# Long Time Behavior of First Order Mean Field Games on Euclidean Space

arXiv: https://arxiv.org/abs/1809.09057 · HTML: https://mlsanity.com/paper/1809.09057/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Sep 2018.

## Built on by (most cited first, 2 of 2)

- [Decoding Mean Field Games from Population and Environment Observations By Gaussian Processes](https://mlsanity.com/paper/2312.06625/) (2023)
- [Learning Surrogate Potential Mean Field Games via Gaussian Processes: A Data-Driven Approach to Ill-Posed Inverse Problems](https://mlsanity.com/paper/2502.11506/) (2025)

## Abstract (arXiv)

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 Tn 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 Tn. 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 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.