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Discrete stochastic maximal regularity

Foivos Evangelopoulos-Ntemiris, Mark Veraar

TL;DR

<3-5 sentence high-level summary> This work develops a comprehensive framework for discrete stochastic maximal regularity (DSMR) in time-discretized parabolic SPDEs and establishes a precise equivalence with the continuous stochastic maximal L^p-regularity (SMR) for a broad class of schemes, including exponential Euler and stable rational methods. The authors build a robust toolkit based on sectorial operators, $H^∞$-calculus, and $R$-boundedness to prove DSMR, permanence, and weighted extrapolation properties, and they derive a sharp maximal estimate in the trace space $D_A(1- rac{1}{p},p)$ that incorporates parabolic smoothing. A key contribution is showing that DSMR for a scheme is necessary and sufficient for SMR of the generator, enabling transfer of continuous-time results to the discrete setting and yielding wide-ranging DSMR consequences across Hilbert and L^q spaces. The results pave the way for improved convergence and stability analyses of time-discrete schemes for SPDEs and provide deep insights into discrete stochastic smoothing phenomena.

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 $p \in [2,\infty)$.

Discrete stochastic maximal regularity

TL;DR

<3-5 sentence high-level summary> This work develops a comprehensive framework for discrete stochastic maximal regularity (DSMR) in time-discretized parabolic SPDEs and establishes a precise equivalence with the continuous stochastic maximal L^p-regularity (SMR) for a broad class of schemes, including exponential Euler and stable rational methods. The authors build a robust toolkit based on sectorial operators, -calculus, and -boundedness to prove DSMR, permanence, and weighted extrapolation properties, and they derive a sharp maximal estimate in the trace space that incorporates parabolic smoothing. A key contribution is showing that DSMR for a scheme is necessary and sufficient for SMR of the generator, enabling transfer of continuous-time results to the discrete setting and yielding wide-ranging DSMR consequences across Hilbert and L^q spaces. The results pave the way for improved convergence and stability analyses of time-discrete schemes for SPDEs and provide deep insights into discrete stochastic smoothing phenomena.

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 -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 and with respect to a power weight. Furthermore, employing the -functional calculus, we derive a powerful discrete maximal estimate in the trace space norm for .

Paper Structure

This paper contains 32 sections, 32 theorems, 185 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Formulation of our main results
  3. Preliminaries
  4. Sectorial operators, functional calculus, and interpolation
  5. Functional calculus
  6. Standard estimates for semigroups
  7. Fractional powers
  8. Approximation schemes
  9. Stochastic integration
  10. Discrete stochastic maximal regularity
  11. Definition and basic properties
  12. Equivalence of continuous and discrete ${\rm SMR}$
  13. ${\rm SMR}$ implies discrete ${\rm SMR}$ for exponential Euler
  14. Equivalence of discrete ${\rm SMR}$ for different schemes
  15. Discrete SMR implies SMR
  16. ...and 17 more sections

Key Result

Theorem 1.2

Let $q\in [2, \infty)$ and suppose that $X_0$ is isomorphic to a closed subspace of $L^q(\mathcal{O})$ with $(\mathcal{O}, \Sigma,\mu)$ a $\sigma$-finite measure space. Suppose that Assumption assum:mainDSMR holds. Then for any $p\in [2, \infty)$ the following are equivalent:

Figures (1)

  • Figure 1: Flowchart of the proof of Theorem \ref{['thm:mainequiv']}.

Theorems & Definitions (65)

  • Theorem 1.2: Characterization of Discrete Stochastic Maximal $\ell^p$-regularity
  • Corollary 1.3: Discrete Stochastic Maximal $\ell^p$-regularity for free
  • Theorem 1.4: Maximal estimate with parabolic regularization
  • Proposition 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5: One-sided Burkholder-Davis--Gundy inequality
  • ...and 55 more