Mean Field Games in Hilbert Spaces with Degenerate Diffusion: A Viscosity Solution Approach

TL;DR

This paper proves existence and uniqueness of solutions for a degenerate second-order mean field game system in a Hilbert space, using viscosity solutions and fixed-point methods.

Contribution

It extends classical fixed-point and viscosity solution techniques to a degenerate MFG system in infinite-dimensional Hilbert spaces, addressing key uniqueness challenges.

Findings

Established existence of solutions via fixed-point approach.

Proved uniqueness using viscosity solutions and monotonicity methods.

Addressed degeneracy in the Fokker--Planck equation with new adjoint equations.

Abstract

We study a degenerate second order mean field game (MFG) system in a Hilbert space $H$ which couples a Fokker--Planck equation describing the evolution of probability measures on $H$ with a Hamilton--Jacobi--Bellman (HJB) equation for the value function. Our main result establishes existence and uniqueness of solutions to this coupled system. Solutions of the HJB equation are interpreted in the viscosity sense. For existence, we extend the classical fixed-point approach based on Tikhonov's theorem to our setting. A central difficulty in this approach is proving uniqueness for the corresponding linear degenerate Fokker--Planck equation. To address this issue, we introduce a class of suitable adjoint equations and employ viscosity solution techniques to construct sufficiently regular solutions. Uniqueness for the full MFG system is then obtained via an adaptation of the Lasry--Lions monotonicity method.