The fragmentation process of an infinite recursive tree and Ornstein-Uhlenbeck type processes

TL;DR

This paper studies a natural fragmentation process of an infinite recursive tree, revealing Markovian properties and connections to Ornstein-Uhlenbeck processes despite non-exchangeability.

Contribution

It introduces a novel destruction process for infinite recursive trees and analyzes its Markovian fragmentation dynamics with non-standard dislocation measures.

Findings

The process is Markovian with a specific splitting rates measure.

Normalization of block weights leads to connections with Ornstein-Uhlenbeck processes.

The dislocation measure does not satisfy usual integrability conditions.

Abstract

We consider a natural destruction process of an infinite recursive tree by removing each edge after an independent exponential time. The destruction up to time t is encoded by a partition $\Pi$(t) of N into blocks of connected vertices. Despite the lack of exchangeability, just like for an exchangeable fragmentation process, the process $\Pi$ is Markovian with transitions determined by a splitting rates measure r. However, somewhat surprisingly, r fails to fulfill the usual integrability condition for the dislocation measure of exchangeable fragmentations. We further observe that a time-dependent normalization enables us to define the weights of the blocks of $\Pi$(t). We study the process of these weights and point at connections with Ornstein-Uhlenbeck type processes.