Time periodic solutions of first order mean field games from the perspective of Mather theory

TL;DR

This paper proves the existence of non-trivial time periodic solutions in first order mean field games using Mather theory, and explores their convergence over large times.

Contribution

It introduces a novel approach connecting Mather theory with mean field games to establish periodic solutions and analyzes their long-term behavior.

Findings

Existence of non-trivial time periodic solutions under certain conditions.

Large time convergence of solutions to these periodic solutions.

Application of Mather theory to mean field games.

Abstract

In this paper, the existence of non-trivial time periodic solutions of first order mean field games is proved. It is assumed that there is a non-trivial periodic orbit contained in the Mather set. The whole system is autonomous with a monotonic coupling term. Moreover, the large time convergence of solutions of first order mean field games to time periodic solutions is also considered.