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

Panrui Ni

TL;DR

This work addresses the existence of non-trivial time-periodic solutions for first-order mean field games with a monotone, spatially homogeneous coupling and a fixed horizon. It leverages weak KAM and Aubry–Mather theory to relate long-time dynamics to Mather’s minimal measures, proving the existence of time-periodic states when a non-trivial periodic orbit lies in the Mather set. The results establish regularity, uniqueness up to constants, and Lipschitz dependence of the spectral parameter $c(m_T)$, along with large-time convergence of solutions to the periodic regime. A concrete torus example illustrates a smooth time-periodic solution, highlighting the connection between MFG dynamics and minimal-measure geometry. These findings deepen the understanding of long-time behavior in MFGs by tying it to Mather theory and the structure of minimal measures.

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.

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

TL;DR

This work addresses the existence of non-trivial time-periodic solutions for first-order mean field games with a monotone, spatially homogeneous coupling and a fixed horizon. It leverages weak KAM and Aubry–Mather theory to relate long-time dynamics to Mather’s minimal measures, proving the existence of time-periodic states when a non-trivial periodic orbit lies in the Mather set. The results establish regularity, uniqueness up to constants, and Lipschitz dependence of the spectral parameter , along with large-time convergence of solutions to the periodic regime. A concrete torus example illustrates a smooth time-periodic solution, highlighting the connection between MFG dynamics and minimal-measure geometry. These findings deepen the understanding of long-time behavior in MFGs by tying it to Mather theory and the structure of minimal measures.

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.
Paper Structure (8 sections, 10 theorems, 73 equations)

This paper contains 8 sections, 10 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['m1']}
  3. Basic properties of (\ref{['ME1']})-(\ref{['ME3']})
  4. Proof of Theorem \ref{['m2']}
  5. Proof of Item (1)
  6. Proof of Item (2)
  7. Proof of Item (3)
  8. Proof of Item (4)

Key Result

Theorem 1

Assume (H1)-(H3)(F) and $m_T\in\mathfrak N$, then there is $c(m_T)\in\mathbb R$ such that the system (ME1)-(ME3) has a non-trivial time periodic solution with period $\tau$ when $c=c(m_T)$.

Theorems & Definitions (22)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.1
  • Theorem 1
  • Theorem 2
  • Remark 1.2
  • Example 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 12 more