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Theses

Limites d'échelles pour des modèles cinétiques stochastiques

Abstract : This thesis aims at providing an understanding of certain scaling limits for kinetic models perturbed with some random noise, where the limiting object remains of stochastic nature, governed by a stochastic partial differential equation. In the first chapter, the transition from a mesoscopic to a macroscopic description is studied through a kinetic system of equations – corresponding to the behavior of a “spray” of particles embedded in an ambient fluid perturbed by a mixing Markov process. Under a suitable scaling, relying on the perturbed test function method, we establish the convergence of the density of particles to a hydrodynamic limit which can be expressed as the solution of a stochastic conservation equation driven by a Wiener process.Next, we focus on stochastic kinetic equations derived from biological models of collective motion. This study is split into two different works, devoted to distinct models. In chapter 2, we first examine the mean-field limits of a few different particle systems which correspond to random perturbations of the classical Cucker-Smale model. Then, in chapter 3, we establish the existence of martingale solutions for some more advanced model, which allows local interactions between individuals.
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Submitted on : Monday, July 20, 2020 - 5:01:10 PM
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Angelo Rosello. Limites d'échelles pour des modèles cinétiques stochastiques. Equations aux dérivées partielles [math.AP]. École normale supérieure de Rennes, 2020. Français. ⟨NNT : 2020ENSR0021⟩. ⟨tel-02903157⟩

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