Discrete stochastic maximal regularity

In this paper, we investigate discrete regularity estimates for a broad class of temporal numerical schemes for parabolic stochastic evolution equations. We provide a characterization of discrete stochastic maximal ${\ell }^p$ -regularity in terms of its continuous counterpart, thereby establishing a unified framework that yields numerous new discrete regularity results. Moreover, as a consequence of the continuous-time theory, we establish several important properties of discrete stochastic maximal regularity such as extrapolation in the exponent p and with respect to a power weight. Furthermore, employing the $H^{\infty }$ -functional calculus, we derive a powerful discrete maximal estimate in the trace space norm $D_A(1-\frac{1}{p},p)$ for $p\in [2,\infty )$ .