Villemonais, Denis;
Watson, Alexander R;
(2025)
A quasi-stationary approach to the long-term asymptotics of the growth-fragmentation equation.
The Annals of Applied Probability
, 35
(2)
pp. 1233-1297.
10.1214/24-AAP2142.
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Abstract
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 quasi-stationary 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 growth-fragmentation equation in a wide range of situations.
| Type: | Article |
|---|---|
| Title: | A quasi-stationary approach to the long-term asymptotics of the growth-fragmentation equation |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1214/24-AAP2142 |
| Publisher version: | https://doi.org/10.1214/24-AAP2142 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | Cell division equation; Feynman–Kac formula; Growth-fragmentation equation; piecewise-deterministic Markov processes; quasi-stationary distribution; transport equations |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10203541 |
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