On the discretisation in time of the stochastic Allen-Cahn equation

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