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A Probabilistic Look at Conservative Growth-Fragmentation Equations

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Séminaire de Probabilités XLIX

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2215))

Abstract

In this note, we consider general growth-fragmentation equations from a probabilistic point of view. Using Foster-Lyapunov techniques, we study the recurrence of the associated Markov process depending on the growth and fragmentation rates. We prove the existence and uniqueness of its stationary distribution, and we are able to derive precise bounds for its tails in the neighborhoods of both 0 and + ∞. This study is systematically compared to the results obtained so far in the literature for this class of integro-differential equations.

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Acknowledgements

The author wants to thank Pierre Gabriel for fruitful discussions about growth-fragmentation equations, as well as Eva Löcherbach, Florent Malrieu and Jean-Christophe Breton for their precious help and comments. The referee is also warmly thanked for his constructive remarks. This work was financially supported by the ANR PIECE (ANR-12-JS01-0006-01), and the Centre Henri Lebesgue (programme “Investissements d’avenir” ANR-11-LABX-0020-01).

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Authors and Affiliations

  1. Université de Lorraine, IECL, UMR 7502, Vandœuvre-lès-Nancy, France

    Florian Bouguet

  2. CNRS, IECL, UMR 7502, Vandœuvre-lès-Nancy, France

    Florian Bouguet

  3. Inria, Villers-lès-Nancy, France

    Florian Bouguet

Authors
  1. Florian Bouguet
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Correspondence to Florian Bouguet .

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Editors and Affiliations

  1. Laboratoire de Mathématiques de Versailles, Université Versailles Saint-Quentin, Versailles, France

    Catherine Donati-Martin

  2. Institut Elie Cartan de Lorraine, Vandoeuvre-les-Nancy, France

    Antoine Lejay

  3. Laboratoire de Mathématiques de Versailles, Université Versailles Saint-Quentin, Versailles, France

    Alain Rouault

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Bouguet, F. (2018). A Probabilistic Look at Conservative Growth-Fragmentation Equations. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIX. Lecture Notes in Mathematics(), vol 2215. Springer, Cham. https://doi.org/10.1007/978-3-319-92420-5_2

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