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May 2017 Markovian growth-fragmentation processes
Jean Bertoin
Bernoulli 23(2): 1082-1101 (May 2017). DOI: 10.3150/15-BEJ770

Abstract

Consider a Markov process ${X}$ on $[0,\infty)$ which has only negative jumps and converges as time tends to infinity a.s. We interpret $X(t)$ as the size of a typical cell at time $t$, and each jump as a birth event. More precisely, if $\Delta{X}(s)=-y<0$, then $s$ is the birthtime of a daughter cell with size $y$ which then evolves independently and according to the same dynamics, that is, giving birth in turn to great-daughters, and so on. After having constructed rigorously such cell systems as a general branching process, we define growth-fragmentation processes by considering the family of sizes of cells alive a some fixed time. We introduce the notion of excessive functions for the latter, whose existence provides a natural sufficient condition for the non-explosion of the system. We establish a simple criterion for excessiveness in terms of ${X}$. The case when ${X}$ is self-similar is treated in details, and connexions with self-similar fragmentations and compensated fragmentations are emphasized.

Citation

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Jean Bertoin. "Markovian growth-fragmentation processes." Bernoulli 23 (2) 1082 - 1101, May 2017. https://doi.org/10.3150/15-BEJ770

Information

Received: 1 May 2015; Revised: 1 September 2015; Published: May 2017
First available in Project Euclid: 4 February 2017

zbMATH: 1375.60129
MathSciNet: MR3606760
Digital Object Identifier: 10.3150/15-BEJ770

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

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Vol.23 • No. 2 • May 2017
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