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2019 Long time behavior of the master equation in mean field game theory
Pierre Cardaliaguet, Alessio Porretta
Anal. PDE 12(6): 1397-1453 (2019). DOI: 10.2140/apde.2019.12.1397

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.

Citation

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Pierre Cardaliaguet. Alessio Porretta. "Long time behavior of the master equation in mean field game theory." Anal. PDE 12 (6) 1397 - 1453, 2019. https://doi.org/10.2140/apde.2019.12.1397

Information

Received: 13 October 2017; Revised: 24 July 2018; Accepted: 18 October 2018; Published: 2019
First available in Project Euclid: 12 March 2019

zbMATH: 07061130
MathSciNet: MR3921309
Digital Object Identifier: 10.2140/apde.2019.12.1397

Subjects:
Primary: 35B40, 35F21

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.12 • No. 6 • 2019
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