Maximal Lp-Lq regularity of finite element methods for stochastic parabolic equations in polygonal domains
Abstract
We prove that a spatial semidiscrete finite element method preserves the stochastic maximal regularity of the heat equation on bounded convex polygonal and polyhedral domains in dimensions two and three. A main difficulty is to establish suitable Green-function estimates for stochastic convolutions, whose paths generally lack classical time derivatives. Besides, the failure of the Burkholder-Davis-Gundy (BDG) inequality at the endpoint p=∞ creates additional difficulties. Our approach leverages a hidden symmetry structure of the discrete Green's function, which enables the use of a key integration-by-parts type formula. Then the discrete stochastic maximal Lp-Lp regularity with p∈ [2,∞) is obtained from the discrete local energy estimates and interpolation inequalities. To extend further the discrete stochastic maximal Lp-Lq regularity with the full range of integrability exponents, i.e., p∈ (2,∞), q∈ [2,∞) or p=q=2, we also show that the discrete convolution kernel satisfies the stochastic Hörmander's condition, allowing the application of a stochastic version of the Calderón-Zygmund theory.
Related
- Maximal Regularity and Existence for Superquadratic Parabolic Hamilton--Jacobi Equations: The Endpoint case
- Stochastic maximal Lp-regularity for non-autonomous evolution equations with fractional derivative in UMD spaces
- Interior regularity of mixed local-nonlocal parabolic semilinear equations
- Parabolic Poincaré inequalities and maximal function estimates for systems of partial differential equations
- Local regularity for solutions to singular and degenerate quasilinear parabolic equations
BibTeX
@misc{bai2026maximal,
title = {{Maximal $L^p$-$L^q$ regularity of finite element methods for stochastic parabolic equations in polygonal domains}},
author = {Genming Bai and Jianbo Cui and Buyang Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25275},
eprint = {2610.25275},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25275},
}
RIS
TY - UNPB
TI - Maximal L^p-L^q regularity of finite element methods for stochastic parabolic equations in polygonal domains
AU - Genming Bai
AU - Jianbo Cui
AU - Buyang Li
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25275
UR - https://arxiv.org/abs/2610.25275
AB - We prove that a spatial semidiscrete finite element method preserves the stochastic maximal regularity of the heat equation on bounded convex polygonal and polyhedral domains in dimensions two and three. A main difficulty is to establish suitable Green-function estimates for stochastic convolutions, whose paths generally lack classical time derivatives. Besides, the failure of the Burkholder-Davis-Gundy (BDG) inequality at the endpoint p=∞ creates additional difficulties. Our approach leverages a hidden symmetry structure of the discrete Green's function, which enables the use of a key integration-by-parts type formula. Then the discrete stochastic maximal L^p-L^p regularity with p∈ [2,∞) is obtained from the discrete local energy estimates and interpolation inequalities. To extend further the discrete stochastic maximal L^p-L^q regularity with the full range of integrability exponents, i.e., p∈ (2,∞), q∈ [2,∞) or p=q=2, we also show that the discrete convolution kernel satisfies the stochastic Hörmander's condition, allowing the application of a stochastic version of the Calderón-Zygmund theory.
ER -