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

We consider a family of self-adjoint Ornstein–Uhlenbeck operators LαLα in an infinite dimensional Hilbert space H having the same gaussian invariant measure μ for all α∈[0,1]α∈[0,1]. We study the Dirichlet problem for the equation λφ−Lαφ=fλφ−Lαφ=f in a closed set K, with f∈L2(K,μ)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 View the MathML sourceWα1,2(K,μ)) of the Dirichlet problem; 2) the meaning of the Dirichlet boundary condition. Concerning regularity, we are able to prove interior View the MathML sourceWα2,2 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 View the MathML sourceWα1,2(H,μ). In the second case we exploit the Malliavinʹs theory of surface integrals which is recalled in Appendix A of the paper, then we are able to give a meaning to the trace of φ at ∂K and to show that it vanishes, as it is natural.