Capacities, removable sets and $L^p$-uniqueness on Wiener spaces

TL;DR

This paper establishes the equivalence of two capacity notions in Wiener spaces, providing criteria for $L^p$-uniqueness of the Ornstein-Uhlenbeck operator based on the size of removed sets.

Contribution

It introduces a new criterion for $L^p$-uniqueness linked to capacities and Gaussian measures, connecting nonlinear truncation operators to potential theory in Wiener spaces.

Findings

Equivalence of two capacity types in Wiener spaces.

Criterion for $L^p$-uniqueness based on set codimension.

Connection between capacities and Gaussian Hausdorff measures.

Abstract

We prove the equivalence of two different types of capacities in abstract Wiener spaces. This yields a criterion for the $L^p$-uniqueness of the Ornstein-Uhlenbeck operator and its integer powers defined on suitable algebras of functions vanishing in a neighborhood of a given closed set $\Sigma$ of zero Gaussian measure. To prove the equivalence we show the $W^{r,p}(B,\mu)$-boundedness of certain smooth nonlinear truncation operators acting on potentials of nonnegative functions. We also give connections to Gaussian Hausdorff measures. Roughly speaking, if $L^p$-uniqueness holds then the 'removed' set $\Sigma$ must have sufficiently large codimension, in the case of the Ornstein-Uhlenbeck operator for instance at least $2p$.