Self-adjointness of some infinite-dimensional elliptic operators and application to stochastic quantization

We consider an operator (K) over circle phi = L phi-[CDU(x), D phi] in a Hi lbert space H, where L is an Ornstein-Uhlenbeck operator, U epsilon W-1,W-4 (H, mu) and mu is the invariant measure associated with L. We show that (K) over circle is essentially self-adjoint in the space L-2(H,nu) where nu is the "Gibbs" measure nu(dx) = Z(-1)e(-2U(x))dx. An application to Stochasti c quantization is given.