Weak solutions to the master equation of a potential mean field game

The purpose of this work is to introduce a notion of weak solution to the master equation of a potential mean field game and to prove that existence and uniqueness hold under quite general assumptions. Remarkably, this is shown to hold true 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 HJB equation. Here, the master equation is indeed regarded as an infinite dimensional system set on the space of probability measures. To make the analysis easier, we assume that the coefficients are periodic and accordingly that the probability measures are defined on the torus. This allows to represent probability measures through their Fourier coefficients. Most of the analysis then consists in rewriting the master equation and the corresponding HJB equation for the mean field control problem lying above the mean field game as PDEs set on the Fourier coefficients themselves.

Joint work with A. Cecchin (Ecole Polytechnique, France)