This study demonstrates that the boundedness of the \( H^\infty \)-calculus for the negative discrete Laplace operator is independent of the spatial mesh size. Using this result, we deduce the discrete stochastic maximal \( L^p \)-regularity estimate for a spatial semidiscretization. Furthermore, we derive (nearly) sharp error estimates for the semidiscretization under the general spatial \( L^q \)-norms.

翻译：本研究证明了负离散拉普拉斯算子的\( H^\infty \)演算有界性与空间网格尺寸无关。利用该结果，我们推导出空间半离散格式的离散随机极大\( L^p \)正则性估计。此外，我们在一般空间\( L^q \)范数下得到了该半离散格式的（近乎）最优误差估计。