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

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.