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Processus auto-stabilisants dans un paysage multi-puits

Julian Tugaut 1, 2
INRIA Lorraine, CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, INPL - Institut National Polytechnique de Lorraine, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : The subject of my thesis is the McKean-Vlasov diffusion. The motion of the process is subject to three concurrent forces: the gradient of a confining potential V, some Brownian motion with a constant coefficient of diffusion and the so-called self-stabilizing term which is equal to the convolution between the derivative of a convex potential F and the own law of the process (which represents the average tension between all the trajectories). There are many results if V is convex. The purpose is to extend these in the general case especially when the landscape contains several wells. Essential differences are found. The first chapter proves the strong existence of a solution on the set of the positive reals. The second one deals with the stationary measure(s). Particularly, the existence and the non-uniqueness are highlighted under weak assumptions when the diffusion coefficient is sufficiently small. The critical value under which several measures appear is also examinated. Chapter three and four are assigned to the asymptotic analysis in the small-noise limit of these measures. It is proved that each family of stationary measures has a limiting value. Moreover, it is a finite combination of Dirac measures. Chapter five connects the self-stabilizing process and some mean-field systems. In one hand, it is stressed that a uniform (with respect to the time) propagation of chaos is not possible. In an other hand, by making a little modification of the interacting system, it is proved that a half-uniform propagation of chaos holds. With three different methods, the uniqueness problem (of the stationary measures) is studied in Chapter six. The first method derives from the computations of Chapter four for the asymmetric measures and for the symmetric ones if V''(0)+F''(0) is not equal to 0. The second one uses the half-uniform propagation of chaos and is applied for the symmetric measures if V''(0)+F''(0) is nonnegative. The last method is classical and direct but can only be used for the symmetric measures if V''(0)+F''(0)>0. Chapter seven is devoted to the study of the long-time behavior. In one hand, a convergence's result is provided in a simple case by using the half-uniform propagation of chaos. In the other hand, a large deviations principle is highlighted by using results closed to those of Freidlin and Wentzell. Various asymptotic lemmas are proved in Annex A and some classical results of the Freidlin-Wentzell theory are recalled in Annex B.
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Julian Tugaut. Processus auto-stabilisants dans un paysage multi-puits. Mathématiques [math]. Université Henri Poincaré - Nancy 1, 2010. Français. ⟨NNT : 2010NAN10047⟩. ⟨tel-01748560v2⟩



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