On the existence and uniqueness of solutions to time-dependent fractional MFG

TL;DR

This paper proves existence and uniqueness of solutions for time-dependent fractional Mean Field Game systems across all fractional orders, distinguishing between classical and distributional solutions depending on the fractional parameter.

Contribution

It introduces a novel functional framework for fractional MFGs and establishes existence and uniqueness results for all fractional orders, including subcritical regimes.

Findings

Existence of solutions via vanishing viscosity method.

Classical solutions for s>1/2, distributional solutions for s≤1/2.

Uniqueness under monotonicity and short time conditions.

Abstract

We establish existence and uniqueness of solutions to evolutive fractional Mean Field Game systems with regularizing coupling, for any order of the fractional Laplacian $s\in(0,1)$. The existence is addressed via the vanishing viscosity method. In particular, we prove that in the subcritical regime $s>1/2$ the solution of the system is classical, while if $s\leq 1/2$ we find a distributional energy solution. To this aim, we develop an appropriate functional setting based on parabolic Bessel potential spaces. We show uniqueness of solutions both under monotonicity conditions and for short time horizons.