Long-time behavior of first-order mean field games on Euclidean space

The aim of this paper is to study the long-time behavior of solutions to deterministic mean field games systems on Euclidean space. This problem was addressed on the torus Tn in Cardaliaguet (Dyn Games Appl 3:473–488, 2013), where solutions are shown to converge to the solution of a certain ergodic mean field games system on Tn. By adapting the approach in Fathi and Maderna (Nonlinear Differ Equ Appl NoDEA 14:1–27, 2007), we identify structural conditions on the Lagrangian, under which the corresponding ergodic system can be solved in Rn. Then, we show that time-dependent solutions converge to the solution of such a stationary system on all compact subsets of the whole space.