PreprintCC BY 4.0arXiv
Sharp regularity and small ball probabilities for the stochastic heat equation on bounded domains
Abstract
We consider the stochastic heat equation ∂t u(t,x) = Δu(t,x) + Ẇα(t,x) on a bounded Lipschitz domain with zero Dirichlet boundary condition and zero initial condition, where Ẇα is a Gaussian noise that is white in time and whose spatial covariance is the kernel of (−Δ)−α with α>0. We prove that a unique pointwise defined mild solution exists if and only if α>d/2−1. In this case, if in addition the domain is C2, we also establish spatial and temporal Holder regularity of the solution. When d/2−1<α<d/2, we show that the Holder exponents are optimal and obtain exact local and uniform moduli of continuity, a Chung-type law of the iterated logarithm, and sharp small ball probability estimates for the solution.
Related
- Maximal regularity for the Stokes operator in bounded C1 domains
- Optimal regularity and fine asymptotics for very fast diffusion equations in bounded domains
- Boundary regularity for fractional porous medium-type equations in bounded domains
- Boundary pointwise regularity for the p-Laplace equation on C1,Dini domains
- Geometry property of stochastic parabolic equations and Prékopa-Leindler inequality on bounded domains
BibTeX
@misc{hu2026sharp,
title = {{Sharp regularity and small ball probabilities for the stochastic heat equation on bounded domains}},
author = {Jingwu Hu and Cheuk Yin Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08718},
eprint = {2609.08718},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08718},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp regularity and small ball probabilities for the stochastic heat equation on bounded domains
AU - Jingwu Hu
AU - Cheuk Yin Lee
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08718
UR - https://arxiv.org/abs/2609.08718
AB - We consider the stochastic heat equation ∂_t u(t,x) = Δu(t,x) + Ẇ_α(t,x) on a bounded Lipschitz domain with zero Dirichlet boundary condition and zero initial condition, where Ẇ_α is a Gaussian noise that is white in time and whose spatial covariance is the kernel of (−Δ)^−α with α>0. We prove that a unique pointwise defined mild solution exists if and only if α>d/2−1. In this case, if in addition the domain is C^2, we also establish spatial and temporal Holder regularity of the solution. When d/2−1<α<d/2, we show that the Holder exponents are optimal and obtain exact local and uniform moduli of continuity, a Chung-type law of the iterated logarithm, and sharp small ball probability estimates for the solution.
ER -