A note on first order quasi-stationary Mean Field Games

TL;DR

This paper studies first-order quasi-stationary Mean Field Games using weak KAM theory, focusing on conditions that ensure continuity of key structures like the Peierls barrier and Aubry set, especially for perturbed mechanical Hamiltonians.

Contribution

It introduces assumptions on the Hamiltonian and coupling cost to guarantee continuity properties in first-order quasi-stationary Mean Field Games, expanding understanding of their ergodic behavior.

Findings

Continuity of the Peierls barrier over time

Continuity of the Aubry set over time

Applicability to perturbed mechanical Hamiltonians

Abstract

Quasi-stationary Mean Field Games models consider agents who base their strategies on current information without forecasting future states. In this paper we address the first-order quasi-stationary Mean Field Games system, which involves an ergodic Hamilton-Jacobi equation and an evolutive continuity equation. Our approach relies on weak KAM theory. We introduce assumptions on the Hamiltonian and coupling cost to ensure continuity of the Peierls barrier and the Aubry set over time. These assumptions, though restrictive, cover interesting cases such as perturbed mechanical Hamiltonians.