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

Sugita, Hiroshi

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)

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