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François Bolley; Cédric Villani; École normale supérieure (Lyon).
|Beschreibung:||1 vol. (263 p.) ; 30 cm.|
|Verfasserangabe:||François Bolley ; sous la direction de Cédric Villani.|
In this thesis we study some particle approximation methods of solutions to partial differential equations giving the macroscopic state of some physical systems. They consist in introducing a large number N of fictive particles evolving according to a system of ordinary or stochastic differential equations, in some sense easier to solve than the macroscopic equation; the state of this system is given by a probability measure called empirical measure. The validity of the method is given by the convergence, as N tends to infinity, of this empirical measure towards the original macroscopic solution, called mean field limit. We mainly look for explicit estimates on this convergence, thus quantifying the accuracy of the approximation. In this framework we study the approximation of Vlasov and Euler transport equations by some deterministic interacting particle systems. The convergence of the method turns into a stability issue on solutions, which we solve by some contraction type properties for some (Wasserstein) distances linked with measure optimal transportation. We also derive a similar contraction property for scalar conservation laws. We also study the approximation of McKean-Vlasov equations by stochastic particle systems. We give error bounds on it by means of coupling technics, propagation of chaos estimates and deviation or concentration inequalities. In a more systematic way we consider such concentration inequalities on probability measures, and their links with transport inequalities (between Wasserstein distances and entropy) and logarithmic Sobolev inequalities. In particular we derive such inequalities for some classes of laws of dependent variables.