Volume 25, 2021
|Page(s)||258 - 285|
|Published online||04 June 2021|
A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates
Institut für Mathematik, Universität Zürich,
* Corresponding author: email@example.com
Accepted: 3 May 2021
The growth-fragmentation equation models systems of particles that grow and reproduce as time passes. An important question concerns the asymptotic behaviour of its solutions. Bertoin and Watson (2018) developed a probabilistic approach relying on the Feynman-Kac formula, that enabled them to answer to this question for sublinear growth rates. This assumption on the growth ensures that microscopic particles remain microscopic. In this work, we go further in the analysis, assuming bounded fragmentations and allowing arbitrarily small particles to reach macroscopic mass in finite time. We establish necessary and sufficient conditions on the coefficients of the equation that ensure Malthusian behaviour with exponential speed of convergence to the asymptotic profile. Furthermore, we provide an explicit expression of the latter.
Mathematics Subject Classification: 34K08 / 35Q92 / 47D06 / 47G20 / 45K05 / 60G51 / 60J99
Key words: Growth-fragmentation equation / transport equations / cell division equations / one parameter semigroups / spectral analysis / Malthus exponent / Feynman-Kac formula / piecewise deterministic Markov processes
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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