Discrete stochastic maximal regularity

TL;DR

This paper develops a unified framework for discrete stochastic maximal regularity in numerical schemes for parabolic stochastic equations, extending continuous-time results and deriving new estimates using advanced functional calculus.

Contribution

It characterizes discrete stochastic maximal regularity via continuous theory, introduces new regularity results, and applies $H^$-calculus for powerful trace space estimates.

Findings

Established discrete stochastic maximal regularity for various schemes.

Proved extrapolation properties in exponent p and weights.

Derived maximal estimates in trace space norms using $H^$-calculus.

Abstract

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 $D_A(1-\frac1p,p)$ for $p \in [2,\infty)$.