Boundary regularity of stochastic PDEs

TL;DR

This paper investigates the boundary regularity of solutions to stochastic PDEs, showing that under certain conditions, solutions are Hölder continuous up to the boundary, contrasting with known irregularities in some cases.

Contribution

The authors establish positive boundary regularity results for semilinear SPDEs on $C^1$ domains, providing conditions under which solutions exhibit Hölder continuity at the boundary.

Findings

Solutions are Hölder continuous up to the boundary under mild regularity assumptions.

The paper provides a positive counterpart to known irregular boundary behaviors.

Boundary regularity depends on coefficient regularity and domain smoothness.

Abstract

The boundary behaviour of solutions of stochastic PDEs with Dirichlet boundary conditions can be surprisingly - and in a sense, arbitrarily - bad: as shown by Krylov, for any $\alpha>0$ one can find a simple $1$-dimensional constant coefficient linear equation whose solution at the boundary is not $\alpha$-H\"older continuous. We obtain a positive counterpart of this: under some mild regularity assumptions on the coefficients, solutions of semilinear SPDEs on $C^1$ domains are proved to be $\alpha$-H\"older continuous up to the boundary with some $\alpha>0$.