Symmetric jump processes: localization, heat kernels, and convergence

TL;DR

This paper studies symmetric pure jump processes, providing local exit probability estimates, regularity results for harmonic functions and heat kernels, and analyzing their convergence behavior.

Contribution

It introduces new local estimates and regularity results for symmetric jump processes, along with convergence analysis, advancing understanding of their probabilistic and analytical properties.

Findings

Established local exit probability bounds

Proved Hölder continuity of heat kernels and harmonic functions

Demonstrated convergence of sequences of symmetric jump processes

Abstract

We consider symmetric processes of pure jump type. We prove local estimates on the probability of exiting balls, the H\"older continuity of harmonic functions and of heat kernels, and convergence of a sequence of such processes.