Joint work with Bertrand Cloez.
The Crump-Young model consists of two fully coupled stochastic processes modeling the substrate and micro-organisms dynamics in a chemostat. The substrate evolves follo-wing an ordinary di˙erential equation, depending on the micro-organisms number. Micro-organims are modeled though a pure jump process whose the jump rates depend on the substrate concentration. It is known that this model extincts almost surely in the sense that micro-organism population vanishes. However, despite its simplicity, the long-time behavior of this process is not well understood. The existence (but not the uniqueness) of quasi-stationary distribution (QSD) as well as some regularity properties of these QSD were proved in the literature. Nevertheless, the long-time behavior of the process before extinction was, until now, unknown. In this work, we prove that there exists a unique QSD and that, conditionally on the non-extinction, the Crump-Young model converges exponentially fast to this QSD. Conver-gence of quasi-stationary distribution is usually proved though Hilbert techniques. Ho-wever, our process is not reversible making these techniques di°cult to deal with. To overcome this problem, we use the recent results of Champagnat-Villemonais and Bansaye-Cloez-Gabriel-Marguet, which are a generalization of usual techniques to prove conver-gence to stationary distribution. The proof is mainly based on hitting time estimates and Lyapunov functions bounds. Due to the deterministic part, the dynamics of the Crump-Young model is highly degenerated. The proof then consists of technical sharp estimates.
De Nick Barton
De Quentin Griette
De Ada Altieri
De Mete Demircigil
De Elisa Thébault
De Jean-François Arnoldi
De Michael Scott
De Florence Bansept
De Apolline Louvet
De Vincent Bansaye
De Sébastien Lion
De Philipp Messer
De Raphael Forien
De Nicolas Vauchelet
De David MacLeod
De Cheryl Mentuda
De Léo Girardin
De Coralie Fritsch
De Adrian Lam
De Diana Fusco
De Léonard Dekens
De Oana Carja
De Sepideh Mirrahimi
De Olivier Cotto
De Benoît Henry
De Himani Sachdeva
De Florian Patout
De Ignacio Madrid Canales
De Denis Roze
De Julien Berestycki
De Sylvie Ferrari
De Vasilis Dakos
