Sugita, Hiroshi Positive generalized Wiener functions and potential theory over abstract Wiener spaces. (English) Zbl 0737.46038 Osaka J. Math. 25, No. 3, 665-696 (1988). 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) Cited in 1 ReviewCited in 37 Documents 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.) Keywords:Malliavin’s calculus; Gaussian measure in a real separable Banach space; abstract Wiener spaces; Ornstein–Uhlenbeck projector; Ornstein-Uhlenbeck semigroup; capacities related to Sobolev spaces; positive generalized Wiener functionals are measures; infinite-dimensional analog of the Riesz potential; equilibrium measures; generalized Wiener functionals; square- exponential integrability; Wiener homogeneous chaos decomposition; Sobolev spaces PDF BibTeX XML Cite \textit{H. Sugita}, Osaka J. Math. 25, No. 3, 665--696 (1988; Zbl 0737.46038)