Propagation of chaos for order statistics of particle systems with mean-field interaction

This thesis focuses on the study of the asymptotic behavior of the order statistics of real-valued diffusive particles with mean-field drift interaction, with a particular emphasis on the extremes of N mean-field interacting particle systems, following the spirit of extreme value theory. The main contribution of this thesis is presented in two parts, outlined in Chapter 4 and Chapter 5. 

Chapter 4 presents the study of the asymptotic behavior of the normalized maxima of real-valued diffusive particles with mean-field drift interaction. Our main result establishes propagation of chaos, demonstrating that in the large population limit, the normalized maxima behave like those arising in an i.i.d. system where each particle follows the associated McKean–Vlasov limiting dynamics. Unlike classical propagation of chaos, where convergence to an i.i.d. limit holds for any fixed number of particles but not all particles simultaneously, our result accounts for the dependence of the maximum on all particles. Our proof involves a change of measure argument that relies on a delicate combinatorial analysis of the iterated stochastic integrals present in the chaos expansion of the Radon–Nikodym density. 

Chapter 5 focuses on the asymptotics of the point process induced by the same interacting particle system with mean-field drift interaction. Under suitable assumptions, we establish propagation of chaos for this point process. It has the same weak limit as the point process induced by i.i.d. copies of the solution of a limiting McKean–Vlasov equation. The weak limit is a Poisson point process whose intensity measure is related to classical extreme value distributions. In particular, this yields the limiting distribution of the normalized upper order statistics.