Rough Paths applications to McKean-Vlasov equations with common noise

In this talk we give a brief introduction to the theory of rough paths, that was developed by T. Lyons and studied by many others in the past decades. Then we discuss applications to McKean-Vlasov equations with common noise. Rough path theory builds on the theory of Young integration and aims to make sense of differential equations driven by a continuous path (but cadlag generalizations are possible) that need not be differentiable or of bounded variation. The driving signal can be Hölder continuous for any strictly positive Hölder exponent, in the talk we will only focus on path or regularity between 1/3 and 1/2. Many classical results of stochastic analysis can be recovered using rough path theory. We consider a McKean-Vlasov diffusion with "common" noise perturbed by a deterministic rough paths. These types of law-dependent equations arise as a mean-field limit of systems of interacting particles subject to a common noise that acts on each particle. Finally we give an outlook into applications to filtering theory.