Asymptotics for the small fragments of the fragmentation at nodes

Abstract

We consider the fragmentation at nodes of the Lévy continuous random tree introduced in a previous paper. In this framework we compute the asymptotic behaviour of the number of small fragments at time $θ$. This limit is increasing in $θ$ and discontinuous. In the $α$-stable case the fragmentation is self-similar with index $1/α$, with $α∈(1,2)$, and the results are close to those Bertoin obtained for general self-similar fragmentations but with an additional assumption which is not fulfilled here.