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Published in:
17-07-2019
Long-Time Behavior of First-Order Mean Field Games on Euclidean Space
Published in: Dynamic Games and Applications | Issue 2/2020
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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 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 \({\mathbb {T}}^n\). 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 \({\mathbb {R}}^{n}\). Then, we show that time-dependent solutions converge to the solution of such a stationary system on all compact subsets of the whole space.