Long-time behaviour of 1st order Mean Field Games systems with non-monotone interactions
speaker: Martino Bardi (Università degli Studi di Padova)abstract: We consider deterministic Mean Field Games (MFG) with a cost functional continuous with respect to the distribution of the agents and bounded away from 0 for large states of the generic player. We first show that the static MFG with such a cost has an equilibrium, and we build from it a solution of the ergodic MFG system of 1st order PDEs with the same cost. The proof is based on 1. Earlier related results use stronger assumptions on the data 2, 3. Next we address the long-time limit of the solutions to finite horizon MFG with cost functional satisfying various additional assumptions. The parabolic system arising when the agents are affected by independent non-degenerate noise was treated by Cardaliaguet, Lasry, Lions, and Porretta. For deterministic MFG the problem was studied by Cardaliaguet on the torus 2 and by Cannarsa et al. in the whole space 3, and both papers use in a crucial way a monotonicity condition on the cost functional. This condition strengthens the classical one by Lasry and Lions that describes a preference for less crowded areas and implies uniqueness of the solution to the MFG system. Our main assumption, instead, is about the set of minima of the cost, motivated by possible applications to global optimization, as in 1. It allows the aggregation of the agents and the existence of multiple solutions. We prove the convergence of the distribution of the agents and of the value function to a solution of the ergodic MFG system as the horizon of the game tends to infinity. This research is part of an ongoing project with Hicham Kouhkouh (Univ. Aachen). 1 M. Bardi, H. Kouhkouh: An Eikonal equation with vanishing Lagrangian arising in Global Optimization, Appl. Math. Optim. 87, 49 (2023). 2 P. Cardaliaguet: Long time average of first order mean field games and weak KAM theory, Dyn. Games Appl. 3 (2013), 473 - 488. 3 P. Cannarsa, W. Cheng, C. Mendico, K. Wang: Long-time behavior of first-order mean field games on Euclidean space, Dyn. Games Appl. 10 (2020), 361 - 390.
timetable:
Mon 8 May, 14:00 - 14:30, Aula Dini
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