Weak solutions of the master equation for Mean Field Games with no idiosyncratic noise

TL;DR

This paper develops a framework for weak solutions of the master equation in Mean Field Games without idiosyncratic noise, proving existence, uniqueness, and consistency with classical solutions.

Contribution

It introduces a novel notion of weak solutions for the master equation in the absence of idiosyncratic noise and establishes their well-posedness and equivalence with measure-based definitions.

Findings

Existence and uniqueness of weak solutions are proven.

Weak solutions are consistent with classical solutions when smooth.

An equivalent measure-based formulation is provided for the first-order case.

Abstract

We introduce a notion of weak solution of the master equation without idiosyncratic noise in Mean Field Game theory and establish its existence, uniqueness up to a constant and consistency with classical solutions when it is smooth. We work in a monotone setting and rely on Lions' Hilbert space approach. For the first-order master equation without idiosyncratic noise, we also give an equivalent definition in the space of measures and establish the well-posedness.