This study investigates the boundedness of the \( H^\infty \)-calculus for the negative discrete Laplace operator, subject to homogeneous Dirichlet boundary conditions. The negative discrete Laplace operator is implemented using the finite element method, and we establish that its \(H^\infty\)-calculus is uniformly bounded with respect to the spatial mesh size. Using this finding, we derive a discrete stochastic maximal \(L^p\)-regularity estimate for a spatial semidiscretization. Furthermore, we provide a nearly optimal pathwise uniform convergence estimate for this spatial semidiscretization under a wide range of spatial \(L^q\)-norms.

翻译：本研究探讨了在齐次Dirichlet边界条件下，负离散Laplace算子的$H^\infty$演算的有界性。负离散Laplace算子采用有限元方法实现，我们证明了其$H^\infty$演算关于空间网格尺寸是一致有界的。利用这一结果，我们导出了一个空间半离散格式的离散随机极大$L^p$-正则性估计。此外，我们在广泛的$L^q$空间范数下，为该空间半离散格式提供了近乎最优的路径一致收敛估计。