Gradient contractivity of a rescaled resolvent on domains in Wiener spaces

Abstract

Given an abstract Wiener space $(X,γ,H)$, we consider an open set $O\subseteq X$ which satisfies certain smoothness and mean-curvature conditions. We prove that the rescaled resolvent operator associated to the Ornstein-Uhlenbeck operator with homogeneous Dirichlet boundary conditions on $O$ is gradient contractive in $L^p(X,γ)$ for every $p\in(1,\infty)$. This is the Gaussian counterpart of an analogous result for the rescaled resolvent operator associated to the Laplace operator $Δ$ in $L^p$ with respect to the Lebesgue measure, $p\in[1,\infty)$, with homogeneous Dirichlet boundary conditions on a bounded convex open set $O\subseteq \mathbb R^n$.