An Introduction to Analysis on Wiener Space

102 Pages·1995·1.19 MB·English

by  Ali Süleyman Üstünel (auth.)

Lecture Notes in Mathematics 0161
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Ali Stileyman Usttinel
An Introduction to
Analysis on Wiener Space
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Author
Ali Stileyman lJsttinel
ENST, Drpt. Rrseaux
46, rue Barrault
F-75013 Paris, France
e-mail: [email protected]
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Siileyman:
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An introduction to analysis on Wiener space / All SOleyman
Ustt~nel, - Berlin ; Heidelberg ; New York ; Barcelona ;
Budapest ; Hong Kong ; London ; Milan ; Paris ; Tokyo :
Springer, 1995
(Lecture notes in mathematics ; Vol 1610)
ISBN 3-540-60170-8
NE: GT
Mathematics Subject Classification (1991): 60H07, 60H05, 60H15, 46F25, 81T,
81Q
ISBN 3-540-60170-8 Springer-Verlag Berlin Heidelberg New York
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Introduction
The following pages are the notes from a seminar that I gave during the
spring and some portion of the summer of 1993 at the Mathematics Institute
of Oslo University. The aim of the seminars was to give a rapid but rigorous
introduction for the graduate students to Analysis on Wiener space, a subject
which has grown up very quickly these recent years under the impulse of the
Stochastic Calculus of Variations of Paul Malliavin (cf. 12).
Although some concepts are in the preliminaries, I assumed that the students
had already acquired the notions of stochastic calculus with semimartingales,
Brownian motion and some rudiments of the theory of Markov processes. A
small portion of the material exposed is our own research, in particular, with
Moshe Zakai. The rest has been taken from the works listed in the bibliography.
The first chapter deals with the definition of the (so-called) Gross-Sobolev
derivative and the Ornstein-Uhlenbeck operator which are indispensable tools of
the analysis on Wiener space. In the second chapter we begin the proof of the
Meyer inequalities, for which the hypercontractivity property of the Ornstein-
Uhlenbeck semigroup is needed. We expose this last topic in the third chapter,
then come back to Meyer inequalities, and complete their proof in chapter IV.
Different applications are given in next two chapters. In the seventh chapter we
study the independence of some Wiener functionals with the previously devel-
oped tools. The chapter VIII is devoted to some series of moment inequalities
which are important for applications like large deviations, stochastic differential
equations, etc. In the last chapter we expose the contractive version of Ramer's
theorem as another example of the applications of moment inequalities developed
in the preceding chapter.
During my visit to Oslo, I had the chance of having an ideal environment for
working and a very attentive audience in the seminars. These notes have partic-
ularly profited from the serious criticism of my colleagues and friends Bernt
Oksendal, Tom LindstrOm, Ya-Zhong Hu, and the graduate students of the
Mathematics department. It remains for me to express my gratitude also to
Nina Haraldsson for her careful typing, and, last but not least, to Laurent De-
creusefond for correcting so many errors .
Ali Sfileyman Ostiinel
Contents
Preliminaries 1
1 The Brownian Motion and the Wiener Measure .......... 1
Gross-Sobolev Derivative, Divergence and Ornstein-Uhlenbeck
Operator 9
1 The Construction of V and its properties ............. 10
1.1 Relations with the stochastic integration .......... 12
2 The Ornstein-Uhlenbeck Operator ................. 16
II Meyer Inequalities 19
1 Some Preparations .......................... 19
2 ~(I + f~)-1/2 as the Riesz Transform ............... 21
III Hypercontractivity 27
IV LP-Multipliers Theorem, Meyer Inequalities and Distributions 31
1 LP-Multipliers Theorem ....................... 31
V Some applications of the distributions 41
1 Extension of the Ito-Clark formula ................. 42
2 Lifting of,_q'(~d) with random variables .............. 45
VI Positive distributions and applications 53
1 Capacities and positive Wiener functionals ............. 55
VII Characterization of independence of some Wiener functionals 61
1 Independence of Wiener functionals ................ 61
VIII Moment inequalities for Wiener functionals 69
1 Exponential tightness ........................ 60
2 Coupling inequalities ......................... 73
3 An interpolation inequality ..................... 77
IX Introduction to the theorem of Ramer 81
Bibliography 91
Subject Index 94
Notations 95
Preliminaries
This chapter is devoted to the basic results about the Wiener measure, Brownian
motion, construction of the Ito stochastic integral and the chaos decomposition
associated to it.
1 The Brownian Motion and the Wiener Mea-
sure
1) Let W = C0([O, 1]), w E W, t E [0, 1], define Wt(w) = w(t) (the coordinate
functional). If we note by Bt = a{Ws;s < t}, then there is one and only one
measure tt on W such that
i) tt{W0(w) = }O = 1,
ii) Vf E C~(R), the stochastic process process
'0/1
S"(W,(~))ds
(t,~) ~ S(w,(~)) - ~
is a (Bt, ~)-martingale. ~/ is called the Wiener measure.
2) From the construction we see that for t > s,
E(cid:127)[exp ia(Wt - W,)IB, ] = exp-c~2(t - s),
hence (t, w) ~ Wt(w) is a continuous additive process (i.e.,a process with inde-
pendent increments) and (Wt;t E [0, 1]) is also a continous martingale.
3) Stochastic Integration
Let K : W x [0, ]1 --* R be a step process :
n
Kt(w) = Eai(w). l[t,.t,+,t(t), ai(w) E L2(Bt,) (cid:12)9
i=1
Define
Of
I(K) = K~dW,(w)
2 Preliminaries
as
n -
,+,,W(
1----i
Then we have
i.e. I is an isometry from the adapted step processes into L~(p), hence it has a
unique extension as an isometry from
L2(0,1 x W,A, dt x dp) ;~ L~(p)
where A. denotes the sigma algebra on 0, 1 x W generated by the adapted, left
(or right) continuous processes. I(K) is called the stochastic integral of K and
it is denoted as 1. KjdWs. If we define
It(K) = ~0 t K, dWs
as
1
fo lo,tl(s)K, dWs,
it is easy to see that the stochastic process t ~ It(K) is a continuous, square
integrable martingale. With some localization techniques using stopping times,
I can be extended to any adapted process K such that f: K~(w)ds < c~ a.s.
In this case the process t ~ It(K) becomes a local martingale, i.e., there exists
a sequence of stopping times increasing to one, say (Tn, n E N) such that the
process t ~ hAT. (K) is a (square integrable) martingale.
Application: Ito formula We have following important applications of the
stochastic integration:
a) If f E C2(R) and Mt = f~ grdWr, then
f 1/'
f(Mt) = f(O) + f'(Ms)KsdWs + ~- f"(M,)K~ds.
)b
o~ t 1 0~ t
gt(I(h)) = exp( h, dW, - ~- h~ds)
is a martingale for any h E L 2 0, .1