Heat kernel estimates for Markov processes with jump kernels blowing-up at the boundary

In this paper, we study purely discontinuous symmetric Markov processes on closed subsets of ${\mathbb R}^d$, $d\ge 1$, with jump kernels of the form $J(x,y)=|x-y|^{-d-α}{\mathcal B}(x,y)$, $α\in (0,2)$, where the function ${\mathcal B}(x,y)$ may blow up at the boundary of the state space. This extends the framework developed recently for conservative self-similar Markov processes on the upper half-space to a broader geometric setting. Examples of Markov processes that fall into our general framework include traces of isotropic $α$-stable processes in $C^{1,\rm Dini}$ sets, processes in Lipschitz sets arising in connection with the nonlocal Neumann problem, and a large class of resurrected self-similar processes in the closed upper half-space. We establish sharp two-sided heat kernel estimates for these Markov processes. A fundamental difficulty in accomplishing this task is that, in contrast to the existing literature on heat kernels for jump processes, the tails of the associated jump measures in our setting are not uniformly bounded. Thus, standard techniques in the existing literature used to study heat kernels are not applicable. To overcome this obstacle, we employ recently developed weighted functional inequalities specifically designed for jump kernels blowing up at the boundary.

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