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 $W_2^2(f\mu ,\mu )⩽C_V{\mathrm{Var}}_{\mu }(f)$), see Jourdain \cite{Jourdain} and most recently Ledoux \cite{Ledoux18}. We give two alternative proofs to this fact. In particular, we achieve a smaller $C_V$ 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 $W_2^2(f\mu ,\mu )$ and its centralization $W_2^2(f_c\mu ,\mu )$ for $f_c=\frac{|\sqrt{f}-\mu (\sqrt{f}){|}^2}{{\mathrm{Var}}_{\mu }(\sqrt{f})}$, which may be viewed as some special counterpart of the Rothaus' lemma for relative entropy. Then it yields some new bound of $W_2^2(f\mu ,\mu )$ associated to the variance of $\sqrt{f}$ 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.

17 pages, small mistakes are fixed and the proof of Lemma 3.1 is rewritten due to the referee's comments, one new reference is added