Processus auto-stabilisants dans un paysage multi-puits

Abstract : Self-stabilizing processes are defined as the solutions of stochastic differential equations which drift term contains the gradient of a potential and a term nonlinear in the sense of McKean which attracts the process to its own law distribution. There are many results if the landscape is convex. The purpose of this work is to extend these in the general case especially when the landscape contains contains several wells. Essential differences are found.The first chapter proves the strong existence of a solution. The second one deals with the probability measure of the solution. Particularly, the existence and the non-uniqueness of the stationary measures are highlighted under weak assumptions. Chapter three and four are assigned to the asymptotic analysis in the small noise limit of these measures.Chapter five connects the self-stabilizing process and some particle systems via a « propagation of chaos ». It is thus possible to translate some results from the particle systems to the non-markovian process and reciprocally.Chapter seven is used to study the long time behavior. In one hand, a convergence's result is provided in a simple case. In the other hand, a large deviations principle is highlighted by using the results of Freidlin and Wentzell.
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Julian Tugaut. Processus auto-stabilisants dans un paysage multi-puits. Mathématiques générales [math.GM]. Université Henri Poincaré - Nancy 1, 2010. Français. ⟨NNT : 2010NAN10047⟩. ⟨tel-01748560v1⟩

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