Bakry-Emery meet Villani

Abstract: We revisit Villani's approach to the study of hypocoercive diffusion operators by applying a variant of the Bakry-\'Emery machinery. The method relies on a generalized Bakry-\'Emery type criterion that applies to Kolmogorov type operators. Our approach includes as a special case the kinetic Fokker-Planck equation and allows, in that case, to recover hypocoercive estimates first obtained by Villani.