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<p>This study investigates the boundedness of the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\( H^\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-calculus for the negative discrete Laplace operator under homogeneous Dirichlet boundary conditions. The discrete operator is implemented using the finite element method, and we establish that the boundedness constant of its <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\( H^\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-calculus remains uniformly bounded with respect to the spatial mesh size. Based on this result, we derive a discrete stochastic maximal <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\( L^p \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-regularity estimate for a spatial semidiscretization of a linear stochastic heat equation. Furthermore, within the framework of general spatial <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\( L^q \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation>-norms, we provide a nearly optimal pathwise uniform convergence estimate for this semidiscretization.</p>

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Discrete stochastic maximal \( L^p \)-regularity and convergence of a spatial semidiscretization for a linear stochastic heat equation

  • 期刊论文

  • 发表日期：

    2026年1月19日

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  • Binjie Li,
  • Qin Zhou

摘要

This study investigates the boundedness of the \( H^\infty \) H ∞ -calculus for the negative discrete Laplace operator under homogeneous Dirichlet boundary conditions. The discrete operator is implemented using the finite element method, and we establish that the boundedness constant of its \( H^\infty \) H ∞ -calculus remains uniformly bounded with respect to the spatial mesh size. Based on this result, we derive a discrete stochastic maximal \( L^p \) L p -regularity estimate for a spatial semidiscretization of a linear stochastic heat equation. Furthermore, within the framework of general spatial \( L^q \) L q -norms, we provide a nearly optimal pathwise uniform convergence estimate for this semidiscretization.