<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\ell ^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ℓ</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-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 <i>p</i> and with respect to a power weight. Furthermore, employing the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$H^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-functional calculus, we derive a powerful discrete maximal estimate in the trace space norm <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$D_A(1-\frac{1}{p},p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mi>A</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p \in [2,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>2</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Discrete stochastic maximal regularity

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2026年2月13日

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Foivos Evangelopoulos-Ntemiris,
Mark Veraar

摘要

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 $\ell ^p$ $ℓ^{p}$ -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 $ $H^{\infty}$ -functional calculus, we derive a powerful discrete maximal estimate in the trace space norm $D_A(1-\frac{1}{p},p)$ $D_{A} (1 - \frac{1}{p}, p)$ for $p \in [2,\infty )$ $p \in [2, \infty)$ .