Symmetric Jump Processes and their Heat Kernel Estimates

TL;DR

This paper surveys recent advances in the theory of symmetric jump processes, focusing on heat kernel estimates, regularity properties, and probabilistic methods distinct from classical differential operators.

Contribution

It provides a comprehensive overview of sharp heat kernel estimates and regularity results for symmetric jump processes using probabilistic techniques.

Findings

Sharp two-sided heat kernel estimates established

Holder and Harnack inequalities proven for parabolic functions

Probabilistic methods applied to integro-differential operators

Abstract

We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes(or equivalently, a class of symmetric integro-differential operators). We focus on the sharp two-sided estimates for the transition density functions (or heat kernels) of the processes, a priori Holder estimate and parabolic Harnack inequalities for their parabolic functions. In contrast to the second order elliptic differential operator case, the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic.