The Annals of Probability

Boundary regularity of stochastic PDEs

Máté Gerencsér

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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 $\alpha>0$.

Article information

Ann. Probab., Volume 47, Number 2 (2019), 804-834.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Stochastic partial differential equations Dirichlet problem boundary regularity


Gerencsér, Máté. Boundary regularity of stochastic PDEs. Ann. Probab. 47 (2019), no. 2, 804--834. doi:10.1214/18-AOP1272.

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