Discrete stochastic maximal $ L^p $-regularity and convergence of a spatial semidiscretization for a stochastic parabolic equation

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.