Abstract:

The course aims to provide a panoramic overview of McKean-Vlasov (MKV) stochastic differential equations (SDEs), with a particular focus on models featuring degenerate noise (kinetic-type models) and low-regularity coefficients. The course is divided into three parts:

I. We introduce the class of MKV SDEs and explore their links with nonlinear Fokker–Planck equations and mean-field particle systems. We also present examples that illustrate how these connections can be exploited in both directions.

II. We give an overview of the main results in the Lipschitz setting, including strong well-posedness, propagation of chaos (PoC), and particle system simulation. Additionally, we outline some alternative numerical methods, based on analytical approximations and stochastic gradient descent.

III. We begin with examples of relevant models that involve irregular dependence on the state and/or measure variables, as well as degenerate noise. We then review key PDE techniques used in the non-Lipschitz setting and recall essential semigroup estimates in the kinetic (hypoelliptic) framework. Finally, we present recent results on well-posedness and regularity for a class of MKV SDEs with singular (distributional) drift and degenerate noise.