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Processus cinétiques dans les domaines à bord et quasi-stationnarité

Mouad Ramil 1, 2 
2 MATHERIALS - MATHematics for MatERIALS
CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique, Inria de Paris
Abstract : This thesis is divided into three parts, each one focuses on a different problem related to the study of the Langevin process, described at each time by its position and velocity vectors. We consider here an arbitrary dimension and for the purpose of this work, different tools, probabilistic tools as well as more analytic tools, are combined.The first part focuses on the extension of certain results statisfied in the parabolic theory on smooth bounded domains to the degenerate case of the kinetic Fokker-Planck operator on a domain D, only bounded with respect to its position coordinates. We obtain in this part the existence of a unique classical solution to the kinetic Fokker-Planck equation on a domain D with initial conditions and homogeneous Dirichlet boundary conditions. We also obtain a Harnack inequality as well as a Maximum principle associated to the kinetic Fokker-Planck operator. Finally, we obtain a compactness result on the set of bounded continuous functions of the semigroup of the Langevin process absorbed at the boundary of D.The results prove to be useful in the second part to prove the existence of a unique quasi-stationary distribution (QSD) for the Langevin process on the domain D. We also obtain a weak convergence of the law of the Langevin process conditioned to stay in D during [0,t], when t goes to infinity, towards its QSD. We also consider the obtained QSD and explicit its weak limit when the friction pramater in the equation satisfied by the Langevin process goes to infinity.Finally, we consider in the last part a Markov chain obtained from the successive entry and exit points of a domain for the Langevin process. We then study the stationarity of this Markov chain
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Mouad Ramil. Processus cinétiques dans les domaines à bord et quasi-stationnarité. Mathématiques générales [math.GM]. Université Paris-Est, 2020. Français. ⟨NNT : 2020PESC1038⟩. ⟨tel-03413317v2⟩

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