Growth-fragmentation processes describe the evolution of systems of cells which grow continuously and fragment suddenly; they are used in models of cell division and protein polymerisation. Typically, we may expect that in the long run, the concentrations of cells with given masses increase at some exponential rate, and that, after compensating for this, they arrive at an asymptotic profile. Up to now, this question has mainly been studied for the average behavior of the system, often by means of a natural partial integro-differential equation and the associated spectral theory. However, the behavior of the system as a whole, rather than only its average, is more delicate. In this work, we show that a criterion found by one of the authors for exponential ergodicity on average is actually sufficient to deduce stronger results about the convergence of the entire collection of cells to a certain asymptotic profile, and we find some improved explicit conditions for this to occur.
Les processus de croissance-fragmentation décrivent l’évolution de familles de cellules qui croissent continûment et se divisent soudainement ; ils apparaissent notamment comme modèles pour la division cellulaire et la polymérisation des protéines. Au fur et à mesure que le temps passe, on s’attend à ce que les concentrations de cellules de masse donnée croissent à un taux exponentiel, et qu’une fois ce taux compensé, elles convergent vers un profil asymptotique. Jusqu’à présent, cette question a principalement été étudiée pour le processus moyenné, le plus souvent via l’analyse spectrale d’une équation intégro-différentielle qui est associée naturellement au modèle. Cependant, l’étude du comportement du processus lui-même, et pas seulement de sa moyenne, est plus délicate. Dans ce travail, nous établissons qu’un critère obtenu par l’un des auteurs pour assurer l’ergodicité exponentielle en moyenne est également une condition suffisante pour des résultats de convergence forte (i.e. en probabilité) pour la famille des cellules vers un certain profil asymptotique. Nous donnons par ailleurs des conditions explicites pour que ceci ait lieu.
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Keywords: growth-fragmentation process, Malthusian behavior, intrinsic martingale, branching process
@article{AHL_2020__3__795_0, author = {Bertoin, Jean and Watson, Alexander R.}, title = {The strong {Malthusian} behavior of growth-fragmentation processes}, journal = {Annales Henri Lebesgue}, pages = {795--823}, publisher = {\'ENS Rennes}, volume = {3}, year = {2020}, doi = {10.5802/ahl.46}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/ahl.46/} }
TY - JOUR AU - Bertoin, Jean AU - Watson, Alexander R. TI - The strong Malthusian behavior of growth-fragmentation processes JO - Annales Henri Lebesgue PY - 2020 SP - 795 EP - 823 VL - 3 PB - ÉNS Rennes UR - http://archive.numdam.org/articles/10.5802/ahl.46/ DO - 10.5802/ahl.46 LA - en ID - AHL_2020__3__795_0 ER -
Bertoin, Jean; Watson, Alexander R. The strong Malthusian behavior of growth-fragmentation processes. Annales Henri Lebesgue, Volume 3 (2020), pp. 795-823. doi : 10.5802/ahl.46. http://archive.numdam.org/articles/10.5802/ahl.46/
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