# Weak KAM theory for potential MFG

## TL;DR

This paper extends weak KAM theory to potential mean field games, providing insights into their long-term behavior and establishing the existence of a limit for the value function as time approaches infinity.

## Contribution

It introduces a weak KAM framework for potential MFGs, analyzing their asymptotic properties and connecting the ergodic constant with the Hamilton-Jacobi equation.

## Key findings

- Existence of a limit for the value function as time tends to infinity.
- A mean field limit for the ergodic constant.
- Characterization of long-term behavior of potential MFG systems.

## Abstract

We develop the counterpart of weak KAM theory for potential mean field games. This allows to describe the long time behavior of time-dependent potential mean field game systems. Our main result is the existence of a limit, as time tends to infinity, of the value function of an optimal control problem stated in the space of measures. In addition, we show a mean field limit for the ergodic constant associated with the corresponding Hamilton-Jacobi equation.