The structure of combinatorial Markov processes

TL;DR

This paper characterizes the structure of exchangeable combinatorial Markov processes, showing they can be constructed via iterated random Lipschitz functions and project onto limit spaces, unifying various stochastic processes.

Contribution

It provides a general framework for understanding exchangeable Feller processes on combinatorial spaces, extending known results to a broader class of processes.

Findings

All exchangeable combinatorial Feller processes share structural features.

Processes can be constructed via iterated random Lipschitz functions.

Jump measures decompose explicitly under certain conditions.

Abstract

Every exchangeable Feller process taking values in a suitably nice combinatorial state space can be constructed by a system of iterated random Lipschitz functions. In discrete time, the construction proceeds by iterated application of independent, identically distributed functions, while in continuous time the random functions occur as the atoms of a time homogeneous Poisson point process. We further show that every exchangeable Feller process projects to a Feller process in an appropriate limit space, akin to the projection of partition-valued processes into the ranked-simplex and graph-valued processes into the space of graph limits. Together, our main theorems establish common structural features shared by all exchangeable combinatorial Feller processes, regardless of the dynamics or resident state space, thereby generalizing behaviors previously observed for exchangeable coalescent and fragmentation processes as well as other combinatorial stochastic processes. If, in addition, an exchangeable Feller process evolves on a state space satisfying the $n$-disjoint amalgamation property for all $n\geq1$, then its jump measure can be decomposed explicitly in the sense of L\'evy--It\^o--Khintchine.