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 $L^p$-regularity estimate, a pathwise uniform convergence rate with the general spatial $ L^q $-norms is derived.

翻译：本文分析了带有乘法噪声的三维随机Allen-Cahn方程的全离散格式。该离散化采用Euler格式进行时间离散，有限元法进行空间离散。通过推导离散随机卷积的稳定性估计，并利用该稳定性估计与离散随机极大$L^p$-正则性估计，得到了关于一般空间$L^q$范数的轨道一致收敛速率。