Second order local minimal-time Mean Field Games

TL;DR

This paper analyzes a mean field game model where agents aim to minimize their expected escape time from a domain under drift constraints, establishing existence, asymptotic behavior, and boundary conditions for the solutions of the associated PDE system.

Contribution

It proves existence of solutions for finite time horizons and characterizes the long-time behavior and boundary conditions for the mean field game with drift constraints.

Findings

Existence of solutions for finite time horizon T.

Asymptotic analysis as T approaches infinity.

Characterization of long-time limit as a stationary problem.

Abstract

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.