Abstract.
The theory of Mean Field Games (MFG) has been developed in the last two decades by economists, engineers, and mathematicians in order to study decision making in very large populations of “small" interacting agents. The approach by Lasry and Lions leads to a system of nonlinear partial differential equations, the solution of which can be used to approximate the limit of an N-player Nash equilibrium as N tends to infinity. This course will be mainly focused on deterministic models in Euclidean space. These problems are associated with a first order pde system coupling a Hamilton-Jacobi equation, depending on the distribution of players, with a continuity equation driving such a distribution by the optimal feedback provided by the first equation. We will first prove the existence of solutions to the MFG system by a fixed point argument. Then, we will discuss uniqueness under some monotonicity condition for the coupling functions. Finally, we will study the long time behaviour of solutions following the approach of weak KAM theory. For this last topic, a survey of the main notions and results will be given by Cristian Mendico within the course.
Prerequisites: basics of optimal control theory, viscosity solutions of Hamilton-Jacobi equations, properties of semiconcave functions.