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Stochastic analysis. (English) Zbl 0878.60001
Grundlehren der Mathematischen Wissenschaften. 313. Berlin: Springer. xi, 342 p. (1997).
The famous Malliavin calculus in the author’s interpretation is presented in the book. The book is divided into five parts. Part I, “Differential calculus on Gaussian probability spaces”, contains the description of Gaussian probability space, Ornstein-Uhlenbeck semigroup, abstract Wiener space, Gross-Stroock Sobolev spaces over a Gaussian probability space. A hierarchy of Sobolev spaces \(D^p_r\) is defined, and an approximation procedure for the space \(D^p_r\) by Sobolev spaces over finite-dimensional spaces is considered. It permits to extend the usual rules of differential calculus on \(\mathbb{R}^d\). The notions of smooth vector field and its divergence are introduced, where the divergence is a natural generalization of Cameron-Martin representation. The smoothness of the law of \(\mathbb{R}^d\)-valued random variables defined on a Gaussian probability space is investigated by the help of the procedure of “lifting up” and “pushing down” through some smooth non-degenerate map and integration by parts.
Part II, “Quasi-sure analysis”, is devoted to nonlinear potential theory and its application to Sobolev spaces in infinite dimensions. The construction of quasi-sure analysis on a numerical model is described. The two main ingredients are hierarchy of capacities and redefinition of a smooth function. The tightness of capacities on an increasing family of compact sets is established. Watanabe generalized functionals of finite energy and their \((p,r)\)-potential are introduced. The existence of equilibrium potential of any Borel set with positive \((p,r)\)-capacity is proved, and it follows that the class of Borel sets of \((p,r)\)-capacity zero coincides with the class of Borel sets which are not charged by any measure of finite energy. For a compact set \(K\) its \((p,r)\)-charge is defined and the equality of charge with capacity is proved. To show that quasi-sure analysis is intrinsic, the covariance of numerical quasi-sure analysis under a change of basis is established. A precise Gaussian probability space and its differential geometry complete the Part II.
Part III, “Stochastic integrals”, is devoted to probability space of white noise. The author comes from the general things to particular, i.e. he considers white noise stochastic integrals as divergences, constructs Itô-Wiener multiple integrals, presents Zakai-Nualart-Pardoux version of Skorokhod integral and then goes to Itô’s theory of stochastic integration, probability space of Brownian motion and its filtration, chaos expansion in terms of iterated Itô stochastic integrals and other more or less traditional subjects.
Part IV, “Stochastic differential equations”, demonstrates the following approach to the subject: the theory of stochastic differential equations (SDE) is a limiting case of the theory of ordinary differential equations (ODE). The approximating ODE are constructed from the control map. In its turn, control map is generated by the driving vector fields and its structure depends upon the Lie algebra generated by these fields. The main limit theorem states that the Itô map is the limit of control map, and the existence theorem for SDE is obtained as a by-product of limit theorem, in the same way as principle of transfer from ODE to SDE. This principle permits to differentiate the solution of SDE according to its initial value. Reduced variation, Bismut identity and Hörmander hypoellipticity under degenerate hypotheses are considered.
Part V, “Stochastic analysis in infinite dimensions”, deals, in particular, with Ornstein-Uhlenbeck flow, that is considered first on a finite-dimensional Gaussian space and is translated then to an abstract Wiener space. Path spaces and their tangent spaces are studied in the last chapter.
In general, the book contains multitude of concepts, notions, ideas and deep relations between them. It is rather interesting for reading, especially due to nonstandard approach to well-known subjects and clear description of unknown ones.

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60H07 Stochastic calculus of variations and the Malliavin calculus
60G15 Gaussian processes
60H05 Stochastic integrals
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J45 Probabilistic potential theory