This paper studies the convergence of a spatial semidiscretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. For non-smooth initial values, the regularity of the mild solution is investigated, and an error estimate is derived with the spatial $ L^2 $-norm. For smooth initial values, two error estimates with the general spatial $ L^q $-norms are established.

翻译：本文研究带有乘性噪声的三维随机Allen-Cahn方程空间半离散格式的收敛性。针对非光滑初值，我们研究了其温和解的正则性，并在空间$L^2$范数意义下推导了误差估计。对于光滑初值，我们建立了两种基于一般空间$L^q$范数的误差估计。