On the discretisation in time 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 \(d\le 3\), and study the semidiscretisation in time of the equation by an Euler type split-step method. We show that the method converges strongly with a rate \(O(\Delta t^{\frac12}) \). By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.