Weak and Renormalized Solutions to a Hypoelliptic Mean Field Games System

.We study the well-posedness of a degenerate, hypoelliptic Mean Field Games system with local coupling and Hamiltonians which either are Lipschitz or grow quadratically in the gradient. In the former case, we prove the existence and uniqueness of weak solutions, while in the latter we study the same question for renormalized solutions. 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 nondegenerate Mean Field Games system.KeywordsMFGhypoellipticMSC codes35Q9191A1335H10