Mass concentration for Ergodic Choquard Mean-Field Games

Abstract: We study concentration phenomena in the vanishing viscosity limit for second-order stationary Mean-Field Games systems defined in the whole space $\mathbb{R}^N$ with Riesz-type aggregating nonlocal coupling and external confining potential. In this setting, every player of the game is attracted toward congested areas and the external potential discourages agents to be far away from the origin. Focusing on the mass-subcritical regime $N-\gamma'<\alpha<N$, we study the behavior of solutions in the vanishing viscosity limit, namely when the diffusion becomes negligible. First, we investigate the asymptotic behavior of rescaled solutions as $\varepsilon\to0$, obtaining existence of classical solutions to potential free MFG systems with Riesz-type coupling. Secondly, we prove concentration of mass around minima of the potential.