A Unified Monotonicity Framework for Mean-Field Games
In this talk, we discuss how monotone operator methods provide a unified approach to existence, uniqueness, and regularity in MFGs.
Overview
Mean-field games (MFGs) provide a flexible framework for modeling strategic interactions in large populations of agents, and are typically characterized by a coupled system of Hamilton-Jacobi (HJ) and Fokker-Planck (FP) equations. In this talk, we present a line of work that combines the structure of mean-field game systems with tools from monotone operator theory to develop a unified variational framework for a broad class of problems. This approach yields the existence of weak solutions under minimal structural assumptions and applies to several important settings, including first- and second-order models, degenerate problems, and congestion-type interactions.
To bridge these results with classical literature, we introduce a weak-strong uniqueness principle, proving that monotonicity-induced weak solutions coincide with sufficiently regular strong solutions. Finally, we demonstrate how different regularization strategies yield improved regularity and, in several cases, stronger notions of solutions. These results show that monotone operator methods provide a unified approach to existence, uniqueness, and regularity in MFGs.
Presenters
Brief Biography
Rita Ferreira is a Research Scientist at KAUST in the research group of Prof. Diogo Gomes, within the CEMSE Division. Rita received her dual Ph.D. degree in Mathematics at Carnegie Mellon University (CMU), USA, and at the New University of Lisbon (UNL), Portugal, in 2011. She was supervised by Prof. Irene Fonseca (CMU) and Prof. Luísa Mascarenhas (UNL). Before joining KAUST in 2014, Rita was a Visiting Assistant Professor at the Department of Mathematics of the Faculdade de Ciências e Tecnologia of the New University of Lisbon, Portugal, and a Postdoctoral Research Fellow at Instituto Superior Técnico of the University of Lisbon, Portugal.
