Ergodicity and invariant measure approximation of the stochastic Cahn-Hilliard equation via an explicit fully discrete scheme

TL;DR

This paper develops a fully discrete numerical scheme for the stochastic Cahn-Hilliard equation driven by white noise, proving ergodicity, invariant measure approximation, and long-time statistical properties with strong convergence guarantees.

Contribution

It introduces an explicit fully discrete scheme combining finite differences and a tamed exponential Euler method, extending ergodic theory and providing convergence and invariant measure approximation results.

Findings

Proved the existence of a unique invariant measure in a regular state space.

Established uniform-in-time moment bounds and strong convergence rates for the numerical scheme.

Demonstrated the scheme's geometric ergodicity and accurate invariant measure approximation.

Abstract

This paper investigates the stochastic Cahn-Hilliard equation (SCHE) driven by additive space-time white noise. We first refine the analytical ergodic theory by proving that the continuum equation admits a unique invariant measure in the more regular state space H_\alpha, extending the classical result of Da Prato and Debussche (1996) on the negative Sobolev space $\dot{H}^{-1}_\alpha$. To approximate long-time behaviour, we introduce an explicit fully discrete scheme that combines a finite-difference spatial discretization with a strongly tamed exponential Euler method in time. Uniform-in-time moment bounds in the $L^\infty$-norm are established for the numerical solution, and a uniform strong convergence estimate with an explicit rate is derived for the fully discrete approximation. Exploiting a mass-preserving minorization tailored to Neumann boundary conditions, we further show that the numerical scheme is geometrically ergodic and possesses a unique invariant measure, together with polynomial-order error bounds for approximating the exact invariant measure. Strong laws of large numbers are proved for both the continuous and discrete systems, ensuring almost-sure convergence of temporal averages to the corresponding ergodic limits. Numerical experiments corroborate the theoretical findings, including the long-time strong convergence and the accuracy of invariant measure approximation. Overall, the results provide a complete analytical and numerical framework for investigating the long-time statistical behaviour of the SCHE.