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.
Keywords: Growth-fragmentation equation, transport equations, cell division equations, one parameter semigroups, spectral analysis, Malthus exponent, Feynman-Kac formula, piecewise deterministic Markov processes
@article{PS_2021__25_1_258_0,
author = {Cavalli, Benedetta},
title = {A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates},
journal = {ESAIM: Probability and Statistics},
pages = {258--285},
year = {2021},
publisher = {EDP-Sciences},
volume = {25},
doi = {10.1051/ps/2021008},
mrnumber = {4268042},
zbl = {1470.60209},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2021008/}
}
TY - JOUR AU - Cavalli, Benedetta TI - A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates JO - ESAIM: Probability and Statistics PY - 2021 SP - 258 EP - 285 VL - 25 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ps/2021008/ DO - 10.1051/ps/2021008 LA - en ID - PS_2021__25_1_258_0 ER -
%0 Journal Article %A Cavalli, Benedetta %T A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates %J ESAIM: Probability and Statistics %D 2021 %P 258-285 %V 25 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ps/2021008/ %R 10.1051/ps/2021008 %G en %F PS_2021__25_1_258_0
Cavalli, Benedetta. A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates. ESAIM: Probability and Statistics, Tome 25 (2021), pp. 258-285. doi: 10.1051/ps/2021008
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