Contributors: Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)); Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité); Dynamical Interconnected Systems in COmplex Environments (DISCO); Inria Saclay - Ile de France; Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire des signaux et systèmes (L2S); CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS); Laboratoire des signaux et systèmes (L2S); CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS); Institut Camille Jordan (ICJ); École Centrale de Lyon (ECL); Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL); Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon); Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS); Modélisation mathématique, calcul scientifique (MMCS); Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL); Public grant as part of the ``Investissement d'avenir'' project, reference ANR-11-LABX-0056-LMH, LabEx LMH, PGMO project VarPDEMFG.; French IDEXLYON project Impulsion ``Optimal Transport and Congestion Games'' PFI 19IA106udl.; Hadamard Mathematics LabEx (LMH), grant number ANR-11-LABX-0056-LMH in the ``Investissement d'avenir'' project.; ANR-16-CE40-0015,MFG,Jeux Champs Moyen(2016); ANR-11-LABX-0056,LMH,LabEx Mathématique Hadamard(2011)
نبذة مختصرة : International audience ; The paper considers a forward-backward system of parabolic PDEs arising in a Mean Field Game (MFG) model where every agent controls the drift of a trajectory subject to Brownian diffusion, trying to escape a given bounded domain $\Omega$ in minimal expected time. Agents are constrained by a bound on the drift depending on the density of other agents at their location. Existence for a finite time horizon $T$ is proven via a fixed point argument, but the natural setting for this problem is in infinite time horizon. Estimates are needed to treat the limit $T\to\infty$, and the asymptotic behavior of the solution obtained in this way is also studied. This passes through classical parabolic arguments and specific computations for MFGs. Both the Fokker--Planck equation on the density of agents and the Hamilton--Jacobi--Bellman equation on the value function display Dirichlet boundary conditions as a consequence of the fact that agents stop as soon as they reach $\partial\Omega$. The initial datum for the density is given, and the long-time limit of the value function is characterized as the solution of a stationary problem.
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