Long time behavior in mean field games systems

Mean field games systems were introduced by J-M. Lasry and P.-L. Lions 
to describe Nash equilibria in multi-agents dynamic optimization. In the 
simplest model, those are forward-backward systems coupling 
Hamilton-Jacobi with Fokker-Planck equations.
In this talk I will discuss the long time behavior of second order 
systems in the periodic case under suitable stability conditions. I will 
go through the main features that appear in the study of the long time 
limit: the ergodic behavior, the effects of the forward-backward 
structure, the exponential turnpike property of the underlying control 
problem and the link with the vanishing discount limit in infinite 
horizon problems. Eventually, I will explain how the problem can be 
lifted by looking at the long time convergence of the master equation 
and the feedback operator.