A Probabilistic Look at Conservative Growth-Fragmentation Equations
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.
KeywordsGrowth-fragmentation Markov process Stationary measure Tail of distribution Foster-Lyapunov criterion.
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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