Behaviour near the extinction time in self-similar fragmentation chains

Suppose we have a collection of blocks, which gradually split apart as time goes on. Each block waits an exponential amount of time with parameter given by its size to some power alpha, independently of the other blocks. Every block then splits randomly, but according to the same distribution. In this talk, I will focus on the case where alpha is negative, which means that smaller blocks split faster than larger ones. This gives rise to the phenomenon of loss of mass, whereby the smaller blocks split faster and faster until they are reduced to ''dust''. Indeed, it turns out that the whole state is reduced to dust in a finite time, almost surely (we call this the extinction time). A natural question is then: how do the block sizes behave as the process approaches its extinction time? The answer turns out to involve a somewhat unusual ''spine'' decomposition for the fragmentation, and Markov renewal theory.

This is joint work with Bénédicte Haas (Paris-Dauphine).