Lipschitz partition processes

TL;DR

This paper introduces Lipschitz partition processes, a new family of Markov processes on set partitions with bounded blocks, constructed via Poisson point processes, ensuring Markovian consistency and exchangeability.

Contribution

It explicitly constructs Lipschitz partition processes using Poisson point processes and characterizes exchangeable cases with a novel set-valued matrix operation.

Findings

Constructed Lipschitz partition processes explicitly.

Ensured Markovian consistency through construction.

Characterized exchangeable processes with a new matrix operation.

Abstract

We introduce a family of Markov processes on set partitions with a bounded number of blocks, called Lipschitz partition processes. We construct these processes explicitly by a Poisson point process on the space of Lipschitz continuous maps on partitions. By this construction, the Markovian consistency property is readily satisfied; that is, the finite restrictions of any Lipschitz partition process comprise a compatible collection of finite state space Markov chains. We further characterize the class of exchangeable Lipschitz partition processes by a novel set-valued matrix operation.