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

TL;DR

This paper investigates the long-term behavior of a stochastic Cahn-Hilliard equation with singular logarithmic nonlinearity and degenerate noise, establishing existence, uniqueness, and ergodic properties of its invariant measure.

Contribution

It introduces a novel analysis of the asymptotic properties of a stochastic Cahn-Hilliard equation with singular nonlinearities and degenerate noise, including existence and uniqueness of invariant measures.

Findings

Existence of solutions via polynomial approximation

Existence and uniqueness of invariant measure

Asymptotic strong Feller property and irreducibility

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 approximated sequence of solutions 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.