We prove that the discrete Laplace operator has a bounded $ H^\infty$-calculus,independent of the spatial mesh size. As an application, we obtain the discrete stochastic maximal $ L^p $-regularity estimate for a spatial semidiscretization of a stochastic parabolic equation. In addition, we derive some (nearly) sharp error estimates for this spatial semidiscretization.

翻译：我们证明离散拉普拉斯算子具有与空间网格尺寸无关的有界$H^\infty$演算。作为应用，我们获得了随机抛物方程空间半离散化的离散随机极大$L^p$正则性估计。此外，我们推导了该空间半离散化的一些（近乎）精确误差估计。