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

TL;DR

This paper investigates the long-term behavior of solutions to first order mean field games on Euclidean space, establishing convergence to ergodic solutions under certain conditions, extending previous results from the torus to the entire space.

Contribution

It adapts weak KAM theory to Euclidean space, identifying conditions for ergodic solutions and proving convergence of solutions in unbounded domains.

Findings

Solutions converge to ergodic mean field game solutions on compact subsets.

Structural conditions on the Lagrangian enable solving ergodic systems in .

Time-dependent solutions approach stationary solutions over time.

Abstract

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.