Self-similar fragmentations

Self-similar Fragmentations

We introduce a probabilistic model that is meant to describe an object that falls apart randomly as time passes and fulfills a certain scaling property. We show that the distribution of such a process is determined by its index of self-similarity , a rate of erosion c�0, and a so-called Lévy measure that accounts for sudden dislocations. The key of the analysis is provided by a transformation o...

Asymptotic laws for nonconservative self-similar fragmentations

We consider a self-similar fragmentation process in which the generic particle of mass x is replaced by the offspring particles at probability rate x, with positive parameter α. The total of offspring masses may be both larger or smaller than x with positive probability. We show that under certain conditions the typical mass in the ensemble is of the order t and that the empirical distribution ...

Regularity of formation of dust in self-similar fragmentations

In self-similar fragmentations with a negative index, fragments split even faster as their mass is smaller, so that the fragmentation runs away and some mass is reduced to dust. Our purpose is to investigate the regularity of this formation of dust. Let M(t) denote the mass of dust at time t.We give some sufficient and some necessary conditions for the measure dM to be absolutely continuous. In...

Self-similar fragmentations derived from the stable tree II: splitting at nodes

The genealogy of self-similar fragmentations with negative index as a continuum random tree

We encode a certain class of stochastic fragmentation processes, namely self-similar fragmentation processes with a negative index of self-similarity, into a metric family tree which belongs to the family of Continuum Random Trees of Aldous. When the splitting times of the fragmentation are dense near 0, the tree can in turn be encoded into a continuous height function, just as the Brownian Con...