Heat kernel for reflected jump diffusion on Ahlfors regular domains

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.

💡 Research Summary

The paper investigates reflected jump diffusion processes on Ahlfors‑regular open subsets of a general metric measure space (X,d,m). The ambient space is assumed to satisfy volume doubling (VD) and quasi‑reverse volume doubling (QR VD), allowing for possible atoms. An Ahlfors‑regular set D⊂X is defined by a uniform lower bound on the proportion of the ambient measure inside balls intersected with D. Under these geometric assumptions the authors first prove that D inherits the VD and QR VD properties and that its boundary has zero m‑measure.

The ambient Dirichlet form (E,F) is a symmetric regular non‑local form generated by a jump kernel J(x,y) and an increasing scaling function ϕ satisfying two‑sided power bounds. The kernel is assumed to obey the two‑sided estimate Jϕ (i.e., J(x,y)≈1/