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Markovian growth-fragmentation processes


Bertoin, Jean (2017). Markovian growth-fragmentation processes. Bernoulli, 23(2):1082-1101.

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 $\mathit{X(t)}$ as the size of a typical cell at time $t$, and each jump as a birth event. More precisely, if $\Delta\mathit{X(s)}=−\mathit{y}<0$, 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.

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 $\mathit{X(t)}$ as the size of a typical cell at time $t$, and each jump as a birth event. More precisely, if $\Delta\mathit{X(s)}=−\mathit{y}<0$, 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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:1 March 2017
Deposited On:18 Jan 2018 10:21
Last Modified:19 Mar 2018 08:09
Publisher:International Statistical Institute
ISSN:1350-7265
OA Status:Green
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.3150/15-BEJ770

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