## Bernoulli

- Bernoulli
- Volume 23, Number 2 (2017), 1082-1101.

### Markovian growth-fragmentation processes

#### 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.

#### Article information

**Source**

Bernoulli, Volume 23, Number 2 (2017), 1082-1101.

**Dates**

Received: May 2015

Revised: September 2015

First available in Project Euclid: 4 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1486177393

**Digital Object Identifier**

doi:10.3150/15-BEJ770

**Mathematical Reviews number (MathSciNet)**

MR3606760

**Zentralblatt MATH identifier**

1375.60129

**Keywords**

branching process growth-fragmentation self-similarity

#### Citation

Bertoin, Jean. Markovian growth-fragmentation processes. Bernoulli 23 (2017), no. 2, 1082--1101. doi:10.3150/15-BEJ770. https://projecteuclid.org/euclid.bj/1486177393