Long time behavior of the master equation in mean-field game theory
Cardaliaguet, Pierre; Porretta, Alessio (2019), Long time behavior of the master equation in mean-field game theory, Analysis & PDE, 12, 6, p. 1397-1453. 10.2140/apde.2019.12.1397
TypeArticle accepté pour publication ou publié
Journal nameAnalysis & PDE
Mathematical Sciences Publishers
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Dipartimento di Matematica [Roma II] [DIPMAT]
Abstract (EN)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 the two cases, the asymptotic behavior of the Mean Field Game system is strongly related with the long time behavior of the so-called master equation and with 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 the obtention of new estimates for the exponential rates of convergence of the time-dependent MFG system and the discounted MFG system.
Subjects / KeywordsErgodic limit; Mean Field Games; Long time behavior; weak KAM theory
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