On the long time convergence of potential MFG

TL;DR

This paper investigates the long-term behavior of potential Mean Field Games using weak KAM theory, showing convergence to an ergodic constant and highlighting cases where trajectories do not settle into static equilibria.

Contribution

It demonstrates the convergence of the time-dependent MFG minimization problem to an ergodic constant and identifies conditions where static equilibria are not reached.

Findings

Time-dependent minimization converges to an ergodic constant.

Existence of examples where stationary MFG value exceeds the ergodic limit.

Trajectories may not converge to static equilibria in certain cases.

Abstract

We look at the long time behavior of potential Mean Field Games (briefly MFG) using some standard tools from weak KAM theory. We first show that the time-dependent minimization problem converges to an ergodic constant $-\lambda$, then we provide a class of examples where the value of the stationary MFG minimization problem is strictly greater than $-\lambda$. This will imply that the trajectories of the time-dependent MFG system do not converge to static equilibria.