Combinatorial Levy processes

TL;DR

This paper introduces combinatorial Levy processes evolving on countable structures, characterizing their properties through sigma-finite measures and Poisson point processes, with implications for understanding complex combinatorial dynamics.

Contribution

It provides the first comprehensive characterization of combinatorial Levy processes on general state spaces, including exchangeable cases linked to Poisson point processes.

Findings

Characterization of finite and infinite state space processes via sigma-finite measures

Explicit description of exchangeable processes as Poisson point processes

Analysis of structural features through projections into limiting object spaces

Abstract

Combinatorial Levy processes evolve on general state spaces of countable combinatorial structures. In this setting, the usual Levy process properties of stationary, independent increments are defined in an unconventional way in terms of the symmetric difference operation on sets. In discrete time, the description of combinatorial Levy processes gives rise to the notion of combinatorial random walks. These processes behave differently than random walks and Levy processes on other state spaces. Standard examples include processes on sets, graphs, and n-ary relations, but the framework permits far more general possibilities. The main theorems characterize both finite and infinite state space combinatorial Levy processes by a unique sigma-finite measure. Under the additional assumption of exchangeability, we obtain a more explicit characterization by which every exchangeable combinatorial Levy process corresponds to a Poisson point process on the same state space. Associated behavior of the projection into a space of limiting objects reflects certain structural features of the covering process.