Exchangeable Markov processes on graphs: Feller case

TL;DR

This paper characterizes exchangeable Feller processes on countable graphs through a measure-based framework, detailing their discrete and continuous-time dynamics, and showing their projection to graph limits.

Contribution

It provides a comprehensive measure-theoretic description of exchangeable Feller processes on graphs, including a Lévy–Itô decomposition for continuous-time cases.

Findings

Transition laws are determined by a measure on array space.

Discrete-time processes are constructed from exchangeable i.i.d. functions.

Continuous-time processes have a Lévy–Itô decomposition of jump measures.

Abstract

The transition law of every exchangeable Feller process on the space of countable graphs is determined by a $\sigma$-finite measure on the space of $\{0,1\}\times\{0,1\}$-valued arrays. In discrete-time, this characterization amounts to a construction from an independent, identically distributed sequence of exchangeable random functions. In continuous-time, the behavior is enriched by a L\'evy--It\^o-type decomposition of the jump measure into mutually singular components that govern global, vertex-level, and edge-level dynamics. Furthermore, every such process almost surely projects to a Feller process in the space of graph limits.