The local Turnpike Property in Mean Field Control and Games with quadratic Hamiltonian

We study the local stability properties of solutions to ergodic and discounted mean field games systems, as the time horizon $T \to +\infty$, around stationary equilibria, when the Hamiltonian is quadratic. We replace the usual monotonicity of the coupling term with a weaker, local assumption on the stationary equilibrium (that need not be unique), stemming from a second-order strict positivity condition. This new stability assumption, together with a symmetry property of the system, allows us to derive an exponential turnpike property for those solutions that are close to the stationary one, whenever the spatial domain $Ω$ is either the flat torus $\mathbb{T}^n$ or $\mathbb{R}^n$. Finally, through a fixed-point argument, we establish the actual existence of stable solutions, both on the finite horizon $[0,T]$ and on the infinite horizon, in the periodic setting $Ω=\mathbb{T}^n$, provided that the initial (and terminal) data are close enough to the stationary equilibrium.