Well-posedness of mean field games master equations involving non-separable local Hamiltonians

TL;DR

This paper establishes short-time classical solutions for a class of mean field games master equations with non-separable local Hamiltonians and measure-dependent terms, expanding the theoretical understanding of such equations.

Contribution

It provides the first construction of solutions without requiring convexity or monotonicity assumptions on the Hamiltonians.

Findings

Solutions exist for smooth Hamiltonians in Sobolev spaces.

No additional structural conditions like convexity are needed.

Results apply to measures with densities in suitable Sobolev spaces.

Abstract

In this paper we construct short time classical solutions to a class of master equations in the presence of non-degenerate individual noise arising in the theory of mean field games. The considered Hamiltonians are non-separable and $local$ functions of the measure variable, therefore the equation is restricted to absolutely continuous measures whose densities lie in suitable Sobolev spaces. Our results hold for smooth enough Hamiltonians, without any additional structural conditions as convexity or monotonicity.