Sur quelques systèmes d’équations aux dérivées partielles de type champ moyen

He main objective of this thesis is thestudy of two new models of collective behavior invol-ving an infinite number of interacting particles. Thefirst model focuses on a phenomenon of chemo-taxis, namely the formation of trails in ant colonies.The second model addresses the structure of stra-tegic interactions within the framework of mean fieldgame theory. This thesis adopts a mean field-type ap-proach, which consists of replacing the complexity ofmicroscopic interactions within finite groups of par-ticles by an averaged interaction described throughfields. In accordance with this approach, the distribu-tion of an infinite cloud of particles is represented bya density. The description of certain phenomena re-quires the introduction of auxiliary fields, which, toge-ther with this density, obey evolutionary laws expres-sed through partial differential equations (PDEs).Chemotaxis models aim to describe the collective mo-tion of a large population of particles, insects, or bac-teria that use chemical signals to coordinate their mo-vement in space. The density of this population is mo-deled by a Fokker–Planck-type equation, coupled witha second equation that governs the evolution of thechemoattractant concentration.Mean field game theory seeks to model Nash equili-bria in systems with infinitely many players. A typicalmean field game system is described by a coupled setof equations: a Fokker–Planck equation for the evolu-tion of the players’ density and a Hamilton–Jacobi–Bellman equation representing the value function ofa representative player. This theory has proven fruit-ful for modeling a wide range of phenomena in fieldssuch as finance, economics, and crowd dynamics.For each of the phenomena under consideration, wederive a new model and establish the well-posednessof the associated partial differential equations. In thecase of the chemotaxis model, we further analyzequalitative properties of the long-time behavior of thePDE system. Finally, the last chapter of the thesis pre-sents an existence and uniqueness result for stochas-tic differential equations on Riemannian manifolds ofbounded geometry