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

### Multifractal spectra of fragmentation processes

Auteur(s):

Code(s) de Classification MSC:

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

Résumé: Let $(S(t),t\geq 0)$ be a homogeneous fragmentation of $\I$ with no loss of mass. For $x \in \I$, we say that the fragmentation speed of $x$ is $v$ if and only if, as time passes, the size of the fragment that contains $x$ decays exponentially with rate $v$. We show that there is $\vtyp>0$ such that almost every point $x \in \I$ has speed $\vtyp$ and that, nonetheless, for $v$ in a certain range, the random set $\mathcal{G}_v$ of points of speed $v$, is dense in $\I$, and we compute explicitly the spectrum $v \rightarrow \text{Dim} (\mathcal{G}_v)$ where $\text{Dim}$ is the Hausdorff dimension.

Mots Clés: Fragmentation ; Galton-Watson trees ; Multifractal spectra

Date: 2002-11-22

Prépublication numéro: PMA-777

Pdf file : PMA-777.pdf