Long time and large scale behaviour of a few collisional dynamics

Abstract : This thesis contains three independent parts, each one of which is dedicated to the study of a particle system, following either a deterministic or a stochastic dynamics, and in which interactions only occur at collisions. Part I contains a numerical and theoretical study of nonequilibrium steady states of the Complete Exchange Model, which was introduced by physicists in order to understand heat transfer in some porous materials. Part II is dedicated to a system of Brownian particles evolving on the real line and interacting through their ranks. The long time and mean-field behaviour of this system is described, then the results are applied to the study of a model of equity market called the mean-field Atlas model. Part III introduces a multitype version of the particle system studied in the previous part, which allows to approximate parabolic systems of nonlinear partial differential equations. The small noise limit of of this system is called multitype sticky particle dynamics and now approximates hyperbolic systems. A detailed study of this dynamics provides stability estimates in Wasserstein distance for the solutions of these systems.
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Julien Reygner. Long time and large scale behaviour of a few collisional dynamics. General Mathematics [math.GM]. Université Pierre et Marie Curie - Paris VI, 2014. English. ⟨NNT : 2014PA066471⟩. ⟨tel-01087575v2⟩



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