Dirichlet Operators via Stochastic Analysis

We give sufficient conditions for essential self-adjointness of symmetric differential operators (Dirichlet operators) associated with classical Dirichlet forms given by probability measures on Hilbert spaces. Under an additional assumption of log-concavity for the measures we prove the existence of a gap at the lower end of spectrum of the Dirichlet operator and the ergodicity of the corresponding semi-group. We consider also applications to the study of stochastic dynamics in some classical lattice systems.