Stationary focusing Mean Field Games

TL;DR

This paper investigates stationary viscous Mean-Field Games with local, decreasing, and unbounded coupling, analyzing how system parameters influence the existence and uniqueness of solutions in ergodic game models.

Contribution

It provides new insights into the existence, non-existence, and uniqueness of solutions for stationary MFG systems with various couplings and Hamiltonian behaviors, using Sobolev regularity and blow-up techniques.

Findings

Existence of solutions depends on the dimension, coupling, and Hamiltonian at infinity.

Under certain conditions, solutions are unique.

New existence results for MFG systems with local increasing coupling.

Abstract

We consider stationary viscous Mean-Field Games systems in the case of local, decreasing and unbounded coupling. These systems arise in ergodic mean-field game theory, and describe Nash equilibria of games with a large number of agents aiming at aggregation. We show how the dimension of the state space, the behavior of the coupling and the Hamiltonian at infinity affect the existence and non-existence of regular solutions. Our approach relies on the study of Sobolev regularity of the invariant measure and a blow-up procedure which is calibrated on the scaling properties of the system. In very special cases we observe uniqueness of solutions. Finally, we apply our methods to obtain new existence results for MFG systems with competition, namely when the coupling is local and increasing.