Rate of Converrgence for ergodic continuous Markov processes : Lyapunov versus Poincare

We study the relationship between two classical approaches for quantitative ergodic properties : the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincar\'e type). We show that they can be linked through new inequalities (Lyapunov-Poincar\'e inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic Fokker-Planck equation recently studied by H\'erau-Nier, Helffer-Nier and Villani is in particular discussed in the final section.