Statistical estimation for collective dynamics ; Estimation statistique pour les dynamiques collectives

This thesis lies at the crossroads of non-parametric and parametric Statistics of processes, and parabolic partial differential equations analysis. The aim of the present work was to obtain informations about the flow of probability measures solution of a nonlinear FokkerPlanck-Kolmogorov equation, known as McKean-Vlasov equation, and about the drift coefficient of this equation, thanks to the continuous observation of the solution of a linear system of interacting diffusions as an approximation in a mean-field limit, i.e. as the number of particles in this system grows to infinity. In a first part, we present our results when the drift coefficient depends on an unknown parameter. We follow the estimation program of Ibragimov and Hasminskii : we show that our model verifies the local asymptotic normality property, and we deduce from it precise properties of convergence and asymptotic normality of the maximum likelihood estimator of the parameter. We give a new criterium for verifying that this statistical problem is non-degenerate and identifiable. In a second part, we present our results when the drift coefficient is unknown, but lies in a certain class of functions characterized by their regularity : the local Holder functions. This part relies on a new Bernstein type concentration inequality, and on nonparametric statistics tools : we build adaptive kernel estimators of the density solution of the McKean-Vlasov equation and of its drift, and we show that these estimators are optimal in a minimax sense for a pointwise risk. Finally, we construct an estimator of the interaction force in the case of a Vlasov type drift by means of a deconvolution method. ; Cette thèse se situe à l’interface de la Statistique, paramétrique et non-paramétrique, des processus stochastiques, et de l’analyse des équations aux dérivées partielles paraboliques. Le fil conducteur des travaux présentés ici a été d’obtenir des informations sur le flot de mesures de probabilités, solution d’une équation de Fokker-Planck-Kolmogorov ...