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 $\mathit{X(t)}$ 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.