an:01380631
Zbl 0974.46044
??st??nel, Ali S??leyman; Zakai, Moshe
Transformation of measure on Wiener space
EN
Springer Monographs in Mathematics. Berlin: Springer. xiii, 297 p. (2000).
00381456
2000
b
46G12 46-02 28-02 60-02 60H05 60G15 60G30 28C20 60H25 60G35 46T12 60H07
transformation of Wiener measure; abstract Wiener space; stochastic control theory; Malliavin calculus; canonical Gaussian cylinder set measure; Radon-Nikod??m derivative; Ramer's formula; Girsanov theorem; infinite-dimensional flows; Rademacher class of Wiener functionals; shifts; rotations; quasi-sure analysis; Sard inequality; Radon-Nikod??m derivatives; Ornstein-Uhlenbeck semigroup
The book is devoted to transformations of Wiener measures on the abstract Wiener space relative to different types of mappings including shifts, rotations and flows. For this, mappings of Sobolev type \(\mathbb{D}_{p,k}\) in Malliavin's sense of quasi-sure analysis, satisfying definite conditions, are used. There are given many results such as the classical ones of Cameron and Martin and also recent ones including investigations of the authors. The crucial role plays the Girsanov theorem and Theorem 2.7.1 that the class of measures representable by shifts (\(T= I_w+ u\), where \(u\) is an \(H\)-valued random variable for an abstract Wiener space \((W,H,\mu)\)) is dense under the total variation norm in the class of probability measures on \(W\) which are equivalent to \(\mu\). Applications to the Sard inequality and infinite-dimensional flows are given.
For example, in accordance with Theorem 5.3.1 for definite flows \(\varphi_{st}\) of the form
\[
\varphi_{st}(\omega)= \omega+ \int^t_s X_r(\varphi_{sr}(\omega)) dr,
\]
where \(X\) is a measurable map from \(\mathbb{R}_+\times W\) in \(H\), \(0\leq s\leq t\leq T\), \(r\in \mathbb{R}_+\), the Radon-Nikod??m density is given by the following equation
\[
{d\varphi^*_{st}\mu\over d\mu}= \exp\Biggl(\int^t_s (\delta X_r)(\psi_{rt}) dr\Biggr),
\]
where \(\psi_{st}\) is an inverse flow, \(\delta X_r\) is the first-order Wiener integral.
Generalized Radon-Nikod??m derivatives of measures are studied with the help of the Rademacher class of Wiener functionals and the Ornstein-Uhlenbeck semigroup.
One of the chapters contains results of the authors about a class of mappings on \(W\) preserving the Wiener measure. Finally, a measure theortic degree on the Wiener space is presented. Its relations with the Leray-Schauder degree and applications to absolute continuity are outlined.
Sergey L??dkovsky (Moskva)