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March 2019 Boundary regularity of stochastic PDEs
Máté Gerencsér
Ann. Probab. 47(2): 804-834 (March 2019). DOI: 10.1214/18-AOP1272

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 $\alpha>0$.

Citation

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Máté Gerencsér. "Boundary regularity of stochastic PDEs." Ann. Probab. 47 (2) 804 - 834, March 2019. https://doi.org/10.1214/18-AOP1272

Information

Received: 1 May 2017; Revised: 1 March 2018; Published: March 2019
First available in Project Euclid: 26 February 2019

zbMATH: 07053557
MathSciNet: MR3916935
Digital Object Identifier: 10.1214/18-AOP1272

Subjects:
Primary: 35R60 , 60H15

Keywords: boundary regularity , Dirichlet problem , Stochastic partial differential equations

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 2 • March 2019
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