Growth-fragmentation processes and bifurcators

TL;DR

This paper investigates growth-fragmentation processes modeling cell systems, revealing conditions under which different processes share the same distribution and showing that their law is determined by a cumulant function and self-similarity index.

Contribution

It introduces the concept of bifurcators to characterize when two growth-fragmentation processes have the same law, and establishes how their distribution is uniquely determined by a cumulant function and self-similarity.

Findings

Two processes can have the same distribution if they are bifurcators.

The law of a self-similar growth-fragmentation is uniquely determined by a cumulant function and self-similarity index.

Bifurcation times are key to understanding process equivalence.

Abstract

Markovian growth-fragmentation processes introduced by Bertoin model a system of growing and splitting cells in which the size of a typical cell evolves as a Markov process $X$ without positive jumps. We find that two growth-fragmentation processes associated respectively with two processes $X$ and $Y$ (with different laws) may have the same distribution, if $(X,Y)$ is a bifurcator, roughly speaking, which means that they coincide up to a bifurcation time and then evolve independently. Using this criterion, we deduce that the law of a self-similar growth-fragmentation is determined by a cumulant function $\kappa$ and its index of self-similarity.