Monotone solutions to mean field games master equation in the L2-monotone setting

TL;DR

This paper extends the concept of monotone solutions to the mean field game master equation to the displacement monotone setting, establishing existence, uniqueness, and stability results without differentiability assumptions.

Contribution

It introduces a new framework for monotone solutions under displacement monotonicity, broadening the applicability of MFG theory and providing foundational results for existence and stability.

Findings

Existence of monotone solutions under local regularity and strong monotonicity.

Uniqueness and stability of solutions without differentiability assumptions.

Extension to mean field control and forward-backward systems.

Abstract

This paper is concerned with extending the notion of monotone solution to the mean field game (MFG) master equation to situations in which the coefficients are displacement monotone, instead of the previously introduced notion in the flat monotone regime. To account for this new setting, we work directly on the equation satisfied by the controls of the MFG. Following previous works, we define an appropriate notion of solution under which uniqueness and stability results hold for solutions without any differentiability assumption with respect to probability measures. Thanks to those properties, we show the existence of a monotone solution to displacement monotone mean field games under local regularity assumptions on the coefficients and sufficiently strong monotonicity. Albeit they are not the focus of this article, results presented are also of interest for mean field games of control and general mean field forward backward systems. In order to account for this last setting, we use the notion of L2--monotonicity instead of displacement monotonicity, those two notions being equivalent in the particular case of MFG.