The Annals of Probability

Behavior near the extinction time in self-similar fragmentations II: Finite dislocation measures

Christina Goldschmidt and Bénédicte Haas

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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 [Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 338–368], 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.

Article information

Ann. Probab., Volume 44, Number 1 (2016), 739-805.

Received: September 2013
Revised: September 2014
First available in Project Euclid: 2 February 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces 60G18: Self-similar processes

Self-similar fragmentations extinction time scaling limits invariant measure Markov renewal theory spine decomposition


Goldschmidt, Christina; Haas, Bénédicte. Behavior near the extinction time in self-similar fragmentations II: Finite dislocation measures. Ann. Probab. 44 (2016), no. 1, 739--805. doi:10.1214/14-AOP988.

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