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(∆tγ) for any γ < 1 2. We also prove that the scheme converges uniformly in the strong Lp-sense but with no rate given. 1