Self-stabilizing processes in a multi-wells landscape
Processus auto-stabilisants dans un paysage multi-puits
Résumé
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.
Les processus auto-stabilisants sont définis comme des solutions d'équations différentielles stochastiques dont le terme de dérive contient à la fois le gradient d'un potentiel ainsi qu'un terme non-linéaire au sens de McKean qui attire le processus vers sa propre loi de distribution. On dispose de nombreux résultats lorsque l'environnement est convexe. L'objet de ce travail est de les étendre autant que possible au cas général notamment lorsque le paysage contient plusieurs puits. Des différences fondamentales sont constatées. Le premier chapitre prouve l'existence d'une solution forte. Le second s'intéresse aux lois de probabilités d'une telle solution. En particulier, l'existence et la non-unicité des mesures stationnaires sont mises en évidence sous des hypothèses faibles. Les chapitres trois et quatre sont affectés au comportement de ces mesures lorsque le coefficient de diffusion tend vers 0. Le chapitre cinq met en relation le processus auto-stabilisant avec des systèmes particulaires via une "propagation du chaos". Il est ainsi possible de transposer certains résultats du système de particules sur le processus non-markovien et réciproquement. Le chapitre six est dédié au dénombrement exact des mesures stationnaires. Le chapitre sept est employé pour l'étude du comportement en temps long. D'une part, un résultat de convergence dans un cas simple est fourni. D'autre part, un principe de grandes déviations est mis en évidence par l'utilisation des résultats de Freidlin et Wentzell.
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