Weak Solutions to the Master Equation of Potential Mean Field Games

The purpose of this work is to introduce a notion of weak solution to the master equation of potential mean field games and to prove that existence and uniqueness hold under quite general assumptions. Remarkably, this is achieved without any monotonicity constraint on the coefficients. The key point is to interpret the master equation in a conservative sense and then to adapt to the infinite dimensional setting earlier arguments for hyperbolic systems deriving from a Hamilton-Jacobi-Bellman equation. Here, the master equation is indeed regarded as an infinite dimensional system set on the space of probability measures and is formally written as the derivative of the Hamilton-Jacobi-Bellman equation associated with the mean field control problem lying above the mean field game. To make the analysis easier, we assume that the coefficients are periodic, which allows to represent probability measures through their Fourier coefficients. Part of the analysis then consists in rewriting the master equation and the corresponding Hamilton-Jacobi-Bellman equation for the mean field control problem as partial differential equations set on the Fourier coefficients themselves. In the end, we establish existence and uniqueness of functions that are displacement semi-concave in the measure argument and that solve the Hamilton-Jacobi-Bellman equation in a suitable generalized sense. In parallel, we construct a suitable probability measure on the space of probability measures, for which we prove a Rademacher-type theorem and under which the optimal trajectories of the associated mean field control problem are shown to be unique for almost every starting point. Back to the mean field game, we get existence and uniqueness of possibly discontinuous functions that solve the master equation in an appropriate weak sense and that satisfy a weak one-sided Lipschitz inequality. We hope to prove in future contributions that those solutions are the ones that arise when passing to the limit in large games, which is an important conjecture in the domain.