On the domain of elliptic operators defined in subsets of Wiener spaces

Abstract

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\hat{I}¼$. The associated Cameron-Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\to \mathbb{R}$ and $G:X\to \mathbb{R}$. We let $\hat{I}½=e^{-U}\hat{I}¼$ and $\hat{I}\text{c◯}=G^{-1}(-\infty ,0]$. In this paper we are interested in the domain of the the self-adjoint operator associated with the quadratic form \begin{gather} (ψ,φ)\mapsto \int_Ω\langle\nabla_Hψ,\nabla_Hφ\rangle_Hdν\qquadψ,φ\in W^{1,2}(Ω,ν).\qquad\qquad (\star) \end{gather} In particular we obtain a complete characterization of the Ornstein-Uhlenbeck operator on half-spaces, namely if $U\equiv 0$ and $G$ is an affine function, then the domain of the operator defined via $(\star )$ is the space \[\{u\in W^{2,2}(Ω,μ)\,|\, \langle\nabla_H u(x),\nabla_H G(x)\rangle_H=0\text{ for }ρ\text{-a.e. }x\in G^{-1}(0)\},\] where $\ddot{I}$ is the Feyel-de La Pradelle Hausdorff-Gauss surface measure.

arXiv admin note: text overlap with arXiv:1609.07337