An introduction to analysis on Wiener space
The monograph gives a short and self-contained exposition of analysis of Wiener functionals with various applications. The main milestones are the Malliavin calculus and the theorem of Ramer about the equivalence of measures for Wiener process \(w\) and \(w+ u(w)\) with non-adapted \(\dot u\). There are Preliminaries and nine chapters. In Preliminaries the notions of Wiener process and abstract Wiener space are introduced, Cameron- Martin and P. Lévy's theorems and Wiener chaos decomposition are formulated. In the first chapter Gross-Sobolev derivative (stochastic gradient \(\nabla\)) on classical Wiener space (close notions are Malliavin derivative, Zakai derivative) and divergence operator \(\delta= \nabla^*\) w.r.t. the Wiener measure \(\mu\) are introduced. Connections with the stochastic integral are given: \(\delta h= \int \dot h dw\). The Ornstein-Uhlenbeck operator \(\mathcal L\) is introduced, \(-{\mathcal L}\) being the infinitesimal operator of Ornstein-Uhlenbeck process. The identity \(\delta\circ\nabla= L\) is proved. Meyer's inequalities i.e. the equivalence of the norms \(|\nabla (I+ {\mathcal L}^{- 1/2}\varphi|_p\) and \(|\varphi|_p\) are proved in Chapters 2-4 with the help of the hypercontractivity property of Ornstein-Uhlenbeck process. In Chapters 5-6 various applications are given, namely, to Itô-Clark representation, to non-degenerate random variables in the Malliavin calculus sense, to diffusion process filtering, to capacities and Itô formula for generalized functions, to positive functionals. The independence of multiple Wiener integrals is discussed in Chapter 7. A useful exponential inequality is proved in Chapter 8 for a functional with a bounded derivative which generalizes Fernique's inequality. Poincaré type inequalities are also derived as a consequence of so- called coupling inequalities. One of them is as follows: \[ E\exp(\varphi- E\varphi)\leq E\exp ((\pi^2/8)|\nabla\varphi|^2_H). \] One more interpolation type inequality is proved for \(L_p\)-norms of \(\nabla F\), \(F\) and \(\nabla^2 F\) using Meyer's inequalities. The last Chapter 9 is deovted to the Ramer's theorem which says that the process \(w+ u(w)\) is a Brownian motion under a certain new probability measure given Brownian motion \(w\) and anticipated functional \(u\). The main assumption is that the norm \(|\nabla u|\) is small enough a.e. The analogue of Girsanov exponent has the representation \[ \Lambda_u= |\text{det}_2(I+ \nabla u)|\exp(- \delta u- |u|^2_H/2), \] \(\text{det}_2\) being the Carleman-Fredholm determinant. The exposed theorem is a special case of a result of the author and M. Zakai.
scientific article; zbMATH DE number 803282