Behavior near the extinction time in self-similar fragmentations II: Finite dislocation measures
Résumé
We study a Markovian model for the random fragmentation of an object. At each time, the state consists of a collection of blocks. 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 into sub-blocks whose relative sizes are distributed according to the so-called dislocation measure. We focus here on the case where alpha < 0. In this case, small blocks split intensively and so the whole state is reduced to "dust" in a finite time, almost surely (we call this the extinction time). In this paper, we investigate how the fragmentation process behaves as it approaches its extinction time. In particular, we prove a scaling limit for the block sizes which, as a direct consequence, gives us an expression for an invariant measure for the fragmentation process. In an earlier paper, we considered the same problem for another family of fragmentation processes, the so-called stable fragmentations. The results here are similar, but we emphasize that the methods used to prove them are different. Our approach in the present paper is based on Markov renewal theory and involves a somewhat unusual "spine" decomposition for the fragmentation, which may be of independent interest.