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

Abstract: In this article, we establish embeddings {`a} la Lions and transfer of regularity {`a} la Bouchut for a large scale of kinetic spaces. We use them to identify a notion of weak solutions to Kolmogorov-Fokker-Planck equations with (local or integral) diffusion and rough (measurable) coefficients under minimal requirements. We prove their existence and uniqueness for a large class of source terms, first in full space for the time, position and velocity variables and then for the kinetic Cauchy problem on infinite and finite time intervals.