On the backward Euler approximation of the stochastic Allen-Cahn equation

Abstract

We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretization in time of the equation by an implicit Euler method. We show that the method converges pathwise with a rate O(Δ t ^γ) for any γ < ½. We also prove that the scheme converges uniformly in the strong L ^p -sense but with no rate given.