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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 (Δ t γ ) for any γ < ½. We also prove that the scheme converges uniformly in the strong L p -sense but with no rate given.
Kovács et al. (Mon,) studied this question.