On the backward Euler approximation 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 semidiscretization in time of the equation by an implicit Euler method. We show that the method converges pathwise with a rate $O(\Delta t^{\gamma})$ for any $\gamma<\frac12$. We also prove that the scheme converges uniformly in the strong $L^p$-sense but with no rate given.