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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>ℓ</mml:mi> <mml:mi>p</mml:mi></mml:msup> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>H</mml:mi> <mml:mi>∞</mml:mi></mml:msup> </mml:math> -functional calculus, we derive a powerful discrete maximal estimate in the trace space norm <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>D</mml:mi> <mml:mi>A</mml:mi></mml:msub> <mml:mrow><mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mfrac><mml:mn>1</mml:mn> <mml:mi>p</mml:mi></mml:mfrac> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> .