The poincaré inequality and quadratic transportation-variance inequalities

Abstract

It is known that the Poincaré inequality is equivalent to the quadratic transportation-variance inequality (namely W22 (fµ, µ) ≤ CV Varµ(f)), see Jourdain [10] and most recently Ledoux [12]. We give two alternative proofs to this fact. In particular, we achieve a smaller CV than before, which equals the double of Poincaré constant. Applying the same argument leads to more characterizations of the Poincaré inequality. Our method also yields a by-product as the equivalence between the logarithmic Sobolev inequality and strict contraction of heat flow in Wasserstein space provided that the Bakry-Émery curvature has a lower bound (here the control constants may depend on the curvature bound). Next, we present a comparison inequality between W22 (fµ, µ) and its centralization W22 (fcµ, µ) for fc =|√f−µ(√f)|2 Var√, which may be viewed as some special counterpart ofµ(f) the Rothaus’ lemma for relative entropy. Then it yields some new bound of W22 (fµ, µ) associated to the variance of√f rather than f. As a by-product, we have another proof to derive the quadratic transportation-information inequality from Lyapunov condition, avoiding the Bobkov-Götze’s characterization of the Talagrand’s inequality.