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

Erich Baur,Jean Bertoin

doi:10.1214/ejp.v20-3866

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 $\mathbb{N}$ 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 ${\bf r}$. However, somewhat surprisingly, ${\bf r}$ 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.

Highlights

The purpose of this work is to investigate various aspects of a simple and natural fragmentation process on an infinite tree, which turns out to exhibit some rather unexpected features.we first construct a tree T with set of vertices N = {1, . . .} by incorporating vertices one after the other and uniformly at random
If we view the fragmentation of T up to time t as a Bernoulli bond-percolation with parameter e−t, the blocks of Π(t) are the percolation clusters
Π(t) gets finer as t increases, and it is seen from the fundamental splitting property of random recursive trees that the process Π = (Π(t) : t ≥ 0) is Markovian

Summary

Introduction

The purpose of this work is to investigate various aspects of a simple and natural fragmentation process on an infinite tree, which turns out to exhibit some rather unexpected features. Because Π resembles homogeneous fragmentations, but with splitting rate measure r which does not fulfill the integral condition of the former, and because the notion (2) of the weight of a block depends on the time t, one might expect that X should be an example of a so-called compensated fragmentation which was recently introduced in [6]. This is not exactly the case, we shall see that X fulfills closely related properties.

Destruction of T and fragmentation of partitions

The process of the weights

Asymptotic behaviors