Self-similar fragmentations derived from the stable tree I

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

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...

Behavior near the extinction time in self-similar fragmentations I: the stable case

The stable fragmentation with index of self-similarity α ∈ [−1/2, 0) is derived by looking at the masses of the subtrees formed by discarding the parts of a (1 + α)–stable continuum random tree below height t, for t ≥ 0. We give a detailed limiting description of the distribution of such a fragmentation, (F (t), t ≥ 0), as it approaches its time of extinction, ζ. In particular, we show that tF ...

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 ...

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...