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

## TL;DR

This paper characterizes the domain of elliptic operators associated with quadratic forms in subsets of Wiener spaces, focusing on the Ornstein-Uhlenbeck operator on half-spaces with explicit boundary conditions.

## Contribution

It provides a complete characterization of the operator's domain in Wiener spaces, especially for half-spaces, with explicit boundary conditions involving the Hausdorff-Gauss measure.

## Key findings

- Explicit domain description for Ornstein-Uhlenbeck operator on half-spaces.
- Boundary conditions involve the Hausdorff-Gauss surface measure.
- Results extend understanding of elliptic operators in infinite-dimensional spaces.

## Abstract

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron-Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\rightarrow\mathbb{R}$ and $G:X\rightarrow \mathbb{R}$. We let $\nu=e^{-U}\mu$ and $\Omega=G^{-1}(-\infty,0]$. In this paper we are interested in the domain of the the self-adjoint operator associated with the quadratic form \begin{gather} (\psi,\varphi)\mapsto \int_\Omega\langle\nabla_H\psi,\nabla_H\varphi\rangle_Hd\nu\qquad\psi,\varphi\in W^{1,2}(\Omega,\nu).\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 \[\{u\in W^{2,2}(\Omega,\mu)\,|\, \langle\nabla_H u(x),\nabla_H G(x)\rangle_H=0\text{ for }\rho\text{-a.e. }x\in G^{-1}(0)\},\] where $\rho$ is the Feyel-de La Pradelle Hausdorff-Gauss surface measure.