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Analytic properties of infinite-dimensional distributions. (English. Russian original) Zbl 0708.60049
Russ. Math. Surv. 45, No. 3, 1-104 (1990); translation from Usp. Mat. Nauk 45, No. 3(273), 3-83 (1990).
The paper is an expanded review on the modern theory of smooth measures in infinite-dimensional spaces. The theory of random processes, basically diffusion ones, gives here one of the main sources of examples. The analytical properties of measures, i.e. properties of differentiability, are studied. Subspaces of differentiability are investigated, that is subspaces “along” which the measure has derivatives. The smoothness of product-measures, stable measures, Gaussian measures and diffusion process distributions is studied.
The last part of the paper is devoted to the connection of smooth measures theory with Malliavin calculus. It is explained that Malliavin calculus may be naturally considered as a part of smooth measures theory. The basic notions of Malliavin calculus are expounded in terms of distributions. The smoothness of some functionals of diffusion processes, namely, quadratic functionals and integral type functionals, is investigated. The review contains a list of open problems and 435 references.
Reviewer: A.Yu.Veretennikov

MSC:
60H07 Stochastic calculus of variations and the Malliavin calculus
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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