Mass concentration for Ergodic Choquard Mean-Field Games

TL;DR

This paper analyzes how solutions to certain mean-field game systems with nonlocal interactions behave as diffusion vanishes, showing mass concentrates near potential minima in the limit.

Contribution

It establishes the existence of classical solutions in the vanishing viscosity limit for Riesz-type coupled MFG systems and demonstrates mass concentration near potential minima.

Findings

Existence of classical solutions in the zero-viscosity limit.

Mass concentrates around minima of the external potential.

Behavior characterized in the mass-subcritical regime.

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.