  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.
