Electronic Journal of Probability

The Poincaré inequality and quadratic transportation-variance inequalities

Yuan Liu

Full-text: Open access


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

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

Received: 18 June 2019
Accepted: 8 December 2019
First available in Project Euclid: 3 January 2020

Permanent link to this document

Digital Object Identifier

Primary: 26D10: Inequalities involving derivatives and differential and integral operators 60E15: Inequalities; stochastic orderings 60J60: Diffusion processes [See also 58J65]

Poincaré inequality transportation-variance inequality quadratic Wasserstein distance quadratic transportation-information inequality

Creative Commons Attribution 4.0 International License.


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

Export citation


  • [1] Bakry, D., Barthe, F., Cattiaux, P., Guillin, A.: A simple proof of the Poincaré inequality for a large class of measures including the logconcave case, Electron. Commun. Probab. 13 (2008), 60–66.
  • [2] Bakry D., Gentil I., and Ledoux M.: Analysis and geometry of Markov diffusion operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 348. Springer, Cham, 2014.
  • [3] Bobkov S. G., Götze F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal. 163 (1999), 1–28.
  • [4] Cattiaux P., Guillin A.: Functional Inequalities via Lyapunov conditions. In Optimal transportation, Theory and applications, London Mathematical Society Lecture Notes Series, 413, 274–287. Cambridge Univ. Press, 2014.
  • [5] Cattiaux P., Guillin A., and Wu L.-M.: A note on Talagrands transportation inequality and logarithmic Sobolev inequality, Proba. Theory Relat. Fields 148 (2010), no. 1-2, 285–304
  • [6] Ding Y.: A note on quadratic transportation and divergence inequality, Statist. Probab. Lett. 100 (2015), 115–123.
  • [7] Evans L. C.: Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.
  • [8] Gozlan N., Roberto C., and Samson P. M.: A new characterization of Talagrand’s transport-entropy inequalities and applications, Ann. Probab. 39 (2011), no. 3, 857–880.
  • [9] Guillin A., Léonard C., Wu L.-M., and Yao N.: Transportation information inequalities for Markov processes, Probab. Theory Relat. Fields 144 (2009), no. 3-4, 669–696.
  • [10] Jourdain B.: Equivalence of the Poincaré inequality with a transport-chi-square inequality in dimension one, Electron. Commun. Probab. 17 (2012), no. 43, 1–12.
  • [11] Kuwada K.: Duality on gradient estimates and Wasserstein controls, J. Funct. Anal. 258 (2010), 3758–3774.
  • [12] Ledoux M.: Remarks on some transportation cost inequalities, preprint (2018), see the website https://perso.math.univ-toulouse.fr/ledoux/publications-3/.
  • [13] Liu Y.: A new characterization of quadratic transportation-information inequalities, Probab. Theory Related Fields 168 (2017), 675–689.
  • [14] Milman E.: Properties of isoperimetric, functional and transport-entropy inequalities via concentration, Probab. Theory Related Fields 152 (2012), no. 3-4, 475–507.
  • [15] Otto F., Villani C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000), no. 2, 361–400.
  • [16] Villani C.: Topics in Optimal Transportation. Graduate Studies in Mathematics 58, American Mathematical Society, Providence RI, 2003.
  • [17] Villani C.: Optimal Transport: old and new. Grundlehren der Mathematischen Wissenschaf-ten 338, Springer-Verlag, Berlin, 2009.
  • [18] Wang F.-Y.: Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory and Relat. Fields 109 (1997), no. 3, 417–424.
  • [19] Wang F.-Y.: Exponential contraction in Wasserstein distances for diffusion semigroups with negative curvature, Potential Anal., to appear.