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

Abstract

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 Naïm kernel to the topological boundary, and the Doob--Naïm 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 Naïm kernel.

129 pages; new Subsection 5.4 on extensions of the results in Sections 4 and 5 to the case with a weaker CDC considered in arXiv:2410.19201 has been added