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Positive generalized Wiener functions and potential theory over abstract Wiener spaces. (English) Zbl 0737.46038

In the spirit of the Malliavin’s calculus, the author extends many results in the finite-dimensional potential theory, by replacing the Lebesgue measure by a Gaussian measure in a real separable Banach space. Main topics are specified below.
Section 1: An introduction to the Malliavin’s calculus and abstract Wiener spaces; the Wiener homogeneous chaos decomposition and related Sobolev spaces; Ornstein-Uhlenbeck projector \(J_ n\) onto the chaos of degree \(n\).
Section 2: The Ornstein-Uhlenbeck semigroup \(T_ t=\sum_ n e^{- nt}J_ n\).
Section 3: Properties of capacities related to Sobolev spaces;
Theorem: In terms of capacities, any Borel set can be approximated from below by compacta.
Section 4: Relations between positive functionals and capacities;
Theorem: positive generalized Wiener functionals are measures (cf.: positive Schwartz distributions are measures).
Section 5: An infinite-dimensional analog of the Riesz potential; equilibrium measures.
Section 6: Various properties of positive generalized Wiener functionals: continuity of certain embeddings, the square-exponential integrability.
Reviewer: J.Szulga (Auburn)

MSC:

46G12 Measures and integration on abstract linear spaces
60H07 Stochastic calculus of variations and the Malliavin calculus
46F25 Distributions on infinite-dimensional spaces
60J45 Probabilistic potential theory
47N30 Applications of operator theory in probability theory and statistics
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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