A Discrete Weak KAM Method for First-Order Stationary Mean Field Games

Abstract

We provide a discrete weak KAM method for finding solutions of a class of first-order stationary mean field games systems. Especially, such solutions have clear dynamical meaning. First, we discretize Lax–Oleinik equations by discretizing in time, and then we prove the existence of minimizing holonomic measures for mean field games. We obtain two sequences of solutions of discrete Lax–Oleinik equations and minimizing holonomic measures for mean field games and show that converges to a solution of the stationary mean field games systems. Finally, we briefly describe how to implement a discretization in the space variable also.