Motivated by the problem of proposing mean field game models for crowd motion, this talk considers a mean field game in which agents want to leave a bounded domain in minimal time. In order to model congestion, we assume that the maximal speed of an agent depends on the density of agents at their position. Agents' trajectories are perturbed by additive Brownian motions, assumed to be mutually independent.
After presenting the model, its motivation, and previous results on the first-order case, we formally derive the system of PDEs describing equilibria and turn to the study of existence of solutions. The major difficulty in this analysis is the fact that the Hamilton--Jacobi--Bellman equation on the value function is backwards in time but the time horizon is infinite.
In order to tackle this difficulty, we first introduce an artificial time horizon at some finite time T, penalizing agents who do not leave the domain before time T. We prove existence of equilibria in this fixed time horizon setting and provide estimates on the value function independent of the time horizon T, which allows us to address existence of solutions in the infinite time horizon case by a limit argument. We also characterize the asymptotic behavior of equilibria and discuss the assumptions on the regularity of initial data.
This talk is based on a joint work with Romain Ducasse and Filippo Santambrogio