Long time behaviour of Mean Field Games with fractional diffusion

TL;DR

This paper investigates the long-term behavior of mean field games with fractional diffusion, showing that solutions tend to a stationary state exponentially fast when the time horizon is large, under certain conditions.

Contribution

It establishes the existence, uniqueness, and turnpike property of solutions for mean field games with fractional diffusion and specific Hamiltonian and cost conditions.

Findings

Existence and uniqueness of solutions for large time horizons.

Solutions exhibit exponential convergence to a stationary ergodic state.

Turnpike property holds under specified conditions.

Abstract

In this paper we study the long time behaviour of mean field games systems with fractional diffusion, modeling the case that the individual dynamics of the players is driven by independent jump processes and controlled through the drift term, while being confined by an external field in order to guarantee ergodicity. In the case of globally Lipschitz, locally uniformly convex Hamiltonian, and weakly coupled costs satisfying the Lasry-Lions monotonicity condition, we prove that there is a unique solution $(u_T,m_T)$ to the mean field game problem in $(0,T)$ and we show that, if $T$ is sufficiently large, $(u_T,m_T)$ satisfies the so-called turnpike property, namely it is exponentially close to the (unique) stationary ergodic state for any proportionally long intermediate time.