Second order McKean-Vlasov SDEs and kinetic Fokker-Planck-Kolmogorov equations

In this paper, we investigate second-order stochastic differential equations with measurable and density- and distribution-dependent coefficients. By utilizing De Giorgi's method, we establish a maximum principle for kinetic Fokker–Planck–Kolmogorov equations with singular drifts and distribution-valued inhomogeneous term. This result enables us to demonstrate the existence of weak solutions when the coefficients are density-dependent and the drift interaction kernel belongs to mixed $ L^q_t L^p_x $ spaces. Moreover, leveraging the Hölder regularity estimates recently developed in [10], we establish the well-posedness of generalized martingale problems under the assumption that the diffusion coefficients depend only on the position variable (without requiring continuity). Notably, our work appears to be the first to address the well-posedness of kinetic stochastic differential equations with measurable diffusion coefficients.