A probabilistic approach to spectral analysis of growth-fragmentation equations

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.

References in corpus (2)

• Mean field convergence of a model of multiple TCP connections through a buffer implementing RED
• A Probabilistic Look at Conservative Growth-Fragmentation Equations

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• Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate
• The strong Malthusian behavior of growth-fragmentation processes
• Practical criteria for R-positive recurrence of unbounded semigroups
• On spectral gaps of growth-fragmentation semigroups in higher moment spaces
• Strong laws of large numbers for a growth-fragmentation process with bounded cell sizes
• Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostat
• On the Krein-Rutman theorem and beyond
• A growth-fragmentation model related to Ornstein-Uhlenbeck type processes
• On a family of critical growth-fragmentation semigroups and refracted Lévy processes