Compound Poisson Processes: Potentials, Green Measures and Random Times

TL;DR

This paper investigates Green measures for Markov processes with nonlocal jump generators lacking second moments, and extends the analysis to certain time-changed processes, highlighting their potential and Green measure properties.

Contribution

It introduces conditions for the existence of Green measures in nonlocal jump Markov processes and explores their behavior under random time changes, advancing understanding of such stochastic processes.

Findings

Green measures exist under specific Fourier transform conditions.

Extension to time-changed Markov processes with jump generators.

Insights into potentials for nonlocal jump processes.

Abstract

In this paper we study the existence of Green measures for Markov processes with a nonlocal jump generator. The jump generator has no second moment and satisfies a suitable condition on its Fourier transform. We also study the same problem for certain classes of random time changes Markov processes with jump generator.