A semi-discrete approximation for first-order stationary mean field games

TL;DR

This paper introduces a semi-discrete approximation scheme for first-order stationary mean field games with separable Hamiltonians, demonstrating convergence of discrete solutions to the continuous problem.

Contribution

It develops a novel discretization method for Hamilton-Jacobi equations in mean field games and proves the convergence of discrete solutions to the continuous solution.

Findings

Established existence of minimizing holonomic measures.

Proved convergence of discrete solutions to the continuous mean field game solution.

Provided a new approximation scheme for stationary mean field games.

Abstract

We provide an approximation scheme for first-order stationary mean field games with a separable Hamiltonian. First, we discretize Hamilton-Jacobi equations by discretizing in time, and then prove the existence of minimizing holonomic measures for mean field games. At last, we obtain two sequences of solutions $\{u_i\}$ of discrete Hamilton-Jacobi equations and minimizing holonomic measures $\{m_i\}$ for mean field games and show that $(u_i,m_i)$ converges to a solution of the stationary mean field games.