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

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 dis-cretize 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 {ui\} of discrete Lax-Oleinik equations and minimizing holonomic measures {mi\} for mean field games and show that (ui, mi) 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.

Automated summary

We propose a discrete weak KAM method for solving a class of first-order stationary mean field games systems, where the solutions have clear dynamical meaning. By discretizing Lax-Oleinik equations in time, we prove the existence of minimizing holonomic measures for mean field games. We obtain sequences of solutions for discrete Lax-Oleinik equations and minimizing holonomic measures for mean field games, and show that they converge to a solution of the stationary mean field games systems. Additionally, we discuss the implementation of a discretization in the space variable.