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2015 The fragmentation process of an infinite recursive tree and Ornstein-Uhlenbeck type processes
Erich Baur, Jean Bertoin
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Electron. J. Probab. 20: 1-20 (2015). 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.<br /><br />

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Erich Baur. Jean Bertoin. "The fragmentation process of an infinite recursive tree and Ornstein-Uhlenbeck type processes." Electron. J. Probab. 20 1 - 20, 2015. https://doi.org/10.1214/EJP.v20-3866

Information

Accepted: 16 September 2015; Published: 2015
First available in Project Euclid: 4 June 2016

zbMATH: 1333.60186
MathSciNet: MR3399834
Digital Object Identifier: 10.1214/EJP.v20-3866

Subjects:
Primary: 60J25
Secondary: 60J80, 60K35

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