Exit spaces for Cox processes and the P\'olya sum process

TL;DR

This paper constructs Markov processes for Cox and Pólya sum processes to analyze their condensations, identifies their exit spaces as mixtures of extremal processes, and characterizes them as Gibbs processes.

Contribution

It introduces a novel Markov process construction for Cox and Pólya sum processes and characterizes their exit spaces as mixtures of extremal and Gibbs processes.

Findings

Constructed Markov processes with increasing paths for Cox and Pólya sum processes.

Identified the exit spaces as mixtures of extremal processes and Gibbs processes.

Provided a new perspective on the structure of condensations in these processes.

Abstract

For Cox processes we construct a Markov process with increasing paths to couple the condensations of the Cox process in a monotone way. A similar procedure procedure yields an analogue Markov process for the P\'olya sum process. Moreover, we identify the exit spaces of these Markov processes and identify them firstly as mixtures of certain extremal processes, i.e. as a process in a random environment, and secondly as Gibbs processes.