Pathwise existence of solutions to the Implicit Euler method for the stochastic Cahn-Hilliard Equation

TL;DR

This paper proves the pathwise existence and uniqueness of solutions for the implicit Euler method applied to the stochastic Cahn-Hilliard equation with additive noise, relaxing previous step size restrictions.

Contribution

It establishes pathwise existence and uniqueness of the implicit Euler scheme for the stochastic Cahn-Hilliard equation, independent of initial data and Wiener process increments.

Findings

Existence and uniqueness of solutions under a step size restriction

Relaxation of previous step size assumptions for deterministic case

Applicability in spatial domains with smooth boundaries in dimension up to 3

Abstract

We consider the implicit Euler approximation of the stochastic Cahn-Hilliard equation driven by additive Gaussian noise in a spatial domain with smooth boundary in dimension $d\le 3$. We show pathwise existence and uniqueness of solutions for the method under a restriction on the step size that is independent of the size of the initial value and of the increments of the Wiener process. This result also relaxes the imposed assumption on the time step for the deterministic Cahn-Hilliard equation assumed in earlier existence proofs.