# The strong Malthusian behavior of growth-fragmentation processes

## TL;DR

This paper demonstrates that growth-fragmentation processes exhibit strong Malthusian behavior, showing that the entire system converges to an asymptotic profile under certain conditions, extending previous average-based results.

## Contribution

It establishes that a criterion for exponential ergodicity on average implies stronger convergence results for the whole system, with improved explicit conditions.

## Key findings

- System converges to an asymptotic profile.
- Stronger results than average behavior.
- Provides explicit conditions for convergence.

## Abstract

Growth-fragmentation processes describe the evolution of systems of cells which grow continuously and fragment suddenly; they are used in models of cell division and protein polymerisation. Typically, we may expect that in the long run, the concentrations of cells with given masses increase at some exponential rate, and that, after compensating for this, they arrive at an asymptotic profile. Up to now, this question has mainly been studied for the average behavior of the system, often by means of a natural partial integro-differential equation and the associated spectral theory. However, the behavior of the system as a whole, rather than only its average, is more delicate. In this work, we show that a criterion found by one of the authors for exponential ergodicity on average is actually sufficient to deduce stronger results about the convergence of the entire collection of cells to a certain asymptotic profile, and we find some improved explicit conditions for this to occur.