Numerical Ergodicity and Uniform Estimate of Monotone SPDEs Driven by Multiplicative Noise

TL;DR

This paper establishes the exponential ergodicity and convergence properties of numerical schemes for monotone SPDEs driven by multiplicative noise, with applications to the stochastic Allen--Cahn equation, ensuring long-term stability and accuracy.

Contribution

It provides the first rigorous analysis of long-time behavior and invariant measures for numerical schemes solving monotone SPDEs with multiplicative noise, including sharp convergence rates.

Findings

Numerical schemes are exponentially ergodic with unique invariant measures.

Strong convergence of schemes to exact solutions with sharp, time-independent rates.

Numerical invariant measures also exhibit exponential ergodicity.

Abstract

We analyze the long-time behavior of numerical schemes for a class of monotone stochastic partial differential equations (SPDEs) driven by multiplicative noise. By deriving several time-independent a priori estimates for the numerical solutions, combined with the ergodic theory of Markov processes, we establish the exponential ergodicity of these schemes with a unique invariant measure, respectively. Applying these results to the stochastic Allen--Cahn equation indicates that these schemes always have at least one invariant measure, respectively, and converge strongly to the exact solution with sharp time-independent rates. We also show that these numerical invariant measures are exponentially ergodic and thus give an affirmative answer to a question proposed in (J. Cui, J. Hong, and L. Sun, Stochastic Process. Appl. (2021): 55--93), provided that the interface thickness is not too small.