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

Erich Baur (1) , Jean Bertoin (2)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 Institut für Mathematik [Zürich]

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 Π(t) of N into blocks of connected vertices. Despite the lack of exchangeability, just like for an exchangeable fragmentation process, the process Π is Markovian with transitions determined by a splitting rates measure r. However, somewhat surprisingly, 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 Π(t). We study the process of these weights and point at connections with Ornstein-Uhlenbeck type processes.

Mots clés: Random recursive tree; destruction of graphs; fragmentation process; cluster sizes; Ornstein-Uhlenbeck type process

Domaines: Probabilités [math.PR]

Erich Baur, Jean Bertoin. The fragmentation process of an infinite recursive tree and Ornstein-Uhlenbeck type processes. 2014. ⟨hal-01071463⟩