Bakry-Émery meet Villani

We revisit Villani's approach to the study of hypocoercive diffusion operators by applying a variant of the Bakry-Émery machinery. The method relies on a generalized Bakry-Émery 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 and somehow improve hypocoercive estimates first obtained by Villani.