The cut-and-paste process

TL;DR

This paper characterizes exchangeable Feller processes on partitions with bounded blocks, providing a Lévy-Itô type decomposition in continuous time and a simple matrix product description in discrete time.

Contribution

It introduces a novel decomposition of exchangeable Feller processes on partitions, linking continuous and discrete-time dynamics through stochastic matrices.

Findings

Decomposition of jump measure into stochastic matrices and constants

Lévy-Itô representation for continuous-time processes

Simplified product form for discrete-time evolution

Abstract

We characterize the class of exchangeable Feller processes evolving on partitions with boundedly many blocks. In continuous-time, the jump measure decomposes into two parts: a $\sigma$-finite measure on stochastic matrices and a collection of nonnegative real constants. This decomposition prompts a L\'evy-It\^o representation. In discrete-time, the evolution is described more simply by a product of independent, identically distributed random matrices.