Positive generalized Wiener functions and potential theory over abstract Wiener spaces (Q1814050)

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.