Occupation Times for Jump Processes

TL;DR

This paper studies pure jump Markov processes with stable-like jumps, establishing support properties, occupation time bounds, and approximation techniques for resolvents, advancing understanding of their probabilistic behavior.

Contribution

It introduces new support and occupation time results for jump processes with stable-like kernels and provides methods to approximate resolvents with smooth functions.

Findings

Support theorem for jump processes

Lower bounds on occupation times of sets

Approximation of resolvents using smooth functions

Abstract

We consider a class of pure jump Markov processes in $\rr^d$ whose jump kernels are comparable to those of symmetric stable processes. We prove a support theorem, a lower bound on the occupation times of sets, and show that we can approximate resolvents using smooth functions.