A probabilistic approach to spectral analysis of growth-fragmentation equations

TL;DR

This paper introduces a probabilistic method using Feynman--Kac formulas to analyze the long-term behavior of solutions to growth-fragmentation equations, providing new insights into their spectral properties and convergence rates.

Contribution

It develops a novel probabilistic framework for spectral analysis of growth-fragmentation equations, linking solutions to Markov processes and characterizing asymptotic behavior.

Findings

Identifies spectral radius and asymptotic profile via Markov processes

Provides conditions for exponential convergence of solutions

Connects spectral properties to probabilistic representations

Abstract

The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymp-totic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this work, we present a probabilistic approach to the study of this asymptotic behaviour. We use a Feynman--Kac formula to relate the solution of the growth-fragmentation equation to the semigroup of a Markov process, and characterise the rate of decay or growth in terms of this process. We then identify the spectral radius and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growth-fragmentation operator and its dual. In special cases, we obtain exponential convergence.