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

2023/12/13 by Kajino, Naotaka, Murugan, Mathav · 1 citation
#31C25 #31E05 #35J08 #35K08 (Primary) 31B25 #60J45 (Secondary) #60J60 #60J76 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2312.08546

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.

Cited by

• Trace and Extension Theorems for Besov Functions in Doubling Metric Measure Spaces

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