Heat kernel estimates for anisotropic symmetric jump processes

TL;DR

This paper establishes two-sided heat kernel bounds for a class of anisotropic symmetric jump processes with jump kernels supported on unions of cones, advancing understanding of their probabilistic and analytic properties.

Contribution

It provides sharp two-sided heat kernel estimates for anisotropic symmetric jump processes with cone-supported kernels, a novel extension in the study of jump processes.

Findings

Established two-sided heat kernel bounds for anisotropic jump processes.

Extended heat kernel estimates to processes with cone-supported jump kernels.

Enhanced understanding of anisotropic jump process behaviors.

Abstract

We show two-sided bounds of heat kernel for anisotropic non-singular symmetric pure jump Markov process whose jump kernel $J(x,y)$ is comparable to $\frac{{\bf 1}_{\mathcal{V}}(x-y)}{|x-y|^{d+\alpha}}$, where $\mathcal{V}$ is a union of symmetric cones, $0<\alpha<2$ and $x,y\in\mathbb{R}^d$.