Heat kernel estimates for boundary traces of reflected diffusions on uniform domains

We study the boundary trace processes of reflected diffusions on uniform domains. We obtain stable-like heat kernel estimates for such a boundary trace process when the diffusion on the underlying ambient space satisfies sub-Gaussian heat kernel estimates. Our arguments rely on new results of indepe...

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Published in:arXiv.org 2024-07
Main Authors: Kajino, Naotaka, Murugan, Mathav
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Dirichlet problem
Estimates
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description We study the boundary trace processes of reflected diffusions on uniform domains. We obtain stable-like heat kernel estimates for such a boundary trace process when the diffusion on the underlying ambient space satisfies sub-Gaussian heat kernel estimates. Our arguments rely on new results of independent interest such as sharp two-sided estimates and the volume doubling property of the harmonic measure, the existence of a continuous extension of the Na\"im kernel to the topological boundary, and the Doob--Na\"im formula identifying the Dirichlet form of the boundary trace process as the pure-jump Dirichlet form whose jump kernel with respect to the harmonic measure is exactly (the continuous extension of) the Na\"im kernel.
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subjects Dirichlet problem
Estimates
Kernels
title Heat kernel estimates for boundary traces of reflected diffusions on uniform domains
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