Second order local minimal-time Mean Field Games

Romain Ducasse (1) , Guilherme Mazanti (2, 3) , Filippo Santambrogio (4)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 DISCO - Dynamical Interconnected Systems in COmplex Environments
3 L2S - Laboratoire des signaux et systèmes
4 ICJ - Institut Camille Jordan

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.

Mots clés: Mean Field Games; Congestion games; Parabolic PDEs; MFG system; Existence of solutions; Asymptotic behavior

Domaines: Equations aux dérivées partielles [math.AP]

Romain Ducasse, Guilherme Mazanti, Filippo Santambrogio. Second order local minimal-time Mean Field Games. 2021. ⟨hal-02616965v2⟩