Weak solutions to Kolmogorov-Fokker-Planck equations: regularity, existence and uniqueness

We prove existence, uniqueness and regularity of weak solutions of Kolmogorov–Fokker–Planck equations with either local or non-local diffusion in the velocity variable and rough diffusion coefficients or kernels. Our results cover the Cauchy problem and allow a broad class of source terms under minimal assumptions. The core of the analysis is a set of sharp kinetic embeddings à la Lions and transfer-of-regularity results à la Bouchut–H''ormander. We formulate these tools in a homogeneous, scale-invariant form, available for a large range of regularity parameters.