On stationary fractional Mean Field Games

TL;DR

This paper establishes existence results for stationary fractional mean field game systems with fractional exponents greater than 1/2, covering both local and nonlocal couplings under various growth conditions.

Contribution

It provides the first existence results for stationary fractional MFG systems with fractional exponents above 1/2, including cases with nonlocal regularizing potentials and local couplings.

Findings

Existence of solutions for nonlocal regularizing potentials.

Existence of solutions in the subcritical regime for local couplings.

Conditions on growth of coupling and Hamiltonian for unbounded couplings.

Abstract

We provide an existence result for stationary fractional mean field game systems, with fractional exponent greater than 1/2. In the case in which the coupling is a nonlocal regularizing potential, we obtain existence of solutions under general assumptions on the Hamiltonian. In the case of local coupling, we restrict to the subcritical regime, that is the case in which the diffusion part of the operator dominates the Hamiltonian term. We consider both the case of local bounded coupling and of local unbounded coupling with power-type growth. In this second regime, we impose some conditions on the growth of the coupling and on the growth of the Hamiltonian with respect to the gradient term.