  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.
