Ergodic Mean-Field Games with aggregation of Choquard-type

TL;DR

This paper studies second-order ergodic mean-field games with nonlocal aggregation in unbounded space, establishing conditions for existence and nonexistence of classical solutions based on the strength of attraction and diffusion.

Contribution

It introduces a novel analysis of ergodic MFG systems with Choquard-type nonlocal coupling, identifying regimes for solution existence and nonexistence using Pohozaev identities and fixed point methods.

Findings

Nonexistence of regular solutions in supercritical regime without potential.

Existence of solutions in subcritical regime for small masses.

Threshold for solution existence can be infinite in the subcritical case.

Abstract

We consider second-order ergodic Mean-Field Games systems in the whole space $\mathbb{R}^N$ with coercive potential and aggregating nonlocal coupling, defined in terms of a Riesz interaction kernel. These MFG systems describe Nash equilibria of games with a large population of indistinguishable rational players attracted toward regions where the population is highly distributed. Equilibria solve a system of PDEs where an Hamilton-Jacobi-Bellman equation is combined with a Kolmogorov-Fokker-Planck equation for the mass distribution. Due to the interplay between the strength of the attractive term and the behavior of the diffusive part, we will obtain three different regimes for the existence and non existence of classical solutions to the MFG system. By means of a Pohozaev-type identity, we prove nonexistence of regular solutions to the MFG system without potential in the Hardy-Littlewood-Sobolev-supercritical regime. On the other hand, using a fixed point argument, we show existence of classical solutions in the Hardy-Littlewood-Sobolev-subcritical regime at least for masses smaller than a given threshold value. In the mass-subcritical regime we show that actually this threshold can be taken to be $+\infty$.