u)t comme dans la section II.

D'apres le lemme 3.13 et la definition 5.3 de Grorud-Pardoux[2], pour tout {hi, i E IN } systeme orthonorme de H,

i

I.

E

IN

(V+ u)s (hi ® hi) E L2(Q ; K) pour presque tout s dans IR+ . Ainsi

( V + u>s (I. hi ® hi) peut etre defini par

On notera alors : ( Y+...u.)s

I.

(V + u)s (hi ® hi) .

= (V+ u)s (I.

hi ® hi> E L2(n ; K) . iEIN

Soit u E 1. 1,2( K; L), V E 9... 1,2( K) , on definit le crochet non- symetrique C t C { u.v It par: V t ;;:: 0 { u.v h = f (V + u)s «V_ v>s) ds t 0 et on note {u,Wh = f (Y+y)s ds ,

Definition 5.2.

o

Formellement cette ecriture est coherente car si l'identite sur H ; c'est a dire I. hi ® hi .

W etait dans !L1,2(H) , V_W serait C

•

ie IN

Rappelons la definition de l'integrale de Stratonovitch dans le cas hilbertien (cf. Grorud- Pardoux [2]):

Definition 5.3: Soit u un processus mesurable a valeurs dans £(H;K). On dira que u est integrable au sens de Stratonovitch si pour toute suite {Qn , n E IN} de projections orthogonales dans H sur des sous-espaces de dimension finie convergeant fortement vers l'identite, pour toute suite { =( 0 = t n 1 < t n 2 < ... < tn k< '" ) } de suites croissantes non bornees verifiant Sup (tn k+ 1 - tn k) ----70' quand n 00, et pour tout t > 0 la suite n E IN} kSn-l' , t

191

avaleurs dans

de variables aleatoires

K definie par : tnk+l A t = 2. ( , Us ds) Q n (W - W ) k=O tn,k+ 1 - tn,k tn,k A t tn,k+ 1 tn,k

n-I

t

f

1

converge en probabilite dans K vers une limite

independante des suites ( Qn ) et

On notera l'integrale de Stratonovitch de u:

on a

(An}.

J Us 0 dW s . o

+

Si u E n... 1,2( H;K) alors u est integrable au sens de Stratonovitch

Proposition 5,4; et

St =

t

C

t

1

t

J0 Us 0 dW s= J usdWs +- (u,Wh· 0 2

: On a montre (Cf. Grorud-Pardoux [2] Th. 6.3) que sous les hypotheses ci-dessus t t l t I Us 0 dWs = I Us dW s + - I (V+..IDs ds , la definition 5.2 donne Ie resultat .

o

0

2

+

0-

On peut ecrire une formule d'Ito analogue a celle donnee a la Proposition 3.6 , les conditions sont donnees dans Grorud-Pardoux [2J (Proposition 2.4) . II en est de meme pour la formule de changement de variables pour l'integrale de Stratonovitch . Pour finir donnons les analogues de la Proposition 4.1 et du Corollaire 4.2 :

Proposition 5,5 Soit Y E n... 1, 4(J"2(K; L», X un processus d'Ito generalise de la forme

t xt=xo+Iovsds+

I

C

t

oil VE L4(QxlR+;K),XE L4(Q;K) UE n... l ,4(H; K )

Soit une suite de subdivisions ( An = (0 = tn,1 < tn,2 < ... < tn,k< ... ) } de IR+ croissantes non bornees verifiant Sup (tn k+ 1- t n k) ,

n-I

2. (

=

'V t E IR+, la suite t tend vers une limite notee

I

t

0 quand n

1

f'n k+ 1 t

k=O tn,k+l - tn,k

Ona

t

Y SO dX s independante de (An)

o

tOt

10 Yso ax,

A

Ys ds) (X - X ) tn,k A t t n,k+l tn,k

Si de plus XE lL 1,2(K) et (V_X) = u alors en notant t t l t C I v, dX s= I Ysvsds+ I YsusdW s + -

o

Alors:

00 •

=

0

IoYs ax,

1

I

t

Y s (V+u)sds

20

+ 2(Y,Xh.

a

La demonstration est analogue celle de la proposition 4.1.

Je remercie le Professeur E. Pardoux de son aide con stante et les Professeurs H. Korezlioglu et S. Ustiinel de m'avoir invite a parler de ce travail a ce Congres .

+

192

BibJio:raphje

[1]

A. BENSOUSSAN - Filtrage optimal des systemes lineaires . Dunod (1970)

[2] A. GRORUD - E.PARDOUX - Integrales hilbertiennes anticipantes par rapport aun processus de Wiener cylindrique. ( aparaitre dans Applied Mathematics and Optimization) [3] A.GRORUD E.PARDOUX - Calcul Stochastique Anticipatif dans les espaces de Hilbert. eRAS. - t. 307 Serie I; P 1001-1004; (1988). [4] D.NUALART- Non causal stochastic integrals and calculus in Stochastic Analysis and Related Topics .(Proceedings) Silivri, pp. 80 - 129; Lectures Notes in Mathematics, Springer-Verlag (1988); H.Korezlioglu, S.Ustunel, Editors. [5]

D.NUALART- E.PARDOUX - Stochastic calculus with anticipating integrands.

In Probability Theory and Related Fields 78 - pp.535-58l (1988).

[6] D. NUALART- M. ZAKAI- Generalized stochastic integrals and the Malliavin calculus. Probability Theory and Related Fields 73 - pp.255-280 (1986) . [7] D.OCONE E.PARDOUX - A generalized ItO-Ventsell formula: application to a class of anticipating stochastic differential equations. Ann. Institut H. Poincare (a paraitre) . [8] E.PARDOUX - Applications of Anticipating Stochastic Calculus to Stochastic Differential Equations. Second Workshop in Stochastic Analysis and Related TopicsSilivri (1988) Eds H.Korezlioglu, A.S.Ustunel. [9] A.V. SKOROHOD - On a generalization of a stochastic integral. Theory Probability and Appl. 20 - pp.219-233 (1975) . [10] S. USTUNEL - The Itoformula for anticipating processes with non monotonous time scale and the Malliavin calculus - Probability Theory and Related Fields 79 - pp.249-269 (1988) .

Large Deviations and the Functional Levy's modulus for Invariant Diffusions Paolo Baldi* Dipartimento di Matematica Catania (Italy) and Laboratoire de Probabilites Paris (France)

1. Introduction Let B be a m-dirnensional Brownian motion and let us set Cg:T = Co ([0, Tj, R m ) the Banach space of all R m -valued continuous paths x on [0, T] such that x(o) 0, endowed with the uniform norm. For every h > let us consider the random subset C(h) of Cg:1 defined by

°

C(h)

{ xE

cr. () D,T,xt

=

B(s) ,O::;s_
0 the set {g, >,(g) a} is compact in Cr;:T' Moreover for every Borel subset A of Cr;:T

s:

-A(AO)

s:

lim c 2log P{X' E A}

s:

lim e:2 log P {X' E A}

£_0

s: -A(A)

195

Remark. If A C C8;T is any closed set such that >.(x) > a for every x E A and some a::::: a, then also A(A) > a. This comes from the lower semicontinuity of >. and the compactness of its level sets stated in Theorem 2.1.

3. The Brownian motion case Let us consider the class 6 of all nonempty closed subsets of C8;1 endowed with the Hausdorff metric d. We recall that if A, B E 6, then d(A, B) < e if and only if for every x E A there exists y E B such that Ilx - ulloa < e and conversely for every U E B there exists x E A such that Ilx ulloa < e . We set for h

> a l(h)

v2hlogi. For aSs S 1 and C(h) the random element of

h) E C8;1 is the path

:=

t --; B(s + th) - B(s), t E [a, 11

C(h)

l(h) , -

6

-

The main result of this section (see [41, [7]) is the following (the set K is defined in the introduction). Theorem 3.1. lim C(h)

h-O

:=

K

a.s.

in the Hausdorff metric. The theorem will follow from Propositions 3.2 and 3.5 and Lemma 3.3. If S is a subset of C8;I, S" will denote the set of all points of C8;1 at distance at most e from S. Proposition 3.2. For every e > a there exist then

for every n ::::: no,

o:(c:) and no

0:

= no(e) such that if h n = n- OI

0: ::::: 0:0.

Proof. We shall prove that for some

0:

>a

oo

L

P {C(h n ) rt K"} < +00

n=2

and the statement will follow from the Borel-Cantelli Lemma. One has by scaling

P {C(h n ) rt K"}

:=

. B(s P { there exists s S 1 s.t.

+ l(h .hn ) ) -

V

+.) - B(

= P { there exists s S 1 s.t.

:=

P { there exists s S

hn

_ _I 2 log s.t.

(B( I

hn

2log..l h n

hn

B(s)

f/:. K"

n

hn

(B(s +.) - B(s)) f/:.

f/:.

}

=

K"} =

K"} S

196

S

(L

+ 1) P { there exists s S 1 s.t.

+ .) - B(s))

2 log

e

K }

where by A we denote the Borel subset of CO:2

A

{x E CO:2 , x(s +.)

A is closed and for every x E A

x(s)

K< for some s S I}

8+ 1

ds

8

/.

for some 0 S s S 1, since x(s + .) - x(s)

>

1

K 0 such that A(A) 2:

P

{

! + 8 and by Theorem 2.1

1] } Sexp [1+8 -2--log2 hn

1 2log -.L hn

for large n. Thus

and we may now chose a >

k.

Lemma 3.3. For every e > 0 and a> 0 there exists no

= no(w)

such that if h n

= n-O
25. Since one has easily

Jh n - h n + 1 l(hn+t}

I

we have, resuming the computation

which is summable. That enables us to conclude by the Borel-Cantelli Lemma.

Proposition 2.2 and Lemma 2.3 together imply that for every e > 0 there exists h o ho(w) C(h) C(h) . such that l(h) C K 2e for h 2: h o. Thus for every U E l(h) there exists g E K such that

lIu - gil < e. 00

The following statements prove that (C(h)t :::> K for large h, thus concluding the proof of Theorem 3.1.

198

Lemma 3.4. For every E: > 0 and I E K such that

for any 0 :;:; {3 < 1 -

1/11

1/11 < 1

and for h large.

Proof. By scaling and independence of the increments, we have for every TJ > 0

-It 2

Proposition 3.5. For every E:

,})!

=

(l-P {

11100

=

f «})' co


0 there exists h o = ho(w) such that (C(h))' :::J K

for every h :;:; h o . Proof. Since K is compact in Cr;:T' we can chose [i , ... .I» E K such that 1/;1 2 < 1 for i = 1, ... , k and such that the union of the balls centered at Ii and of radius i contains K. Then if h n n-O 0 ({3 smaller than 1 I/d 2 for every i = 1, ... ,k). This last quantity being summable, Borel-Cantelli lemma enables us to state that

for n

no

no(w). We now conclude using Lemma 3.3.

199

4. Invariant Diffnsions of s.c, Nilpotent Lie Gronps In this section we recall the main properties of the Invariant Diffusions of Nilpotent simply connected Lie groups and state a result analogous to Theorem 3.1, with the indications of the changes in the proofs. A nilpotent Lie algebra is said to be graded if it admits the decomposition 9 = VI $ ... $ VI , where the Vi'S are vector subspaces of 9 such that for every i,i [Vi, Vi] c Vi +i , with the understanding VI< = {O} for h > l. Of course this implies that if 91 = 9, 92 = [9,91],"" 9iH = [9, 9i], is the central lower series of 9, then 9i C Vi $ ... $ VI. Every g E 9 may be decomposed uniquely

where gi E Vi. It easy to check that for every a E R the linear transformation

is an endomorphism of the Lie algebra 9. A s.c. nilpotent Lie group G may be identified to its Lie algebra 9 through the exponential mapping. G is said to be graded if such is 9. In this case D cr is an automorphism of G for every aER.

Let Xc, Xl, ... ,X r be vectors in 9 and Xc, X 1>••• , X; the corresponding left invariant vector fields. We may now consider on G the differential operator L = Xo+

(4.1)

1

r

"X2 2 L... ' i=1

It can be proved that a diffusion process Y on G is left invariant under the action of group multiplication if and only if its generator L is of the form (4.1). The diffusion is said to be principal if Xo = 0 and Xi E VI for every i = 1, ... , r. The simplest way to construct such a diffusion process Y is to solve the Stochastic Differential Equation

where

(J

is a matrix having the

XiS

as columns, B is a r-dimensiona! Brownian motion and

bi =

2L 1

1.(g) = inf over all f E H; such that g is a solution of (4.3) (possibly >.(g)

=

the infimum being taken +00) and for a subset a of

Cff,T A(A) = inf >.(g) gEA

then Theorem 2.1 asserts that

< it is also clear that K I is exactly the set of all 9 E Cff,T such that >.(g) st. The proof of Theorem 4.1 goes along the same lines as Theorem 3.1, by proving statements analogous to Propositions 3.2 and 3.5 and Lemmas 3.3 and 3.4. Proposition 4.2. For every S > 0 there exists a and a 2:: a(s) then

a(s) and no

= no(w)

such that if h n

= n- a

for every n 2:: no . Proof. As in the proof of Proposition 3.2 one gets easily that

where A b E Cf,2,,(8

+ '),(8)-1 'f. Ki

Let be , E A and 8 such that -1(') = ,(8 such that if -1 is a solution of

for some 5 S I}

+ '),(8)-1 ¢ Kf.

This means that there exists 0> 0

202

then

If Ii >

1

+ 2fJ.

implies >..(-y) > Proposition 3.2.

+ fJ

Thus >"(1') >

+s

for every 1 E A and then A(A)

Lemma 4.3. For every and a a(1;: then

I;:

+ s.

>

One can now conclude as in

> 0 there exists a = a(l;:) and no = no(w) such that if h n sup

hn+1$h$hn for every n

and by the remark preceding Theorem 4.1 this

d(CI(h),CI(h n ) )
0,

a > 0

-= < S

and

for the

)t>o. The estimate shows that


O is realized as (c.f. [BM2l)

semigroup

Here

8+(0)

denotes all the nonnegative Borel functions on

0,

(X is a d-dimensional Brownian motion defined on a probabilt) ity space {n,F, (Px»' N is a polar set of D, T = inf{t > 0: X D } (or T " co when D = JRd), and t functional of (X ) corresponding to ]1. t In what follows we fix a polar set Define for

t > 0, x

p]1 (t,x,y)

pO(t,x,y) -:IE

For any

f E

and

[Jr0 x

t AT

is the n.0.ditive

N

D,

y

e

satisfying (2.3).

_A]1 spo(t-s,X ,y)dA].ll. s

s

(2.4)

(bounded nonnegative Borel function),

we have

(2. 5)

consequently p]1{t,x,y) is a version of the integral kernel tHJ.l of (e ) t>O' It can be proved that (t,x,y) coincides in

i?

fact with x

qO(t,J(,y) and

y E,

(specified by

[BM2] Th.6.7) for all

But we would like to delay this proof

until Section 6 \vhere a more general situation will be considered. 2.1

Right now we need (2.4) for proving the following fact. Lemma

Let

v

Sand f

B-f° «O,t)x D). Then

217

vx

(2.6 1

E

is the additive functional aarresponding to v and where N is a polar set deponding on 'V. 'V

Proof.

Let

V E 8

1,

Then

-Jot J (Ex D

SAT

Jf

0

e

-All

r

,yldAlllv(dylds r

I,

By an elementary calculation we have

Thus we arrive at

By a routine of monotone class argument, for all Now assume that

and v

8.

(2.6) is verified

v E 8,.

By [BM2] Th.2.1 we can take an

218

incerasing sequence of compact sets

o and

51' v (0- U F

k=l k)

for all

x

O'N

where

v

t

{F l n n>l

such that

-

JoIF n (Xs )dAvs is a polar set(depondins on v)

Fix a point x

D'N

v

and define

C

= {(s,y):

pp(s,x,y)
1,

JJDPIl (s,x,y) fI ctl Fn (s,y)\I (dy)ds t

=

Ex[J

Consequently

t.A T -All

e

o

sfIcnF n

IC(S'y) = 0

O.

dsxv (dy) -a.e.

This enable us to

complete the proof by using monotone convergence theorem.

Remark.

ij

The above argument is also available for a general

Hunt process (symmetric or nonsymmetric) provided it admits a transition density function with respect to some reference measure on the state space.

But we are not going to discuss

this matter in the present paper.

We mention here that recent-

ly Jia-an Yan [Ya] has obtained a generalization of formula (2.6) which is available for a symmetric Hunt process even if there is no transition density function. For GIl+n

\l

5

and

n > 0, we define H\l+a-Green function

as follows:

GIl+n(X,y)

00

J

oe

-nt-..Il p (t,x,y)dt.

By (2.4) we have (denote by

Gn _ Go +n)

(2.7)

219

GlJ+C1(x,y) (2.8)

x

D--.N

and hence

GIJ+a(x,y) x

By (2.6) for

v

5

and

f

B+(O)

e:

(2.9)

D--N.

it is clear that

(2.10) Let us notice that (2.9) and (2.10) were first observed by D. Feyel and A.de la Pradelle [FP].

3.

Compatibility condition Let

on

0

B (D) q

be the totality of Borel functions q.e. defined

+ Bq(D)

with values in

negative elements in elements in

Bq(D)

Bq(D), and

11· Hq

Denote by sense.

Bqb(D)

Notice that

is a Banach space. (GlJ+C1.v)f

=

J

GIJ + C1

D

Bqb(D)

For

lJ,V

lJ

5

the subclasses

0

5 (1J) 1

11·11 q

equipped with the

e:

5

( . , y ) f (y ) V {d Y}

Motivated by T. Sturm's class of for a given

the subset of all

with finite q-norm (Recall q-norm

is defined by (2.2». norm

the subset of all non-

and

C1

> O.

provided it makes

and and

0

1,

we define

C(IJ)

of smooth

measures as follows. (3.1)

220

C (ll )

If

II

== {v E S 1 (ll): inf a>O n>1

v E C(ll), then

v

(GII+ex • v ) n 1 /I

q

< 1}

(3.2)

is said to satisfy the compatibility

condition with respect to

ll.

The following proposition is

evident and hence we ommit the proof. Proposition

3.1.

The following statements are eqUivalent:

v E S1(ll)

(i)

(3.3)

for all

(ii)

t

(iii)

inf IIJJ plJ (s,x,y)v (dy)dsll

(iv)

t IIJJoplJ(S,X,y)\J(dY)dS 11q
0

(3.4)


0

(3.6)

II To obtain conditions such that

\J E C(ll), we need the

following lemma. 3.2.

Lemma

Let

lJ,V E S

and

ex > O.

Then for

q.e. x E 0,

(3.7)

Proof.

ObViously (3.7) is true for

(3.7) is true for

n

= k.

n

We have for

= O.

Suppose now that

q.e. x E 0,

221

Thus by induction (3.7) is true for all n

3.3.

Proposition

e:

(i)

'J

(ii)

II E.[J

(iii)

/lE.[f

C (u

O.

The following statements are equivalent

)

(3 ..8)

r -AIJ-as+A'J e S sdA'J) o s

II

q

tA'r _AlJ +A\) e s S d A'J ) 1/ o s q

We shall

co

for some

a > 0

(3.9)

< co

for some

t > 0

(3.10)


0

(X

(4.1)

dA-] < co.

s

0

No'

s

Consequently by (3.14) we have for all

sup t 0,

(4.2)

co.

By [BM2] Th.2.1 we can take an increasing sequence of compact co

sets

of

{F n } n> 1

and

lim Cap (K'F ) n n-for each n,

D

such that

GK ,

I

IF

d

n for all compact sets

0

U F n) = 0 n=1 Define

1 I

K c: D.

(4.3)

By [F) Lemma 5.16 we have for some polar set

1,

for all

N

1

c: D,

(4.4)

x E

Without loss of generality we can assume that here

is defined by the following condition:

{x E D: U

of

There exists a neighborhood

x

such that

I

U'

I

In the sequel we shall fix a polar set No

c:

and

GK

d

N

I.

(4.5)

(No UN,)

N satisfying (4.1) and (4.4) respectively. 1 Define for f E Bq(D) and t > 0

with

224

ll

e tH f (x)

{4.6}

provided the right hand side makes sense.

As suggested by the tHl.l resutls of (BM2], it is natural to expect that (e )t>O is

a realization of a Schrodinger semigroup. Indeed, in Section 7 tHl.l we shall show the connection between (e }t>O and the IJ. (2 - u ) I D'

Schrodinger operator

study the properties of Let

l.l

n

= II + - I

ing of this section.

(e

F n '\.I

tH ll

But ina first step we should

)t>O

where

Denote by

is specified at the beginn qn(t,x,y) the integral kernel F

II

of

(e

tH n

qn{t,x,y)

It>O

constructed as in (BM2].

increases pointwise as

n

It can be shown that

tends to infinite.

Denote

by

pll (t,x,y)

(4,7)

lim qn(t,x,y),

n--

Obviously

p\.l(t,x,y)

is an integral kernel of

following results are direct consequences of the above construction. 4.1.

Theorem

Let

pll{t,x,y)

be given by (4.7).

Then the

following properties hold:

(4,8)

(11)

(4.9) V t,s > 0;

x,y ED;

225 {iii}

pll {t.,x,y}

(iv)

plJ (t,x,y) > 0

(v)

pll{t,x,y} where

(vi)

If

plJ {t,y ,x} , on

(4.10)

V t,x,y ;

(O,co)X

(D---N) X (D---N)j

is jointly continuous on

K(ll)

is defined by (4.5);

f E Sb{D)

is continuous at

(4.11)

{O,oo)xK {lJ)xK (lJ)

(4.12 ) x E D---N, then (4.13 )

lim JDPll(t,X,y)f(y)d Y = f(x).

UO

Proof.

(i),

(ii),

(iii) and (iv) follow directly from [BM2)

Theorem 6.6.

(v) follows from [BM2)

prove (vi).

Let

f E Sb(D)

(6.16) and (5.4).

be continuous at

We now

x E D---N.

We

have

Since the last

Ex[f(Xt}I{t 0

ing equality holds

(t,x,y)

pll (t,x,y) ,

a.e.y.

and

x E: D, the follow-

232

Letting

e

0, and using dominated convergence theorem we get

Thus by monotone class theorem we arrive at (6.1) under the restriction that Repeated

application of monotone convergence theorem enable

us first to replace the restriction E GK

E GK

d

by

E Sand

and then to remove all the restrictions to complete

d,

the proof.

6.2.

II

Corollary

Proof.

Let

x E

be defined by (6.2).

If

From the proof of Th. 6.1 we know already that

(t,x,y)

for

(t,x,y)

(t,x,y)

a.e.y

By (6.1) we have

pO(t,x,y} -

(dz)ds. (6.2 )

233

For

x E

we define = pO(t,x,y) -

JeIo po (S , X, Z) plJ (t - s , Z, Y) IJ (d Z) dS . t

(6.3) Then by Markov property,

Consequently

plJ (t,x,y) =

symmetry it can be shown

(t,x,y) a.e.y. -IJ p (t,y,x)

Noting that by for all

(t,x,y) , we declare from the explicit formulas (6.2) and (6.3) that for

t,s > 0, x,y E D--N,

(6.4)

plJ (t+s,x,y) . On the other hand, we have for

t,s > 0, x,y E D--N,

(6.5)

plJ (t+s,x,y) . Thus the desired assertion follows from (6.4) and (6.5).

7.

II

The connection with Schr5dinger operator 2

.Let HIJ with the domain V (H IJ ) be the L -generator of IJ tH (e )t>o' We are going to show that HIJ is a realization of the Schr5dinger operator

s -IJ) (2'

with Dirichlet boundary

condition. Let

a

be a fixed positive number satisfying (5.8).

Proposition 3.3, we may assume further that

a

satisfies:

By

234

(7.1)

From (5.8), (7.1), (6.1) and the following identity

We conclude that there exists a polar set a >

N

such that for

a, sup

J"'e-as[Jf pll(s,X,Y)Il*(dy) +Jf pll(s,x,y)dy]ds 0

D

D

:=c

a

< "'. (7.3)

Here and henceforth We define for

11* a >

11 + +11 - .

stands for

a,

(7.4)

Provided the right hand side makes sense. Since satisfiying

(e

tH Il

lie

fll

hence the generator on for

is a self-adjoint Co-semigroup on

)t>O tH Il

such that

L

2
B.

(7.5)

and (7.6)

235

Lenuna

7.1.

Let

C1

>

e

(i)

GlJ + C1 f E LP(O,lJ*)

(ii)

Gil +C1f E

(iii)

!Gll + C1 f ! < ""

Proof.

f E LP (0)

and

with

p

t •

Then

S,

n LP(O,\I)

\IE

q.e.

By Holder inequality we have

(7.7)

Consequently by (7.3) and Fubini's Theorem,

which proves (i).

Taking (5.8) into account, by the same

argument we can prove (ii).

The assertion (iii) is a consequence

of (ii) and [F] Theorem 3.3.2.

II COO (0), where

Denote by

o

is

by definition

C1

Let

(u,v)

be the family of quasi-continuous functions in

o

7.2.

Lenuna

Then

Gll+cL f E

Proof.

Let

C1

>

e

and

fE L

2 (0)

(:=L

2 (D,dx»).

(D) •

Without loss of generality we assume that

f > O.

Let

236

where

{F n}n>1

is

the increasing sequence

of compact sets described at the beginning of Section 4. We lJn+a write u = GlJ+a. f and un f. Then 0 < u < u and G nlim u (x) = u(x) n+ oo n we have

for

By virtue of Lemma 7.1

q.e. x E D.

o. By [BM2] Lemma 3.3 and Theorem 6.5 we have

(7.8)

un E

and

Applying [F] Lemma 1.3.4 and Theorem 5.1.3 we obtain e a ( un' urn ) • 1 inl t' ( u

t+o

Consequently for

n

- e -atPtU, U ) n

m

n < In,

Taking (7.8) into account, the above equality shows that {un}n>1

is a £a.-Cauchy sequence.

Thus

u

by [F)

Th. 3.1.4.

7.3.

Theorem

II Suppose that

lJ

is a Radon measure.

Let

237

(7.10)

Proof.

Let

2 L (D).

0 1 n > 1 we have

IF

= n

Letting

Noticing

by monotone convergence theorem we obtain

that

particular that

L 2 (D,lJ - )

U

u

C

1 - ), (7.11) implies in Lloc{D,lJ

L{oc{D,lJ+).

what we have proved shows that

By virtue of (7.5) and (7.6),

V (HlJ )

C

V and

(7.10)

holds.

To complete the proof we need only to show that To this end let

I DU

u E V,

= f.

We define for

a

>

S

q.e.

t:, + 'V (2' - a -lJ ) u +

Denote by

w

Which implies

U -

cu

'V

u.

w = 0

+ f + UIJ

We have

and hence

o

in

D.

and

238

By Martingale argument it can be proved that the above equation is equivalent to

u(x) =E

Tr,at is

x

[Jr

T

0

e

-A].I-at t (au+f) (Xt)dtJ

u = G].I+a (au + f).

q.e.

Thus we get

V c: V (H].I)

which com-

pletes the proof.

8.

II

Gauge theorem and Dirichlet problem of Schr5dinger equation In [BM2] Section 7 we discussed the eigenvalues and eigen-

tJ. +].1 + -].I - )1 D with functions of the Schr5dinger operator (2 ].I + GK d and ].I S. We point out here that all the assertions therein remain true in the case of ].I Sand + ].I S 1 n c (].I ). We shall freely employ those assertion without

verifying them.

o

Throughout this section we assume that

is a bounded tJ.

domain and consider the 5chr5dinger operator (2 -].I) I D with + ].I + 5 and ].I 51 n c (].I ). All the notations will be the same as in the previous section. 8.1.

Lemma

Let

L 2(D)

such that

> 0 q.e

on

D.

Then the following assertionS are e0.uivalent (i)

There exists a nonpolar set

G c: D

= 0,

lim

nt... (ii)

tJ. sup spec (2 - ].I) I D c

Proof.

Let

v x

G.

o.

betheorthornormal basis of L

which consists of the eigenfunctions of

{A i }i >l

such that

(2'A -].I)

10

2(0)

and let

be the corresponding eigenvalues with Ai + l Ai" 2(0) Suppose that L admits a Fourier expansion given by

239 CD

lP

= i=1

By [BM2) Th.7.5 we can choose a polar set

ailPi.

Nc::O

o

such that

By [BM2] Th.7.11 the first eigenfunction positive

q.e.

Therefore if

= J lP 1 (X) lP (X) dX > o

O.

lP > 0

lP 1 > 0

q.e.

is strictly

then

a

,=

Thus from the proof of [BM2] Th.7.9 we

know that the assertions (i), (ii) are equivalent.

II

The following theorem is a generalization of gauge theorems in feR] and [BHH]. 8.2. (i)

Theorem

The following assertions are equivalent.

There exists a strictly positive bounded function on

ao

such that

-All 'f(X,)] < _All 'f(X,)]!I q

(11)

IIE.[e

(iii)

There exists a not vanishing measure E

x

[Jr

, -All e sdA v ]
the corresponding inner product. Let T be a topological space which is homeomorphic to IRm for m E IN ; BT will be the Borel e­fleld of T. If v is a non­negative Radon measure defined T (or better on BT), we set B" the 8­ring of sets B E 8T such that v(B) < 00. We say that p. :8" ----+ H is a (H­valued) orthogonal measure on T with variance t/ if

< p.(A),p.(b) >= v(A n B), A, BE 8". 2.1 a) b) c)

Remarks p. is o­additive and p.(t/» = O. B" contains all relatively compact subsets of T. If 9 E L2(v), the notion

h9dP., is natural and well­known. d) We have the fundamental property

(2.2)

246

Let T be another space of the same type as T, and W: T ----+ T a Borel application; let J1. be an orthogonal measure on T with variance t/, We set v = v 0 w-1 ,[.t = J1. 0 w-1: that is to say, for B E Bt, we set v(B) = v(W- 1(B)) , [.t(B) = J1.(W- 1(B); [.t is an orthogonal measure on T with variance v. Moreover, if 9 : T ----+ JR, g : T ----+ JR are measurable functions, such that g 9 0 W, we have

and

hgd[.t = 19dJ1. Let I be a real interval and a : I -+ JR locally with bounded variation; do will represent the canonical signed measure associated with a. If J is another real interval and a : I -+ J is continuous, monotonous and surjective, a-I : J -+ I will be defined by

(2.3)

0'-1

2.4 Remark sets such that

(v)

sup{u I a(u)::; v}

There is a countable family (eventually empty) of real intervals (In), da-null

n

In particular

0'-1 0

a is equal to identify function I

----+

J, da - a.e .•

From now on, letter i will always be reserved to elements of {I, 2}, r will be a curve in the first open quarter of the plane whose parametrization is given by an injective and continuous application W = (WI, W2) : JR -+ r will be said increasing if WI and W2 are non-decreasing (resp. decreasing if WI is non-decreasing and W2 non-increasing). In the whole paper, r will supposed to be either increasing or decreasing. We remark that W is a homeomorphism JR -+ r, if r is equipped with the induced topology from JR2. We introduce the natural order on r transmitted by W; obviously it does not depend on W. A subset of I' that is the image through W of a bounded interval will be called segment: such a set will be connected and relatively compact. For r, we use similar notations as for JR: for instance, if x, y E r, [x, y[ will be the segment of r whose left-end and right-end points are respectively x and y (the first one included). We note by 4>; : r -+ JR+ the coordinate functions (t 1,t2 ) >--t ti. We have 4>i = W; 0 w-l; we set 4>;1 = Wi 0 w- 1 where w- 1 has been defined in (2.3). Following remark 2.4, 4>;1 0 4>; is equal to identity excepted on a countable union of segments Un I n ; in particular 4>;1 04>; is equal to the identity, bi-a.e., where bi is the Radon measure on r such that

b;(]t,zJ) = IZi

til, t, z E r : t :::z.

On the other hand, Wi 0 W;I = 4>; 0 4>;1 are the identity functions Ii -+ Ii , I, = ImWi = Im4>i, i = 1,2. r will be called separation line if it splits in two domains D_ and D+ such that EJD_ = EJD+ = r. We note T(r) or simply T the rectangle 4>1(r) x 4>2(r). The reader will see along the whole paper that there is no loss of generality in supposing T(r) =

247

If t E T, t;, i

= 1,2, will stay for component i

of t; R t will denote the rectangle

{sETIOi 1(t;) (or (tI,tz) - h(t;) E LZ(ll;f,v) and g(t) = h;(t/>;(t». By a standard argument, if I' is increasing, we have

if

r

is decreasing, we multiply the second member of second inequality by (_1);-1. Moreover M E Br is a 6;-nul set if and only if ll;M is a z--nul set .•

248

With measures jji and Eli, we associate Jli = jji o'lt and 0i = Elio 'It which are respectively an orthogonal measure and a Radon measure on 13R; 0i is the variance of Jli. We set

1 1 11

(2.6)

gaiX

11

1 =1 =

11

i

gd(jji

+ jjz)

11

gaiX

= 1,2

1Jdjji ,

where x,y E f,g E LZ(f, Eli)' We will now speak about linear Markov properties. Let HI,Hz and H3 be subspaces of the initial Hilbert space H. We will say that H3 is a splitting space for H l and Hz (or H3 splits HI and Hz) if

(2.7)

Hi e H 3 is the closed span of elements Y - PHiY' Y E Hi, where PHaY is the orthogonal projection ofY on Hi, i = 1,2. 2.8 Remark Following [R1], chap. 2 §3., we have: (a) If H 3 splits u, and Hz then H3 :J s, n Hz. (b) The class of splitting spaces H3 of HI and Hz which are contained in HI has a minimal element in the sens of inclusion.•

Therefore, if H l

n Hz splits HI and Hz, it will be the minimal splitting space.

Let X : T -7 H. If ACT, H(A) will denote the closed linear span of X(t), tEA; if 0 is a closed subset of T, we note G(O) the intersection of the family of H(O) where 0 runs over all open subsets of T containing O. We say that X has the sharp linear Markov property (SLMP) with respect to I', if H(f) splits H(D_) and H(D+); X is said to have the germ linear Markov property (GLMP) if G(f) splits H(D_) and H(D+) and the minimal linear Markov property (MLMP) if H(D_) n H(D+) splits H(D_) and H(D+); in this case, the splitting space will also be called M(r). If X is continuous and if the MLMP holds then

(2.9)

H(r)

c M(f)

C G(r) .

and the GLMP also holds. If we use now the language of probability theory, H is the Hilbert space L2(n, I:, P) where (XdtET is a square integrable process, we will say that X has previous linear Markov properties, if t I--> Xt (class of Xd has. We remark that a COIP is a COIHVF. If A, 13, C are o-subflelds of I:, one generally says that C is a splitting O'-field for A and 13 (or C splits A and 13) if A and 13 are conditionally independent given C. We can associate with X a family of e-fields ( the natural filtration of X) in the following way:

(n, I:, P) is a complete probability space. If X

H(S) = O'(X t , t E S), SeT, completed by P-null sets. If S is closed, we note g(S) the intersection of the family of H(O) where 0 runs over all open subsets of T containing S; g(S) is called the germ o-field associated with S.

249

If T' is a separation line for two open sets D _ and D +, then we say that X has the sharp (resp. minimal) Markov property or simply SMP (resp. GMP, MMP) with respect to r if ?t(f) (resp. g(r), ?t(D-) n ?t(D+» splits ?t(D_) and ?t(D+) in the sens of conditional independence. When minimal Markov property holds, ?t(D_) n ?t(D+) is also denoted by M(f).

2.10 Remarks (a) Markov properties above are stronger then linear Markov properties; however, they are equivalent when X is Gaussian. (b) If X is continuous in L2, then SMP => MMP => GMP SLMP => MLMP => GLMP (c) In the Gaussian case, we have ?t(D_) = u(H(D_

g(r)

= u( G(f»

»,

?t(D_) n ?t(D+)

= u(H(D_) n H(D+»

;

d. lemma 3.3 of [M].

§3. Description of boundary spaces Let X : T_H be a COIHVF with associated orthogonal measure; we denote by v the variance of X. iii, I-'i , 6i and 6 i in relation to x, v have been defined in previous sections. Let r be either a decreasing or an increasing separation line splitting T in two domains D_ and D+; let '1J be the parametrization of f. X has the MLMP with respect to r. H(D_ )nH(D+) is the minimal splitting space M(f) for H(D_) and H(D+). If X = (Xt)tET is a Gaussian COIP, ?t(D_) n ?t(D+) is the minimal splitting e-field M(r) for ?t(D_) and ?t(D+). Then the MMP also holds. Moreover, a system of generators for this space M(r) is given by following elements

{Z {)iX , x., z E r , i

(3.1) where where

i: ZO

is a fixed point of f. We recall that

1

= 1,2,

z

zo

{)iX, i

= 1,2, is up to sign equal to X(.6.i),

i = 1,2, is represented by the following picture:

z

z Figure 2

250

If X is a Brownian sheet, previous properties have been proved in [W1], tho 3.1 and 3.2; the same arguments remain valid if we replace white noise measure by X. We remark that generators given in (3.1) can be reduced to

t;[ZOiX, i = 1,2, , z E r, Xzo,

(3.2) where

zO

E

r

is fixed. In fact X z = Xzo +

r oX, where oX = 8"X + 8zX.

i:

We note Hi the closed linear span in H(T) of elements

r OjX.

Jzo

It is easy to see that

(3.3) Therefore (3.4) where [X zO 1is the space generated by X zO • First of all, we discuss the case of increasing lines which is the easiest one. We have of course w(-00) = 0

r

Figure 3 We suppose, as in the figure, that D_ is the subset of T whose boundary with respect to the plane includes Ox-axe. zO

Since Xzo

= loX, and HIl.H2 , we can transform (3.4) into

(3.5) This obviously gives the following proposition. 3.6 Proposition Let r be an increasing separation line. M(r) is the space of Y E H(T) such there are iii E L2(r, ad (uniquely determined) such that

(3.7)

Y

(In this case we will say that (iiI,

=

(2)

1

(iilolX

+ 0'28z X)

represent Y) .•

The case of decreasing separation lines is more difficult; in fact HI and H 2 are not orthogonal. Suppose now r to be decreasing; obviously we have WI (-00) = W2 (+00) = O.

251

3.8 Remark The direct sum of closed Hilbert subspaces is not necessarily closed, cf. [H], §15. However if I' is a bounded and decreasing, 'IJ1(-00) = lim 'IJ1(u) belongs to the closure of T and

X"'O

=

1

u-oo

",0

'1t( -00)

ax E n, + H 2 •

We will even see that M(r)

= s, + H 2 ••

We suppose now that r is decreasing, not necessarily bounded and that the variance of X has a product decomposition that is to say, it is of the type n

L v k , dv k = Pkd (vf e v;) , I S k S n, n E IN ,

(3.9)

k=l

where n.

vt i = 1,2, are Radon continous measures on [0, oo[ and Pk, ..L E Loo(T, dt), 1 S k S Pk

We suppose that D_ and D+ are respectively the inferior and the superior parts.

r

Figure 4

3.10 Proposition If 9 E L1oc(lL,v) and there are measurable functions ai : JR+_JR such that g(tl, t2) = al(td - a2(t2), then (tl, t2) 1--+ ai(ti) E L1oc(D_,v). In particular, if r is bounded, these functions are square integrable.

Proof

In this proof again, i will belong to {1,2}.

(a) First of all, we consider the case v = Vl 0 V2. Since constants are in L1oc(D_,v), we can substitute ai by ai + C, where C is a constant. If v(D_) = 0 then our result is trivial. If D_ is not a v-null set then there is ZO E r such that v(R",o) > O. We introduce the following notations.

R = R",o , R = II X 12 ti, = {(tl,t2) E D_ I ti > z?}

252

Figure 5 Of course vi(Ii) > o. Now, ai belongs to Ll(I" vd; in fact, for instance we have

for vz, a.e, t z E I z . By substracting the constant

we can suppose

r

II

al dVl

O.

I

We can prove now that (tl,tz)

then

l

at(ti)dv(t)

s

l

f---->

ai(ti)lR(tI,tZ) is square integrable. In fact

gZdv; in this way ai are square integrable on R.

Let K be a compact of tL; we have to prove that

i

aidv

There is a bounded interval I such that D z n K

< 00.

c II

X

I; therefore

(3.11)

Then al E Lroc(Dz , v) and through additivity az = 9 - al E Lroc(D z , v). In the same way as for (3.11), az E Lroc(Dbv) and through additivity In conclusion, ai E Lroc (D1 U D2 U R) = Lroc(D-, v).

al

E Lroc(Dl,v).

253 (b) If the proposition holds for gE L1oc(.D_,v), it will also be valid for 9 E L1oc(fL,pdv) 1where p E L't:c(T, dt) such that - E L't:c(T, dt). p

n

(c) Let suppose that v =

2: vk, where v

k

is a positive measure on BT. We remark that

k=l

, k E {I ... n}. Therefore, if the conclusion of the proposition holds for every v k , then it will also hold for u,

9 E L't:cU]_, v) if and only if 9 E L1oc(D_, v k )

(a), (b) and (c) allow us to conclude.• If we slightly modify the proof of proposition (3.10), we obtain the following result.

3.12 Proposition Let v be a measure having a product decomposition on T. Let gn : D__ JR such that gn(t!> t2) = an(tl) - bn(t2) u - a.e. , n E IN, where an, bn : JR+_JR. If s« E L1oCi) i = 1,2, with

(b) M(r)

M.

Proof (a) is a consequence of proposition 3.14. Let Hi be the spaces defined after (3.2). Remark 3.8 and corollary 3.18 show that M(r) = HI + Hz is included in M. On the other hand, using (a), we can verify that M = HI + Hz C M(r) and the result follows.• If f is unbounded, the conclusion (b) of the proposition above is true under certain conditions. Using results of next section, we obtain the following representation theorem. 3.20 Theorem Let X = (Xt)tET a COIHVF with orthogonal measure X; we suppose that the variance v of X is of the type VI 0 Vz where Vi is a continous Radon measure on [0,00[. M(f) is the space of

(3.21)

255 Proof Following corollary 3.18, M(r) C M; it remains to prove that this inclusion is an O. equality; for this sake we verify that M(r).L n M Let Y E M satisfying (3.21) and being orthogonal to generators (3.2) with zO = We use results of next section. al and a2 will be solutions of (4.10) with Al A2 == 0 and A(O) = 0; therefore they will also satisfy (4.15). Because of proposition 4.14, 9 will be zero and so Y = O. This proves M(r) = M .• We will now discuss the description of the sharp space H(r) when I' is a decreasing or increasing separation line. If I' is decreasing, we suppose that dv = pd( VI 0 V2) where p, are VI 0 V2 a.e. bounded and Vi are Radon continous measures on [0,00[. Iff is decreasing, we do not suppose anything on v. If X is a Brownian sheet, [WI] remarks that 'H(f) is strictly included in M(f) even if D_ is so simple as the triangle {(tl,t 2) E T I tl +t2 < I} : if W is a white noise, W(D_) belongs to M(f) but not to 'H(r). Then, in general, H(r) is not a splitting space. In general the SMP does not hold. If X is a process with independent increments, vanishing on the axes and I' is a finite union of horizontal-vertical segments, then 'H(r) M(r) = g(r), see theorem 7.5 of [Rul]. When X is a COIHVF, this can be adapted and we can verify that X has the SLMP with respect to T and H(r) = M(f) = G(r). Let 8 i and 6i be the same measures as in section §2. We recall that

x,yEf,u,vEIR. If Y is represented by (a, a), a E LFoc(f,8 1 ) n LFoc(f, 8 2 ) , we will write Y can now state the representation theorem for the sharp space H(r).

3.22 Theorem 3.23 Remark

H(r) is the space of Y

= n(a).

= neal.

If a is square integrable with respect to 8 1 and 8 2 , then

neal

=

We

lr sex.

cf. proposition 3.14. It is the case when f is bounded or increasing.• Before proving theorem 3.22, we need a lemma. 3.24 Lemma Let E be the subset of H(r) consisting in elements nca). Let Y E H(T) represented by a couple (aI, az). Then Y E E if and only if there are 6 i-nul sets M, such that al = a2 on f\(MI U M z ).

Proof If Y E E, there is a with Y = n(a). Following proposition 3.6 and remark 3.16(b), there is a constant C such that ai = 0.+ C, excepted on a 8i-nul set Mi. Then 0. 1 = 0. 2 on f\(M I U M 2 ) . The converse statement follows if we set

this implies ai

= a , 8i a.e.

a = allr\M, + a21r\M and Y = n(a).•

2 ;

256 Proof (of the theorem). We recall that 1>i and IIi have been defined in section §2. We verify that E is closed. We suppose that I' is decreasing; the increasing case is similar. Let (an) such that n(a n) converge to Y E M(r). By definition

where

= an 01>i.

If we add some constants to the

g(t1,t2) = b1(t d - b2(t2) and a v-nul set M, with

We are allowed to suppose IIiMi = Mi. From section §2, we recall that 1>;1 0 1>i(X)

an,

x, if x (j.

there are bi , i

= 1,2,

t; = UI n where I n

such that

is a segment

n

of I' with 1>ilJn constant. We can see that i; is a 0i nul set: in fact, II;Jn is included in a straight line parallel to the OX3-i axe; since X is continous, the measure v evaluated on straight lines is zero; this implies

n

n

We consider now 1Mi C F such that II i1Mi = Mi. By remark 2.5, 1Mi is a 0i-nul set; we set k, = L, U 1Mi and hi = bi 01>i. If x E f\Ki,

In conclusion there are 0i-nul sets

k,

such that

Since Y is represented by (hI, h2 ) , the above lemma says that Y E E. Therefore E is closed. If z E I', then X z = n(a), where a = l{xEf!x::;;z} • E being closed, H(r) C E. It remains to prove that E C H(r). If I' is increasing, we can apply the same reasoning as theorem 3.10 of [DR] : a standard approximation theorem. If I' is decreasing, we observe that it is enough to study the case v of product type VI Q9 V2. For this sake we will be helped by the next section. We verify that En H(r).L O. Let Y = n(a) E H(f).L. Since (X"'(u), Y) = 0, for all u E JR, a = a 0 'I/J will be a solution of (4.8) with A := 0 and then also of (4.15). Proposition 4.14 shows that Y = 0.• We remark that representation theorems for H(f) and M(f) can be extended to the case where D_ relatively convexe (every horizontal and vertical section is convexe) : cf. Dalang-Russo, private communication. As an application of representation theorems, we give necessary and sufficient conditions for the equality H(r) = M(r); if (Xt}tET is a Gaussian process this corresponds to the H(r) = M(f). We will apply the same procedure as in [DR], tho 3.12.

257

3.25 Theorem Let X = (Xt)tET a COIHVF and v be the variance of the associated orthogonal measure X. We suppose that one of following hypothesis is satisfied. a) I' is increasing b) I' is decreasing and dv = pd(V1 ® V2), where p, are bounded v a.e., Vi is a Radon continous measure on [0,00[. We consider the following measures

6i(B) =

1118ixlr ,B E Br, i

= 1,2.

The following conditions are equivalent.

i i

(a) H(f) = M(r) (b) (c)

81X E H(r), for every segment J of r EhX E H(r), for every segment J of f

(d) 6 1 and 6

Proof

2

are mutually singular.

(a)=>(b) is obvious. If J is a segment of I', (b)=>(c) is a consequence of

In order to prove (c)=>(d), we remark that I' is a countable union of segments I n , n E IN. It is clear that

r EhX = i:r O· 8 By theorem 3.22 we can write r EhX = "R.(a i: i:

IX

n ),

+

r 1· EhX .

i:

for every n E IN.

Proposition 3.6 and remark 3.16(b) show the existence of a constant C such that

an -

0 = C,

61 -

a.e.;

on Jn j then, there are M{', M?: such that 0

an -

= 1 on

1

C,

62 -

a.e. ,

J n \ (M{', M?:). We set M, =

U Mr

and

n

we obtain 0 = 1 on f\(MI U M2 ) , which implies that 6 1 and 6 2 are mutually singular. It remains to prove (d)=>(a). Let Y E M(f). By corollary 3.18, Y is represented by a couple (al,a2). (d) says there are 6i-nul sets with M 1 U M 2 = I'; then, obviously, al = a2 on f\(M1 U M 2 ) . Lemma 3.24 and theorem 3.22 allow us to conclude.• When X is a Brownian sheet measures 6i are equivalent to the variation measures Iii of i t h coordinate, that is to say Iii (]s, t]) = Iti - SiI where s, t E I' with s ::; t. Finally we can state the following proposition (practically theorem 3.12 of [DR]).

3.26 Proposition Let X be a Brownian sheet, I' either an increasing or a decreasing separation line. M(f) = 1i(f) if and only if measures Iii are mutually singular.

258

3.27 Remark If r is a countable union of segments which are parallel to the axes, measures 01 and 62 are mutually singular. However [DR], rem. 3.13 shows an example when 01 and 62 are mutually singular and r does not contain any horizontal or vertical segment.•

§4. Extrapolation: general formulas Let H be a Hilbert space with inner product < " . > and norm II . II. X = (Xt)tET will be a COIHVF with values in H. X will be the associated orthogonal measure with variance u, Measures and 0; have been defined before remark 2.5; however, we recall that

e;

r will be either an increasing or a decreasing separation line which is parametrized by '11 : JR __ T. We will evaluate explicit formulas for linear predictors of X when the function X is known on r (through H(r)) or around r (through M(r». We recall that

Given a subspace G of H(T), we denote by Pa the linear projectors of H(T) on G. We begin treating the case when r is increasing: this is the easiest situation. We recall that M(r) = HI EEl H 2 and H 11.H2 • In order to prepare the first proposition of this section, we set (II(U)

Jo Obviously, we have X'I1(u) =

O;X,uEJR,i=1,2.

+

, U E JR. Let Y E H(T). We note

>, A(U) =< Y,X'I1(u) >,

A;(u) =
.(U)

1 =1 D_

-00

e:

U - - 11[0,

d>.;. 2 , z = 1, .

= de;

dll(tI, t2)jo4>11(tx)+

tt

where

de;(v)a;(v),

{del

+ de 2 }{ j

0

f dll(tI,t 2)j o4>'2 JD+

l(t2)

'I'}

--'--"'-v--' de f

llI(u)],u E

JR,! = j

0

Ill. Then!

=

.•

Calculations become rather difficult when r is a decreasing separation line. In this case the extrapolation with respect to M(r) cannot be reduced to the projection onto two orthogonal subspaces of M(r). From now on, we suppose II of product type 111 0 112 where IIi are continuous Radon measures on [0,00[.

260

= fT9dX,

Let Y E H(T); obviously Y

l

'1'(U)

'1'(0)

..\;(u) =
, ..\(u) =< X'I'(u» Y >,

u E JR. According to corollary 3.18 and theorem 3.22

JE

L 2oc (9d 1

n L21oc (9 2 )

and

a, E

PH(r)Y = 'R(ii) and PM(r)Y is represented by (iiI, ii2)' We note f = J0 III Ill. We recall that r splits T in D_ and D+ where D_ is the inferior part (see Figure 4 before proposition 3.10). At this point we can write

Lfoc(9i) such that

and

ai

=

ai 0

< X'I'(u),I(f,f) > < X W(0),I(a1,a2) > < >

(4.6) (4.7.0) (4.7.i)

= =

..\(u), u E JR, ..\(0)

= ..\(u),

u E JR, i = 1,2,

where

(4.6) can be rewritten as follows: (4.8)

..\(u) =

j

dv(f

0

llI}l(tI) -

f 0 1lI 21(t 2

», u E JR

R.ll(u)

Before going on, we introduce the useful notation 7i(U) = viejO, llIi(u)]), u E JR,i = 1,2. 71 is non-decreasing and 72 non-increasing; in particular 7i have locally bounded variation; moreover 71(-00) = 72(+00) O. Because of locally square integrability properties of iii and we find out that ai c L; and fELl n L 2 where

J,

00,n],d71) , L 2 =

nL

2([- n , 00[, d(-72»'

nElN

(4.6) can be expressed as follows (4.9)

In the same way, system (4.7) can be transformed into:

(4.10)

0)

..\(0)

1)

..\1 (u)

2)

..\2(U)

72(0)

1" l °

[°00 a1 d71 -71(0)

l l-00

100

a2d72

Y1

d71(Yd

u

o d72(Y2)

00

d72(Y2)(a1(Y1) - a2(Y2»

Y2

d71 ((yI)(a1 (yI) - a2(Y2» .

261

We observe that d>"i is absolutely continuous with respect to d'Yi and

(4.11)

We consider the following integral operators on L 1

So(fl, h)(u)

= 'Y2(U)

X

h d'Y1

L2:

+ 'Yl(U)

1

1

00

h d'Y2

00

Sl(h,h)(u)

'Y2(u)h(u) +

S2(/t,h)(u)

-'Yl(u)h(u) +

h d'Y2 hd'Yl

If we derivate right members of equations 1) and 2) in (4.10) with respect to d'Yl and d'Y2' we obtain

0) SO(a1,a2)(0) (4.12)

1)

Sl(al,a2)

2)

S2(al,a2)

>"(0) d>"l d'Y1 - a.e. d'Y1 d>"2 = d'Y2 d'Y2 - a.e.

Let i = 1,2. We remark that d'Yi« If we divide equation (4.9) by 'Yi(U) and furthermore we derive the right member with respect to J:1., we find out that f is solution of "2 the following system (for every i 1,2)

So(f, 1)(0) = >"(0) (4.13.i)

{ Si(f, I)

=

c.

d(

- a.e.

where Oi

d( "0>"-;)

= d( ,3-. 1; )'

i

= 1,2

In order to solve systems (4.12) and (4.13.i), it is useful to study solutions of the homogeneous version of (4.12). 4.14 Proposition

Let (h 1,h2 ) E L 1 x L 2 • It is solution of

if and only if there is a constant C such that hi = C, i = 1,2.

262

Proof If hi = C, i = 1,2, then (hI, h z) is solution of (4.15). Suppose (hI, h z) just solution of the two last equations. We set

(4.16)

for U E JR. We have hi = hi, d'Yi - a.e. Following lemma 5.8 of [DR], we can show the existence of constants k 1 and kz such that

(4.17)

(4.16) implies lim u -+ oo 'Yz(u)h1(u) the help of (4.17) we obtain

= lim u -+ _ oo 'Yl(u)h z(u) = O.

liminfh'z(u)h 1(u)l2:

t t

ZI

"'11 +00

"-+00

liminfh'l(u)hz(u)l2:

'Yz

"-+-00

zl

-00

On the other hand, with

) )

Therefore, if "'11 (+00) or 'Yz( -00) are finite, k2 must be O. If both are not finite, (4.17) shows through a direct calculation that So(h 1,h'2) k2: in conclusion k2 must vanish.• Now we are in the position to state general theorems about extrapolation with respect to spaces M(r) and H(r). We will also indicate an idea of the proof which is contained in sections §5.2, §5.3 of [Ru3]. Let X = (Xt)tET a COIHVF with orthogonal measure X. The variance v of X is supposed to be of product type. Let Y E H(T) of the form J gdX, 9 E LZ(T). For u E JR, we recall that "'Ii( u) = Vi(]O, \II i( u)]), Ai(U) =< A(U)

=


OiX, Y

'1'(0)

X'1'(u) , Y

>,

4.18 Theorem We suppose 9 E Ll(D_,v), where D_ is the inferior domain splitted by PM(nY is represented by (al'i.iz) where

r.

=

az(u)

13i =

al 0

\II( u)

= 131 (u) -

= az o\ll(u) =

12

-132(u) "'11

k and Uo are real constants.

-1"

11/2

+l

u.

A l 'YI1Z

(d'YZ 'YI1Z

U(d

".

'Yz

"'11

+

"'11

12

+k

263

h(u)

= h-

0

h(u)

= h-

0

'1'(u)

G1 = -(u) -

'1'(u)

-G 2 = -(u)

72

71

and G, =

l (1 ) + (1) + +l u

u.

G1d -

72

C1

U

u.

G 2d -

71

C2 ,

d(_A) "18-.

d

(...::tL) 1'3-.

To prove above theorems, we apply the same procedure. Concerning the first one, we know that (aI, a2) is uniquely determined up to a constant; moreover it is solution of system (4.12); proposition 4.14 shows that this system has not more than one solution up to constants; therefore, it is enough to evaluate a particular solution of system (4.12). To do that, we suppose for a while that 71,72,131 and 132 are smooth. Through derivation of first two equations, we have

If we subtract these two equations, we can easily calculate the requested expression of (a1, a2)' It is very important to verify that this expression belongs to L 1 X L 2 and that it really solves system (4.12): for this see section 5 of [DR] or section 5 of [Ru3]. Concerning theorem 4.19 we apply a similar method.•

§5. Evaluation of extrapolation (prediction) operators

Let

r

be either an increasing or a decreasing separation line; it splits T in two domains

D_ and D+. Let X = (Xt)tET a COIHVF with associated orthogonal measure x; the variance v of X is supposed to be of product type VI ® V2 where Vi are continuous Radon measures on [0,00[. These notations will also hold for next section. Here we want to calculate the best linear prediction of elements Y = Xt, t E T, with respect to spaces H(r) and M(r). First, we suppose r to be decreasing; D_ will be supposed to be the inferior domain.

264

Let i = 1,2. We set = 8iX ,u E IR and Ili the orthogonal measures such tha.t lli(]U,VJ) = xt u,v E IRj moreover, we set 'Yi(U) = Vi(}O, '*'i(U)]), U E IR. (a) Consider t E b.: We can suppose t = ('*'1 (a), '*'2 (b)), a:S b. In this case 5.1 Theorem

PH(r)Xt

(5.2)

Xq,(a) = v(R.) { (R ) v q,(a)

+

PM(r)Xt

(5.3)

r la d ( 6

d

(d(1l1

+ 1l2) + Xq,(.)d 1'2

1'2

{ 1 1 (1) + Xl + 1 [Xl (1)

= veRt)

a

6 1'2 d

Xq,(O)

1'1

6

+

a

(r112)( a)

a

-d 11

(..!.)) }

Xq,(O)

X2 -d

-

1'1

+ Xc

C'Yl'Y2)(b)

1'2

(1)]} -

1'1

(b) Consider t E D+; we can suppose t = ('*'l(a), '*'2(b)), b:S a. In this case (5.4)

(5.5)

5.6 Remark (5.2) comes out from the first equation in theorem 4.19 and (5.4) to the second equa.tion. IT we had operated differently, we would have obtained different but equivalent formulas.

------,t

'*'2(b) - - - -

I

I I I

,- - ---+---t

I

I

I

I

I

I

I I I

I

'*'2(a) - - -

I -1- - I

Figure 6

I

265

Proof (a) According to the notations which are before theorems 4.18 and 4.19, we observe that

= AI(U) - AI(a) + Al (a) = ('Y1(u) -II(a))-y2(b)1{u::;a} + AI(a) A2(U) =A2(U) A2(b) + A2(b) = II(a)(12(u)-12(b))l{b::;u} +A2(b), AI(U)

if

U

E JR. By choosing suitable constants, we obtain

a I I( a ){2(b)fb aI(u) =

a2(U)=

{

{

d(

II(a){2(b) (fbU..1..d J1"I'

:U

(..1..) + _I_(b)) "12

: U E]a,b[

"1'"12

.. U > b -

.=.1!.W "I,(b) a II(a){2(b) fb u 11(a){2(b)fb

a

d

a

: U

:uE]a,b[ : U

?: b

Formula (5.3) can be obtained with the help of ordinary manipulations with Stieltjes integrals and orthogonal measures (for instance integrations by parts). In order to prove formula (5.2), we first obtain G 1(u ) = 12(b)1{u::;a}

+ 11(a){2(b) d

l{a