Boundary regularity of stochastic PDEs

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 [SIAM J. Math. Anal. 34 (2003) 1167–1182], for any $\alpha>0$ one can find a simple $1$-dimensional constant coefficient linear equation whose solution at the boundary is not $\alpha$-Hölder continuous. We obtain a positive counterpart of this: under some mild regularity assumptions on the coefficients, solutions of semilinear SPDEs on $\mathcal{C}^{1}$ domains are proved to be $\alpha$-Hölder continuous up to the boundary with some $\alpha>0$.