Monotonicity Methods for Mean Field Games: A Functional Analytic Perspective

Abstract

This presentation explores mean field games (MFGs) through the lens of functional analysis, focusing on the role of monotonicity methods in understanding their properties and deriving solutions. We begin by introducing MFGs as models for large populations of interacting rational agents, illustrating their derivation for deterministic problems. We then examine key questions of the existence and uniqueness of MFG solutions.

Monotonicity operators emerge as a central tool in our analysis. We establish the connection between monotone operators and variational inequalities, showcasing how the latter offers a flexible framework for addressing situations where traditional solutions may not exist. We then investigate the role of Hessian operators, Bregman divergence, and regularization techniques in obtaining solutions to MFGs within this framework.

Building on this foundation, we extend our discussion to the Banach space setting, examining monotone operators between a Banach space and its dual. We present existence theorems and regularization methods tailored to this context. We conclude by exploring the concept of weak-strong uniqueness, which establishes conditions under which weak and strong solutions of MFGs coincide.

Speaker

Diogo Gomes is Professor of Applied Mathematics and Computational Science (AMCS), and from 2018 also Chair of the AMCS Program. Gomes' research interests are in partial differential equations (PDE), namely on viscosity solutions of elliptic, parabolic and Hamilton-Jacobi equations as well as in related mean-field models. Applications of his work include from computer vision to population dynamics and numerical methods.