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On the long time convergence of potential MFG

  • Marco MasoeroEmail author
Article
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Abstract

We look at the long time behavior of potential mean field games (briefly MFG) using some standard tools from weak KAM theory. Potential MFGs are those models where the MFG systems associated can be derived as optimality conditions of suitable optimal control problems on the Fokker–Plank equation. In particular we analyze the relationship between the limit behavior of the time dependent one, whose optimality condition corresponds with the finite horizon MFG system, and the stationary one, whose optimality condition is the ergodic MFG system. We first show that, as the time horizon goes to \(+\infty \), the value of the time dependent optimal control problem converges to a limit \(-\lambda \). Then, if we denote with \(-{\bar{\lambda }}\) the value of the stationary one, in general we have that \(\lambda \ge {\bar{\lambda }}\). Moreover, we provide a class of explicit examples where the strict inequality \(\lambda >{\bar{\lambda }}\) holds true. This will imply that the trajectories of the time-dependent MFG system do not converge to static equilibria.

Keywords

Mean field games Weak KAM theory Ergodicity PDE control 

Mathematics Subject Classification

49J20 37K55 37A99 

Notes

Acknowledgements

I would like to thank Pierre Cardaliaguet (Paris Dauphine) for the fruitful discussions all along this work and Marco Cirant (Università di Padova) for a crucial hint which resulted in Lemma 4.2. Moreover, I wish to express gratitude to the anonymous referees for their very careful reading of the manuscript and their valuable advices.

I was partially supported by the ANR (Agence Nationale de la Recherche) Project ANR-16-CE40-0015-01.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Université Paris-DauphineParisFrance

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