On the long time convergence of potential MFG

We look at the long time behavior of potential mean field games (briefly MFG) using some standard tools from weak KAM theory. Potential MFGs are those models where the MFG systems associated can be derived as optimality conditions of suitable optimal control problems on the Fokker-Plank equation. In particular we analyze the relationship between the limit behavior of the time dependent one, whose optimality condition corresponds with the finite horizon MFG system, and the stationary one, whose optimality condition is the ergodic MFG system. We first show that, as the time horizon goes to +infinity, the value of the time dependent optimal control problem converges to a limit -lambda. Then, if we denote with -(lambda) over bar the value of the stationary one, in general we have that lambda >= (lambda) over bar. Moreover, we provide a class of explicit examples where the strict inequality lambda > (lambda) over bar holds true. This will imply that the trajectories of the time-dependent MFG system do not converge to static equilibria.