Exchangeable Markov processes on graphs: Feller case

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 Lévy--Itô-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.

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