Maximal $L^p$-$L^q$ regularity of finite element methods for stochastic parabolic equations in polygonal domains

Abstract

We prove that a spatial semidiscrete finite element method preserves the stochastic maximal regularity of the heat equation on bounded convex polygonal and polyhedral domains in dimensions two and three. A main difficulty is to establish suitable Green-function estimates for stochastic convolutions, whose paths generally lack classical time derivatives. Besides, the failure of the Burkholder-Davis-Gundy (BDG) inequality at the endpoint $p=\infty$ creates additional difficulties. Our approach leverages a hidden symmetry structure of the discrete Green's function, which enables the use of a key integration-by-parts type formula. Then the discrete stochastic maximal $L^p$-$L^p$ regularity with $p\in [2,\infty)$ is obtained from the discrete local energy estimates and interpolation inequalities. To extend further the discrete stochastic maximal $L^p$-$L^q$ regularity with the full range of integrability exponents, i.e., $p\in (2,\infty), q\in [2,\infty)$ or $p=q=2$, we also show that the discrete convolution kernel satisfies the stochastic Hörmander's condition, allowing the application of a stochastic version of the Calderón-Zygmund theory.