Long time behaviour of mean field games with fractional diffusion

We discuss recent results on the long time behaviour of mean field games systems with fractional diffusion. The dynamics of the players are driven by independent jump processes, controlled through the drift, and confined by an external drift to guarantee ergodicity. In the case of globally Lipschitz, locally uniformly convex Hamiltonians, and weakly coupled and LL monotone costs, we prove that there is a unique solution (u_T,m_T) to the problem in (0,T), and we show that if T is sufficiently large, then (u_T,m_T) satisfies the so-called turnpike property - it is exponentially close to the (unique) stationary ergodic state of the problem.