On Mean Field Games in Infinite Dimension

TL;DR

This paper investigates a mean field game system in an infinite-dimensional Hilbert space, establishing well-posedness, existence, and uniqueness of solutions for coupled parabolic equations with Kolmogorov operators.

Contribution

It extends mean field game theory to infinite dimensions, proving well-posedness and solution existence using fixed point theorems and monotonicity conditions.

Findings

Proved well-posedness of the MFG system in infinite dimensions.

Established existence of solutions via Tikhonov's fixed point theorem.

Demonstrated uniqueness under monotonicity conditions.

Abstract

We study a Mean Field Games (MFG) system in a real, separable infinite dimensional Hilbert space. The system consists of a second order parabolic type equation, called Hamilton-Jacobi-Bellman (HJB) equation in the paper, coupled with a nonlinear Fokker-Planck (FP) equation. Both equations contain a Kolmogorov operator. Solutions to the HJB equation are interpreted in the mild solution sense and solutions to the FP equation are interpreted in an appropriate weak sense. We prove well-posedness of the considered MFG system under certain conditions. The existence of a solution to the MFG system is proved using Tikhonov's fixed point theorem in a proper space. Uniqueness of solutions is obtained under typical separability and Lasry-Lions type monotonicity conditions.