# Discrete stochastic maximal regularity

## TL;DR

This paper develops a unified framework for analyzing discrete regularity in numerical solutions of stochastic evolution equations.

## Contribution

It introduces a characterization of discrete stochastic maximal regularity using continuous-time theory and derives new estimates.

## Key findings

- A unified framework for discrete stochastic maximal ℓp-regularity is established.
- Extrapolation properties in the exponent p and with respect to a power weight are proven.
- A discrete maximal estimate in the trace space norm DA(1−1/p,p) is derived for p ∈ [2, ∞).

## 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 \documentclass[12pt]{minimal}
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\begin{document}$$\ell ^p$$\end{document}ℓp-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 \documentclass[12pt]{minimal}
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\begin{document}$$H^\infty $$\end{document}H∞-functional calculus, we derive a powerful discrete maximal estimate in the trace space norm \documentclass[12pt]{minimal}
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\begin{document}$$D_A(1-\frac{1}{p},p)$$\end{document}DA(1-1p,p) for \documentclass[12pt]{minimal}
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\begin{document}$$p \in [2,\infty )$$\end{document}p∈[2,∞).