Short time solution to the master equation of a first order mean field game system

TL;DR

This paper proves the existence of short-time classical solutions to the master equation in first-order mean field games, broadening the class of Hamiltonians and removing previous restrictions.

Contribution

It extends prior results by establishing short-time solutions without monotonicity or convexity assumptions in potential mean field games.

Findings

Existence of short-time classical solutions proven.

Broader class of Hamiltonians covered.

No monotonicity or convexity restrictions required.

Abstract

The goal of this paper is to show existence of short-time classical solutions to the so called Master Equation of \emph{first order} Mean Field Games, which can be thought of as the limit of the corresponding master equation of a stochastic mean field game as the individual noises approach zero. Despite being the equation of an idealistic model, its study is justified as a way of understanding mean field games in which the individual players'~randomness is negligible; in this sense it can be compared to the study of ideal fluids \cite{gangboberkeleynotes}. We restrict ourselves to potential mean field games but do not impose any monotonicity conditions on the running and initial costs, and we do not require convexity of the Hamiltonian, thus extending the result of \cite{mfgmain} to a considerably broader class of Hamiltonians.