Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### A fragmentation process connected to Brownian motion

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Résumé: Let $(B_s, s \geq 0)$ be a standard Brownian motion and $T$ its first passage time at level $1$. For every $t \geq 0$, we consider ladder time set ${\mathcal L}^{(t)}$ of the Brownian motion with drift $t$, $B_s^{(t)} = B_s + ts$. The decreasing sequence $F(t) = (F_1(t), F_2(t), \ldots)$ of lengths of the intervals of the random partition of $[0,T]$ induced by ${\mathcal L}^{(t)}$ is called the fragmentation at $t$. The central result of this work is that the fragmentation process is Markovian; and more precisely, for $0 \leq t < t'$, $F(t')$ is obtained from $F(t)$ by breaking randomly into pieces each fragment independently of the others. We present analytic expressions for the fragmentation law and study the behavior as $t \geq 0$ varies of a remarkable fragment.