Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60J65 Brownian motion, See also {58G32}
- 60J25 Markov processes with continuous parameter

**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.

**Mots Clés:** *Fragmentation; Brownian motion; excursion*

**Date:** 1999-02-04

**Prépublication numéro:** *PMA-487*