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

### The asymptotic behavior of fragmentation processes

Auteur(s):

Code(s) de Classification MSC:

• 60J25 Markov processes with continuous parameter
• 60G09 Exchangeability

Résumé: The fragmentation processes considered in this work are self-similar Markov processes which are meant to describe the ranked sequence of the masses of the pieces of an object that falls apart randomly as time passes. We investigate their behavior as $t\to\infty$. Roughly, we show that the rate of decay of the $\ell^p$-norm (where $p>1$) is exponential when the index of self-similarity $\alpha$ is $0$, polynomial when $\alpha>0$, whereas the entire mass disappears in a finite time when $\alpha<0$. Moreover, we establish a strong limit theorem for the empirical measure of the fragments in the case when $\alpha>0$. Properties of size-biased picked fragments provide key tools for the study.

Mots Clés: Fragmentation ; self-similar ; scattering rate ; empirical measure

Date: 2001-04-25

Prépublication numéro: PMA-651