Electronic Journal of Probability

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

Erich Baur and Jean Bertoin

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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 />

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 98, 20 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Random recursive tree destruction of graphs fragmentation process cluster sizes Ornstein-Uhlenbeck type process

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Baur, Erich; Bertoin, Jean. The fragmentation process of an infinite recursive tree and Ornstein-Uhlenbeck type processes. Electron. J. Probab. 20 (2015), paper no. 98, 20 pp. doi:10.1214/EJP.v20-3866. https://projecteuclid.org/euclid.ejp/1465067204

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