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

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.