## Electronic Journal of Probability

### The Poincaré inequality and quadratic transportation-variance inequalities

Yuan Liu

#### Abstract

It is known that the Poincaré inequality is equivalent to the quadratic transportation-variance inequality (namely $W_{2}^{2}(f\mu ,\mu ) \leqslant C_{V} \mathrm{Var} _{\mu }(f)$), see Jourdain [10] and most recently Ledoux [12]. 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.

#### Article information

Source
Electron. J. Probab., Volume 25 (2020), paper no. 1, 16 pp.

Dates
Accepted: 8 December 2019
First available in Project Euclid: 3 January 2020

https://projecteuclid.org/euclid.ejp/1578020644

Digital Object Identifier
doi:10.1214/19-EJP403

#### Citation

Liu, Yuan. The Poincaré inequality and quadratic transportation-variance inequalities. Electron. J. Probab. 25 (2020), paper no. 1, 16 pp. doi:10.1214/19-EJP403. https://projecteuclid.org/euclid.ejp/1578020644

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