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

We consider an operator K˚ϕ = Lϕ−: <CDU(x), Dϕ> in a Hilbert space H, where L is an Ornstein–Uhlenbeck operator, U∈W ^1,4(H, μ) and μ is the invariant measure associated with L. We show that K˚ is essentially self-adjoint in the space L ²(H, ν) where ν is the “Gibbs” measure ν(dx) = Z ^−:1 e ^−:2U(x) dx. An application to Stochastic quantization is given. Received: 13 August 1998 / Revised version: 20 September 1999 / Published online: 8 August 2000