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

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We consider a family of self-adjoint Ornstein--Uhlenbeck operators Lα in an infinite dimensional Hilbert space H having the same gaussian invariant measure μ for all α∈ [0,1]. We study the Dirichlet problem for the equation λϕ- Lαϕ= f in a closed set K, with f∈ L²(K, μ). 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 φ (which is by definition in a Sobolev space W¹,²α(K,μ)) of the Dirichlet problem; 2) the meaning of the Dirichlet boundary condition. Concerning regularity, we are able to prove interior W²,²α 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 W¹,²α(H,μ). 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 ϕ at the boundary of K and to show that it vanishes, as it is natural.