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 $O(\hat{I}”t^{\frac{1}{2}})$. By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.

32 pages. Several typos fixed, the statement and proof of Lemma 3.2 are updated and so is the proof of Lemma 5.2. The proof of Lemma 5.1 is omitted