Markovian loop soups: permanental processes and isomorphism theorems

TL;DR

This paper develops a framework for constructing loop soups for general Markov processes, establishing their connection to permanental processes and proving isomorphism theorems linking local times with these processes.

Contribution

It introduces a novel construction of loop soups for Markov processes without transition densities and proves new isomorphism theorems relating local times to permanental processes.

Findings

Loop soups for general Markov processes are constructed.

Permanental processes are shown to be distributionally equivalent to loop soup local times.

New properties of the loop measure are analyzed.

Abstract

We construct loop soups for general Markov processes without transition densities and show that the associated permanental process is equal in distribution to the loop soup local time. This is used to establish isomorphism theorems connecting the local time of the original process with the associated permanental process. Further properties of the loop measure are studied.