Long time average of first order mean field games and weak KAM theory

TL;DR

This paper demonstrates that the long-term average behavior of first-order mean field game solutions is described by an ergodic system, with well-posedness and uniqueness established using weak KAM theory.

Contribution

It introduces a novel connection between long time averages of mean field games and ergodic systems via weak KAM theory, providing new insights into their asymptotic behavior.

Findings

Long time average solutions are governed by an ergodic mean field game system.

Well-posedness and uniqueness of the ergodic system are established.

Weak KAM theory is used to analyze the ergodic properties.

Abstract

We show that the long time average of solutions of first order mean field game systems in finite horizon is governed by an ergodic system of mean field game type. The well-posedness of this later system and the uniqueness of the ergodic constant rely on weak KAM theory.