Pathwise uniform convergence of a full discretization for a three-dimensional stochastic Allen-Cahn equation with multiplicative noise

This paper analyzes a full discretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. The discretization uses the Euler scheme for temporal discretization and the finite element method for spatial discretization. By deriving a stability estimate of a discrete stochastic convolution and utilizing this stability estimate along with the discrete stochastic maximal Lᵖ-regularity estimate, a pathwise uniform convergence rate with the general spatial Lq-norms is derived.