Topics in stochastic differential equations and their applications

We present a selection of research topics concerning the study of stochastic differential equations (SDEs) that arise in social, natural and physical sciences. We discuss two macro-classes: (i) jump-diffusion equations with degenerate behavior of their coefficients, and (ii) mean-field (McKean-Vlasov) diffusion equations. For the class (i) we first describe the main features of the models and the general connection with ultra-parabolic differential operators of Kolmogorov type. We then present some recent developments regarding regularity and asymptotic properties of the transition densities for some specific models. For the class (ii) we first describe the interplay between McKean-Vlasov SDEs, mean-field interacting particle systems, and non-linear Fokker-Planck equations. We then discuss some of the problems related to the existence, uniqueness, asymptotic properties of the solutions, as well as to their numerical approximations. We finally present some recent results in particular cases.