Irregular semi-convex gradient systems perturbed by noise and application to the stochastic Cahn–Hilliard equation

Abstract

We prove essential self-adjointness of Kolmogorov operators corresponding to gradient systems with potentials U such that DU is not square integrable with respect the invariant measure (irregular potentials). An application is given to the Cahn–Hilliard–Cook equation in dimension one. In this case the spectral gap is proved for the correspondig semigroup. We also obtain a log-Sobolev inequality.