Weak and renormalized solutions to a hypoelliptic Mean Field Games system

We establish the existence and uniqueness of weak and renormalized solutions to a degenerate, hypoelliptic Mean Field Games system with local coupling. An important step is to obtain $L^{\mathrm{\infty }}-$bounds for solutions to a degenerate Fokker-Planck equation with a De-Giorgi type argument. In particular, we show existence and uniqueness of weak solutions to Mean Fields Games systems with Lipschitz Hamiltonians. Furthermore, we establish existence and uniqueness of renormalized solutions for Hamiltonians with quadratic growth. Our approach relies on the kinetic regularity of hypoelliptic equations obtained by Bouchut and the work of Porretta on the existence and uniqueness of renormalized solutions for the Mean Field Game system, in the non-degenerate setting.