A quasi-stationary approach to the long-term asymptotics of the growth-fragmentation equation

TL;DR

This paper introduces a new quasi-stationary approach to analyze the long-term behavior of the growth-fragmentation equation, demonstrating exponential convergence to equilibrium and establishing solution existence and uniqueness.

Contribution

It develops a novel method using sub-Markov processes and Lyapunov functions to prove exponential convergence and generalizes previous results in growth-fragmentation models.

Findings

Exponential convergence to the asymptotic profile.

Unified framework for growth-fragmentation analysis.

Existence and uniqueness of solutions in various cases.

Abstract

In a growth-fragmentation system, cells grow in size slowly and split apart at random. Typically, the number of cells in the system grows exponentially and the distribution of the sizes of cells settles into an equilibrium 'asymptotic profile'. In this work we introduce a new method to prove this asymptotic behaviour for the growth-fragmentation equation, and show that the convergence to the asymptotic profile occurs at exponential rate. We do this by identifying an associated sub-Markov process and studying its quasi-stationary behaviour via a Lyapunov function condition. By doing so, we are able to simplify and generalise results in a number of common cases and offer a unified framework for their study. In the course of this work we are also able to prove the existence and uniqueness of solutions to the growth-fragmentation equation in a wide range of situations.