Heat kernel for reflected diffusion and extension property on uniform domains

• 成果类型: Article
• 署名作者: Murugan, Mathav
• 署名单位: University of British Columbia
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01266-4
• 发表日期: 2024
• 页码: 543-599
• 关键词: brownian-motion; dirichlet forms; energy measures; differentiable functions; sobolev extension; STABILITY; BOUNDARY; fractals; inequalities; singularity

摘要

We study reflected diffusion on uniform domains where the underlying space admits a symmetric diffusion that satisfies sub-Gaussian heat kernel estimates. A celebrated theorem of Jones (Acta Math 147(1-2):71-88, 1981) states that uniform domains in Euclidean space are extension domains for Sobolev spaces. In this work, we obtain a similar extension property for metric spaces equipped with a Dirichlet form whose heat kernel satisfies a sub-Gaussian estimate. We introduce a scale-invariant version of this extension property and apply it to show that the reflected diffusion process on such a uniform domain inherits various properties from the ambient space, such as Harnack inequalities, cutoff energy inequality, and sub-Gaussian heat kernel bounds. In particular, our work extends Neumann heat kernel estimates of Gyrya and Saloff-Coste (Asterisque 336:145, 2011) beyond the Gaussian space-time scaling. Furthermore, our estimates on the extension operator imply that the energy measure of the boundary of a uniform domain is always zero. This property of the energy measure is a broad generalization of Hino's result (Probab Theory Relat Fields 156:739-793, 2013) that proves the vanishing of the energy measure on the outer square boundary of the standard Sierpinski carpet equipped with the self-similar Dirichlet form.