Numerical Ergodicity of Stochastic Allen--Cahn Equation driven by Multiplicative White Noise

TL;DR

This paper proves the unique ergodicity of a fully discrete numerical scheme for the stochastic Allen--Cahn equation with multiplicative white noise, ensuring long-term statistical stability and validating results with numerical experiments.

Contribution

It introduces a novel approach combining Lyapunov conditions and regularity analysis to establish ergodicity for fully discrete schemes of monotone SPDEs.

Findings

The scheme is uniquely ergodic for any interface thickness.

Uniform moments' estimates are achieved for the discretized equation.

Numerical experiments confirm the theoretical ergodicity results.

Abstract

We establish the unique ergodicity of a fully discrete scheme for monotone SPDEs with polynomial growth drift and bounded diffusion coefficients driven by multiplicative white noise. The main ingredient of our method depends on the satisfaction of a Lyapunov condition followed by a uniform moments' estimate, combined with the regularity property for the full discretization. We transform the original stochastic equation into an equivalent random equation where the discrete stochastic convolutions are uniformly controlled to derive the desired uniform moments' estimate. Applying the main result to the stochastic Allen--Cahn equation driven by multiplicative white noise indicates that this full discretization is uniquely ergodic for any interface thickness. Numerical experiments validate our theoretical results.