Heat kernel estimates for Markov processes with jump kernels blowing-up at the boundary
Soobin Cho, Panki Kim, Renming Song, Zoran Vondraček
TL;DR
The paper develops sharp two-sided heat kernel estimates for purely discontinuous symmetric Markov processes on general $\kappa$-fat domains $D$ with jump kernels that blow up at the boundary, described by $J(x,y)=|x-y|^{-(d+\alpha)}\mathcal{B}(x,y)$ and a blow-up weight $\Phi$. By embedding boundary blow-up into a weighted Dirichlet-form framework and leveraging recent weighted functional inequalities, the authors establish a regular Dirichlet form, Hardy-type inequalities, parabolic Hölder regularity, and truncated forms, then use Meyer's decomposition and Mosco convergence to derive robust off-diagonal heat-kernel bounds. They prove a complete, sharp two-sided heat kernel estimate for times up to $R_0^{\alpha}$ and extend it globally when $R_0=\infty$, applying the theory to the nonlocal Neumann problem, traces of $\alpha$-stable processes, and resurrected processes in the closed half-space. The results substantially extend boundary-blow-up analyses from half-spaces to general geometric domains and provide precise heat-kernel asymptotics in a broad nonlocal setting with boundary singularities.
Abstract
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.