Weak KAM Approach to First-Order Mean Field Games with State Constraints

We study the asymptotic behavior of solutions to the constrained MFG system as the time horizon T goes to infinity. For this purpose, we analyze first Hamilton–Jacobi equations with state constraints from the viewpoint of weak KAM theory, constructing a Mather measure for the associated variational problem. Using these results, we show that a solution to the constrained ergodic mean field games system exists and the ergodic constant is unique. Finally, we prove that any solution of the first-order constrained MFG problem on [0, T] converges to the solution of the ergodic system as T goes to infinity.

Perturbative weak KAM theory

This chapter explores perturbation aspects of the weak Kolmogorov-Arnold-Moser (KAM) theory. By perturbative weak KAM theory, we mean two things. How do the weak KAM solutions and the Mather, Aubry, and Mañé sets respond to limits of the Hamiltonian? How do the weak KAM solutions change when we perturb a system, in particular, what happens when we perturb (1) completely integrable systems, and (2) autonomous systems by a time-periodic perturbation? The chapter states and proves results in both aspects, as a technical tool for proving forcing equivalence. It derives a special Lipshitz estimate of weak KAM solutions for perturbations of autonomous systems. The proof relies on semi-concavity of weak KAM solution.

Weak KAM Theory and Forcing Equivalence

This chapter describes weak Kolmogorov-Arnold-Moser (KAM) theory and forcing relation. One change from the standard presentation is that one needs to modify the definition of Tonelli Hamiltonians to allow different periods in the t component. The chapter points out an alternative definition of the alpha function, namely, one can replace the class of minimal measures with the class of closed measures. It then considers a dual setting which corresponds to forward dynamic. It also looks at elementary solutions, static classes, and Peierls barrier. In many parts of the proof, the chapter studies the hyperbolic property of a minimizing orbit, for which the concept of Green bundles is very useful.

The Hamilton-Jacobi Equation and Weak KAM Theory

This chapter describes another interesting approach to the study of invariant sets provided by the so-called weak KAM theory, developed by Albert Fathi. This approach can be considered as the functional analytic counterpart of the variational methods discussed in the previous chapters. The starting point is the relation between KAM tori (or more generally, invariant Lagrangian graphs) and classical solutions and subsolutions of the Hamilton–Jacobi equation. It introduces the notion of weak (non-classical) solutions of the Hamilton–Jacobi equation and a special class of subsolutions (critical subsolutions). In particular, it highlights their relation to Aubry–Mather theory.