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

Bauer, Erich; Bertoin, Jean (2015). The fragmentation process of an infinite recursive tree and Ornstein-Uhlenbeck type processes. Electronic Journal of Probability, 20(98):online.

## 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 $\boldsymbol{t}$ is encoded by a partition $\Pi$($\mathit{t}$) 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 $\mathbf{r}$. However, somewhat surprisingly, $\mathbf{r}$ 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 $\Pi$($\mathit{t}$) We study the process of these weights and point at connections with Ornstein-Uhlenbeck type processes.

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 $\boldsymbol{t}$ is encoded by a partition $\Pi$($\mathit{t}$) 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 $\mathbf{r}$. However, somewhat surprisingly, $\mathbf{r}$ 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 $\Pi$($\mathit{t}$) We study the process of these weights and point at connections with Ornstein-Uhlenbeck type processes.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 16 September 2015 27 Jan 2016 10:03 05 Apr 2016 19:18 Institute of Mathematical Statistics 1083-6489 Swiss National Science Foundation grant Publisher DOI. An embargo period may apply. https://doi.org/10.1214/EJP.v20-3866
Permanent URL: https://doi.org/10.5167/uzh-111633

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