Time dependent mean-field games with logarithmic nonlinearities

TL;DR

This paper establishes the existence of classical solutions for time-dependent mean-field games featuring a challenging logarithmic nonlinearity and subquadratic Hamiltonians, addressing previously unresolved mathematical issues.

Contribution

It introduces a novel proof technique combining Lipschitz regularity, Lebesgue space estimates, and a priori bounds to handle the unbounded logarithmic nonlinearity in mean-field games.

Findings

Existence of classical solutions proven for the model

Development of a new analytical approach for unbounded nonlinearities

Extension of mathematical tools for mean-field game analysis

Abstract

In this paper, we prove the existence of classical solutions for time dependent mean-field games with a logarithmic nonlinearity and subquadratic Hamiltonians. Because the logarithm is unbounded from below, this nonlinearity poses substantial mathematical challenges that have not been addressed in the literature. Our result is proven by recurring to a delicate argument, which combines Lipschitz regularity for the Hamilton-Jacobi equation with estimates for the nonlinearity in suitable Lebesgue spaces. Lipschitz estimates follow from an application of the nonlinear adjoint method. These are then combined with a priori bounds for solutions of the Fokker-Planck equation and a concavity argument for the nonlinearity.