McKean-Vlasov Differential Equations: An introduction and a focus on some kinetic models

Abstract

These notes were prepared for a series of lectures delivered at the 43rd Finnish Summer School on Probability and Statistics, held in Lammi, Finland, from May 26 to 30, 2025. They provide an introduction to McKean-Vlasov stochastic differential equations and their connections with non-linear Fokker-Planck equations and mean-field interacting particle systems. Particular attention is devoted to density-dependent equations and to kinetic-type models with degenerate noise. For the latter, we introduce the underlying non-Euclidean geometry and the associated anisotropic and intrinsic Hölder spaces, and present well-posedness and regularity results based on Schauder estimates and superposition principles.