A Probabilistic Look at Conservative Growth-Fragmentation Equations

TL;DR

This paper investigates growth-fragmentation equations through probabilistic methods, establishing conditions for recurrence, existence of stationary distributions, and tail behavior, thereby providing a new analytical perspective on these equations.

Contribution

It introduces a probabilistic approach using Foster-Lyapunov techniques to analyze growth-fragmentation equations, offering new insights into their long-term behavior and tail properties.

Findings

Proves recurrence criteria for the associated Markov process.

Establishes existence and uniqueness of stationary distributions.

Derives bounds for the tails near zero and infinity.

Abstract

In this note, we consider general growth-fragmentation equations from a probabilistic point of view. Using Foster-Lyapunov techniques, we study the recurrence of the associated Markov process depending on the growth and fragmentation rates. We prove the existence and uniqueness of its stationary distribution, and we are able to derive precise bounds for its tails in the neighborhoods of both $0$ and $+\infty$. This study is systematically compared to the results obtained so far in the literature for this class of integro-differential equations.