A QUASI-STATIONARY APPROACH TO THE LONG-TERM ASYMPTOTICS OF THE GROWTH-FRAGMENTATION EQUATION

• 成果类型: Article
• 署名作者: Villemonais, Denis; Watson, Alexander R.
• 署名单位: Universites de Strasbourg Etablissements Associes; Universite de Strasbourg; Institut Universitaire de France; University of London; University College London
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2142
• 发表日期: 2025
• 页码: 1233-1297
• 关键词: markov semigroups; time behavior; STABILITY; relaxation; discrete

摘要

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 quasistationary 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 growthfragmentation equation in a wide range of situations.