Long time behavior of the master equation in mean field game theory. (English) Zbl 1428.35607
Summary: 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.
MSC:
| 35Q89 | PDEs in connection with mean field game theory |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35F21 | Hamilton-Jacobi equations |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 37K55 | Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems |
| 91A16 | Mean field games (aspects of game theory) |
Keywords:
weak KAM theory; nonlinear transport equations; ergodic master equation; time-dependent master equationReferences:
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