Singular SPDEs in domains with boundaries

M. Gerencser, M. Hairer, Probability Theory and Related Fields (2018).

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Journal Article | Epub ahead of print | English
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Abstract
We study spaces of modelled distributions with singular behaviour near the boundary of a domain that, in the context of the theory of regularity structures, allow one to give robust solution theories for singular stochastic PDEs with boundary conditions. The calculus of modelled distributions established in Hairer (Invent Math 198(2):269–504, 2014. https://doi.org/10.1007/s00222-014-0505-4) is extended to this setting. We formulate and solve fixed point problems in these spaces with a class of kernels that is sufficiently large to cover in particular the Dirichlet and Neumann heat kernels. These results are then used to provide solution theories for the KPZ equation with Dirichlet and Neumann boundary conditions and for the 2D generalised parabolic Anderson model with Dirichlet boundary conditions. In the case of the KPZ equation with Neumann boundary conditions, we show that, depending on the class of mollifiers one considers, a “boundary renormalisation” takes place. In other words, there are situations in which a certain boundary condition is applied to an approximation to the KPZ equation, but the limiting process is the Hopf–Cole solution to the KPZ equation with a different boundary condition.
Publishing Year
Date Published
2018-01-01
Journal Title
Probability Theory and Related Fields
Acknowledgement
MG thanks the support of the LMS Postdoctoral Mobility Grant.
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Gerencser M, Hairer M. Singular SPDEs in domains with boundaries. Probability Theory and Related Fields. 2018. doi:10.1007/s00440-018-0841-1
Gerencser, M., & Hairer, M. (2018). Singular SPDEs in domains with boundaries. Probability Theory and Related Fields. https://doi.org/10.1007/s00440-018-0841-1
Gerencser, Mate, and Martin Hairer. “Singular SPDEs in Domains with Boundaries.” Probability Theory and Related Fields, 2018. https://doi.org/10.1007/s00440-018-0841-1.
M. Gerencser and M. Hairer, “Singular SPDEs in domains with boundaries,” Probability Theory and Related Fields, 2018.
Gerencser M, Hairer M. 2018. Singular SPDEs in domains with boundaries. Probability Theory and Related Fields.
Gerencser, Mate, and Martin Hairer. “Singular SPDEs in Domains with Boundaries.” Probability Theory and Related Fields, Springer, 2018, doi:10.1007/s00440-018-0841-1.
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