McKean—Vlaosv SDEs: New results on existence of weak solutions and on propagation of chaos

Robert Crowell has just defended his thesis, on which his talk is based on.
Abstract:  

We consider the existence of weak solutions of McKean-Vlasov SDEs with common noise and the propagation of chaos for the associated weakly interacting finite particle systems. Our strategy consists of two main components allowing us to analyze settings with general nonlinear but uniformly elliptic coefficients possessing only low spatial regularity through a marriage of probabilistic and analytic techniques. First, we explore the emergence of regularity in limit points of McKean-Vlasov particle systems, leading to a priori regularity estimates for large-system limits of the empirical measure flows from finite particle systems. Second, we leverage this regularity to establish the existence of weak solutions for McKean-Vlasov SDEs and to identify more nuanced conditions under which chaos propagates, i.e. under which an asymptotic decoupling of the particles takes place and the dynamics in large systems become conditionally independent in law. Next to its applicability to low-regularity regimes, the approach we take to obtain weak solutions and the propagation of chaos may also be useful in future applications to mean-field games and controlled problems.