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. A key contribution of this work is the introduction of a novel stability estimate for a discrete stochastic convolution, which plays a crucial role in establishing pathwise uniform convergence estimates for fully discrete approximations of nonlinear stochastic parabolic equations. By using this stability estimate in conjunction with the discrete stochastic maximal $L^p$-regularity estimate, the study derives a pathwise uniform convergence rate that concerns the general spatial $L^q$-norms.

翻译：本文分析了一类带乘性噪声的三维随机Allen-Cahn方程的全离散格式。该离散化方案采用时间方向的Euler格式与空间方向的有限元方法。本文的关键贡献在于引入了一个针对离散随机卷积的新型稳定性估计，该估计对建立非线性随机抛物型方程全离散逼近的逐路径一致收敛性估计具有关键作用。通过将该稳定性估计与离散随机极大$L^p$-正则性估计相结合，本研究推导出了涉及一般空间$L^q$范数的逐路径一致收敛速率。