A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates

TL;DR

This paper extends the probabilistic analysis of growth-fragmentation equations to include bounded fragmentation rates, providing conditions for exponential convergence to a stable profile even with particles reaching macroscopic sizes.

Contribution

It advances the understanding of long-time behaviour in growth-fragmentation models by allowing small particles to grow large and deriving explicit asymptotic profiles.

Findings

Established necessary and sufficient conditions for Malthusian behaviour.

Proved exponential convergence to the asymptotic profile.

Derived explicit expression for the asymptotic profile.

Abstract

The growth-fragmentation equation models systems of particles that grow and reproduce as time passes. An important question concerns the asymptotic behaviour of its solutions. Bertoin and Watson ($2018$) developed a probabilistic approach relying on the Feynman-Kac formula, that enabled them to answer to this question for sublinear growth rates. This assumption on the growth ensures that microscopic particles remain microscopic. In this work, we go further in the analysis, assuming bounded fragmentations and allowing arbitrarily small particles to reach macroscopic mass in finite time. We establish necessary and sufficient conditions on the coefficients of the equation that ensure Malthusian behaviour with exponential speed of convergence to the asymptotic profile. Furthermore, we provide an explicit expression of the latter.