Second order mean field games with degenerate diffusion and local coupling

TL;DR

This paper establishes the existence, uniqueness, and stability of weak solutions for a class of second order mean field games systems that include degenerate diffusion and local coupling, broadening the understanding of such models.

Contribution

It introduces a framework for analyzing degenerate second order mean field games with local coupling, proving existence, uniqueness, and stability of weak solutions.

Findings

Weak solutions exist and are unique.

Solutions can be approximated by viscous perturbations.

Solutions are stable with respect to data variations.

Abstract

We analyze a (possibly degenerate) second order mean field games system of partial differential equations. The distinguishing features of the model considered are (1) that it is not uniformly parabolic, including the first order case as a possibility, and (2) the coupling is a local operator on the density. As a result we look for weak, not smooth, solutions. Our main result is the existence and uniqueness of suitably defined weak solutions, which are characterized as minimizers of two optimal control problems. We also show that such solutions are stable with respect to the data, so that in particular the degenerate case can be approximated by a uniformly parabolic (viscous) perturbation.