#### Vol. 12, No. 6, 2019

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Long time behavior of the master equation in mean field game theory

### Pierre Cardaliaguet and Alessio Porretta

Vol. 12 (2019), No. 6, 1397–1453
##### Abstract

Mean field game (MFG) systems describe equilibrium configurations in games with infinitely many interacting controllers. We are interested in the behavior of this system as the horizon becomes large, or as the discount factor tends to $0$. We show that, in these two cases, the asymptotic behavior of the mean field game system is strongly related to the long time behavior of the so-called master equation and to the vanishing discount limit of the discounted master equation, respectively. Both equations are nonlinear transport equations in the space of measures. We prove the existence of a solution to an ergodic master equation, towards which the time-dependent master equation converges as the horizon becomes large, and towards which the discounted master equation converges as the discount factor tends to $0$. The whole analysis is based on new estimates for the exponential rates of convergence of the time-dependent and the discounted MFG systems, respectively.

##### Keywords
mean field games, weak KAM theory, long time behavior
##### Mathematical Subject Classification 2010
Primary: 35B40, 35F21
##### Milestones
Received: 13 October 2017
Revised: 24 July 2018
Accepted: 18 October 2018
Published: 7 February 2019
##### Authors
 Pierre Cardaliaguet Université de Paris-Dauphine PSL Research University CNRS Ceremade Paris France Alessio Porretta Dipartimento di Matematica Università di Roma “Tor Vergata” Roma Italy