Introduction and main concepts

Lyapunov conditions appeared a long time ago. They were particularly well fitted to deal with the problem of convergence to equilibrium for Markov processes; see [23, 38–40] and references therein. They also appeared earlier in the study of large and moderate deviations for empirical functionals of Markov processes (for examples, see Donsker and Varadhan [21, 22], Kontoyaniis and Meyn [33, 34], Wu [47, 48], Guillin [28, 29]), for solving the Poisson equation [24].

Their use to obtain functional inequalities is however quite recent, even if one may afterwards find hint of such an approach in Deuschel and Stroock [19] or Kusuocka and Stroock [35]. The present authors and coauthors have developed a methodology that has been successful for various inequalities: Lyapunov–Poincaré inequalities [4], Poincaré inequalities [3], transportation inequalities for Kullback information [17] or Fisher information [32], super Poincaré inequalities [16], weighted and weak Poincaré inequalities [13], or [18] for super weighted Poincaré inequalities. We finally refer to the forthcoming book [15] for a complete review. For more references on the various inequalities introduced here we refer to [1, 2, 36, 46]. The goal of this short review is to explain the methodology used in these papers and to present various general sets of conditions for this panel of functional inequalities. The proofs will of course be only schemed and we will refer to the original papers for complete statements.