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

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.