Strong laws of large numbers for a growth-fragmentation process with bounded cell sizes

TL;DR

This paper investigates the long-term stochastic behavior of a growth-fragmentation process with bounded cell sizes, establishing strong laws of large numbers and revealing different regimes of almost sure asymptotics.

Contribution

It introduces a growth-fragmentation model with bounded cell sizes and proves new strong laws of large numbers for its almost sure long-term behavior.

Findings

Existence of regimes with distinct almost sure asymptotics

Proof of strong laws of large numbers for the process

Characterization of long-term behavior in bounded cell size systems

Abstract

Growth-fragmentation processes model systems of cells that grow continuously over time and then fragment into smaller pieces. Typically, on average, the number of cells in the system exhibits asynchronous exponential growth and, upon compensating for this, the distribution of cell sizes converges to an asymptotic profile. However, the long-term stochastic behaviour of the system is more delicate, and its almost sure asymptotics have been so far largely unexplored. In this article, we study a growth-fragmentation process whose cell sizes are bounded above, and prove the existence of regimes with differing almost sure long-term behaviour.