Mean Field Games systems under displacement monotonicity

TL;DR

This paper establishes the uniqueness, existence, and regularity of solutions for a broad class of Mean Field Games systems with possibly degenerate noise, under a new, more general displacement monotonicity condition.

Contribution

It introduces a sharper displacement monotonicity condition for non-separable Hamiltonians, ensuring well-posedness of MFG systems without relying on master equations.

Findings

Proves uniqueness of solutions under the new monotonicity condition.

Ensures existence and regularity of solutions for long time horizons.

Provides elementary proof techniques independent of master equation well-posedness.

Abstract

In this note we prove the uniqueness of solutions to a class of Mean Field Games systems subject to possibly degenerate individual noise. Our results hold true for arbitrary long time horizons and for general non-separable Hamiltonians that satisfy a so-called $displacement\ monotonicity$ condition. This monotonicity condition that we propose for non-separable Hamiltonians is sharper and more general than the one proposed in our earlier work written jointly with Gangbo and Zhang. The displacement monotonicity assumptions imposed on the data provide actually not only uniqueness, but also the existence and regularity of the solutions. Our analysis uses elementary arguments and does not rely on the well-posedness of the corresponding master equations.