Heat kernel for reflected jump diffusion on Ahlfors regular domains
Shiping Cao, Zhen-Qing Chen
TL;DR
The paper addresses two-sided heat kernel estimates for reflected jump diffusions on Ahlfors regular domains within general metric measure spaces. The authors develop an extension operator, built via a Whitney-type cover, to transfer the ambient Dirichlet form's energy and $L^2$-norm properties to the reflected setting, and prove energy and capacity bounds that enable a CSJ$(\phi)$-type condition for the reflected form. By linking ${\rm CSJ}(\phi)$ to ${\bf HK}(\phi)$ under (VD) and (QRVD), they show that, for a $D$ that is Ahlfors regular, the active reflected Dirichlet space on $\overline{D}$ is regular and satisfies two-sided heat kernel estimates ${\bf HK}(\phi)$. This extends known results for diffusions and uniform domains to much broader classes of domains and metric spaces, including spaces with atoms via QRVD, and provides tools for boundary-harnack and trace-type analyses in censored/nonlocal settings. The results have significant implications for non-local Dirichlet forms in irregular domains, enriching the theory of reflected jump processes and their heat kernels with concrete analytic and geometric hypotheses.
Abstract
We study reflected jump diffusions on Ahlfors regular domains in general metric measure spaces. Under the condition that the Dirichlet form on the ambient space satisfies a capacity upper bound estimate, we construct an extension operator from the reflected Dirichlet space to the ambient Dirichlet space, with a scale-invariant local bound. Second, we establish the mixed stable-like heat kernel estimates for the reflected jump diffusion, assuming that the process on the ambient space satisfies the same type of heat kernel estimates.