We consider discounted infinite horizon mean field games in the probabilistic weak formulation. Well-posedness of this problem can be shown by properly adapting the finite horizon approach, however, additional care with the previously considered topologies is required due to the appearance of singular measures. Further, we study the long-time asymptotic behavior of mean field games. We show that solutions to the infinite horizon game are approximate solutions for the finite horizon game. Additionally, we provide a tightness result for mean field games of different time horizons and show that finite horizon solutions converge to infinite horizon solutions in a general setting. Under a weakened Lasry-Lions monotonicity condition, we can quantify this convergence.

This is joint work with René Carmona and Ludovic Tangpi.