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

翻译：本文研究带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，我们研究了温和解的正则性，并在空间(L^2)范数框架下推导了误差估计。对于光滑初值情形，我们在一般空间(L^q)范数框架下建立了两个误差估计。