Asymptotic properties of stochastic Cahn–Hilliard equation with singular nonlinearity and degenerate noise

Abstract

We consider a stochastic partial differential equation with a logarithmic nonlinearity with singularities at 1 and −1 and a constraint of conservation of the space average. The equation, driven by a trace-class space–time noise, contains a bi-Laplacian in the drift. We obtain existence of solution for equation with polynomial approximation of the nonlinearity. Tightness of this sequence of approximations is proved, leading to a limit transition semi-group. We study the asymptotic properties of this semi-group, showing the existence and uniqueness of invariant measure, asymptotic strong Feller property and topological irreducibility.