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Preprints, Working Papers, ... Year : 2015

Markovian growth-fragmentation processes

Abstract

Consider a Markov process X on [0, ∞) 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 ∆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, i.e. 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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Dates and versions

hal-01152370 , version 1 (16-05-2015)

Identifiers

  • HAL Id : hal-01152370 , version 1

Cite

Jean Bertoin. Markovian growth-fragmentation processes. 2015. ⟨hal-01152370⟩
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