Limites diffusives pour des équations cinétiques stochastiques

Abstract : This thesis presents several results about stochastic partial differential equations. The main subject is the study of diffusive limits of kinetic models perturbed with a random term. We also present a result about the regularity of a class of stochastic partial differential equations and a result of existence and uniqueness of invariant measures for a stochastic Fokker-Planck equation. First, we give three results of approximation-diffusion in a stochastic context. The first one deals with the case of a kinetic equation with a linear operator of relaxation whose velocity equilibrium has a power tail distribution at infinity. The equation is perturbed with a Markovian process. This gives rise to a stochastic fluid fractional limit. The two remaining results consider the case of the radiative transfer equation which is a non-linear kinetic equation. The equation is perturbed successively with a cylindrical Wiener process and with a Markovian process. In both cases, we are led to a stochastic Rosseland fluid limit. Then, we introduce a result of regularity for a class of quasilinear stochastic partial differential equations of parabolic type whose random term is driven by a cylindrical Wiener process. Finally, we study a Fokker-Planck equation with a noisy force governed by a cylindrical Wiener process. We prove existence and uniqueness of solutions to the problem and then existence and uniqueness of invariant measures to the equation.
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Sylvain de Moor. Limites diffusives pour des équations cinétiques stochastiques. Analyse numérique [math.NA]. Ecole normale supérieure de Rennes - ENS Rennes, 2014. Français. ⟨tel-01010825⟩

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