Numerical Ergodicity of Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise (Communications in Mathematical Research, 2025)
We establish the unique ergodicity of a fully discrete scheme formonotone SPDEs with polynomial growth drift and bounded diffusion coefficients driven by multiplicative white noise. The main ingredient of our methoddepends on the satisfaction of a Lyapunov condition followed by a uniformmoments’ estimate, combined with the regularity property for the full discretization. We transform the original stochastic equation into an equivalentrandom 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 noiseindicates that this full discretization is uniquely ergodic for any interface thickness. Numerical experiments validate our theoretical results.
Keywords: Numerical invariant measure; numerical ergodicity; stochastic Allen-Cahn equation.