On the Dirichlet semigroup for Ornstein -- Uhlenbeck operators in subsets of Hilbert spaces

TL;DR

This paper investigates the Dirichlet problem for Ornstein-Uhlenbeck operators in infinite-dimensional Hilbert spaces, focusing on solution regularity and boundary conditions, using variational methods and Malliavin calculus.

Contribution

It provides new regularity results for solutions and clarifies the meaning of boundary conditions in infinite-dimensional settings.

Findings

Established interior $W^{2,2}_{eta}$ regularity for solutions.

Defined the Dirichlet boundary condition using Malliavin surface integrals.

Extended the solution concept to irregular and regular boundaries.

Abstract

We consider a family of self-adjoint Ornstein--Uhlenbeck operators $L_{\alpha} $ in an infinite dimensional Hilbert space H having the same gaussian invariant measure $\mu$ for all $\alpha \in [0,1]$. We study the Dirichlet problem for the equation $\lambda \phi - L_{\alpha}\phi = f$ in a closed set K, with $f\in L^2(K, \mu)$. We first prove that the variational solution, trivially provided by the Lax---Milgram theorem, can be represented, as expected, by means of the transition semigroup stopped to K. Then we address two problems: 1) the regularity of the solution $\varphi$ (which is by definition in a Sobolev space $W^{1,2}_{\alpha}(K,\mu)$) of the Dirichlet problem; 2) the meaning of the Dirichlet boundary condition. Concerning regularity, we are able to prove interior $W^{2,2}_{\alpha}$ regularity results; concerning the boundary condition we consider both irregular and regular boundaries. In the first case we content to have a solution whose null extension outside K belongs to $W^{1,2}_{\alpha}(H,\mu)$. In the second case we exploit the Malliavin's theory of surface integrals which is recalled in the Appendix of the paper, then we are able to give a meaning to the trace of $\phi$ at the boundary of K and to show that it vanishes, as it is natural.