Behaviour near the extinction time in stable fragmentation processes

The stable fragmentations are particularly nice examples of the
self-similar fragmentation processes introduced by Bertoin.  They are
derived by looking at the masses of the subtrees formed by discarding
the the parts of a stable continuum random tree below height t, for t
\geq 0. Such a fragmentation possesses a parameter alpha which lies
between -1/2 and 0, called the index of self-similarity;
heuristically, any block of mass m splits at a rate proportional to
m^{alpha}.  Since alpha is negative, this means that smaller blocks
split faster than larger ones. Indeed, small blocks split faster and
faster until they are reduced to dust (blocks of size 0).  The whole
state is entirely reduced to dust in some almost surely finite time
zeta, called the extinction time.  We give a detailed limiting
description of the distribution of a stable fragmentation of index
alpha as it approaches its extinction time.

This is joint work with Benedicte Haas of Universite Paris-Dauphine.