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 ${mathbb T}^n$ in [P. Cardaliaguet, {it Long time average of first order mean field games and weak KAM theory}, Dyn. Games Appl. 3 (2013), 473-488], where solutions are shown to converge to the solution of a certain ergodic mean field games system on ${mathbb T}^n$. By adapting the approach in [A. Fathi, E. Maderna, {it Weak KAM theorem on non compact manifolds}, NoDEA Nonlinear Differential Equations Appl. 14 (2007), 1-27], we identify structural conditions on the Lagrangian, under which the corresponding ergodic system can be solved in $mathbb{R}^{n}$. Then we show that time dependent solutions converge to the solution of such a stationary system on all compact subsets of the whole space.
Download