A new characterization of Talagrand’s transport-entropy inequalities and applications. (English) Zbl 1233.60007

The starting point of the paper under review is Talagrand’s transportation cost inequality for \(\mu, \nu\in M^1(\mathbb{R}^k)\), a cost function \(c:\mathbb{R}^k\to\mathbb{R}_+\) (with \(c(0)=\nabla c(0)=0\)) and a constant \(C>0\): \[ T_c(\nu,\mu) \leq C\cdot H(\nu|\mu), \tag{\(T_c(C)\)} \] where the optimal transport costs are defined as
\[ T_c(\nu,\mu) = \inf_\pi \iint c(x-y)d\pi(x,y), \]
the infimum is taken over all \(\pi\in M^1(\mathbb{R}^{2k})\) with marginals \(\nu\) and \(\mu\), respectively, and the relative entropy is respectively defined as \(H(\nu|\mu)=\int \log(d\nu/d\mu) d\nu\) if \(\nu\) is absolutely continuous with respect to \(\mu\), and \(H(\nu|\mu)=\infty\) otherwise, cf., e.g., for Gaussian laws and quadratic cost functions [M. Talagrand, “Transportation cost for Gaussian and other product measures”, Geom. Funct. Anal. 6, No. 3, 587–600 (1996; Zbl 0859.46030)]. These inequalities are closely related to logarithmic Sobolev inequalities, c.f., e.g., [F. Otto and C. Villani, “Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality”, J. Funct. Anal. 173, No. 2, 361–400 (2000; Zbl 0985.58019)].
In the present investigation, the authors prove (among other results of independent interest) the equivalence of \((T_c(C_1))\) with restricted versions of logarithmic Sobolev inequalities (Theorem 1.5), as \[ \text{Ent}_\mu(e^f) \leq \eta/(1-C_2(\eta+K))\int c^*(\nabla f/\eta)e^f d\mu \] and \[ \text{Ent}_\mu(e^f) \leq 2C_3/(1-KC_3)^2 \int |\nabla f|^2 e^f d\mu \] for \(f\) belonging to a class of functions (\(K\) semi convex functions) \(K, C_i,\eta\) denoting suitable positive constants and
\[ \begin{aligned}\text{Ent}_\mu(g)&:=\int g\log g d\mu - \int g d\mu \log\int g d\mu, \\ c^*(u)&:= \sup_{h\in\mathbb{R}^k} \{\langle u,h\rangle -c(h)\}, \end{aligned} \]
and particular cost functions \(c(x):=\sum_1^k\alpha_i(x_i)\) with concave \(\alpha_i\in C^1(\mathbb{R})\), \(\alpha_i(0)=\alpha_i'(0)=0\).
The paper closes with possible extensions of the main results, as extensions to Riemannian manifolds, relations to other types of logarithmic Sobolev inequalities, and to Poincaré’s inequality. (For background, see, e.g., [S. G. Bobkov, I. Gentil and M. Ledoux, “Hypercontractivity of Hamilton-Jacobi equations”, J. Math. Pures Appl. (9) 80, No. 7, 669–696 (2001; Zbl 1038.35020); S. G. Bobkov and M. Ledoux, “Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution”, Probab. Theory Relat. Fields 107, No. 3, 383–400 (1997; Zbl 0878.60014)]).


60E15 Inequalities; stochastic orderings
60F10 Large deviations
26D10 Inequalities involving derivatives and differential and integral operators
Full Text: DOI arXiv


[1] Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C. and Scheffer, G. (2000). Sur les Inégalités de Sobolev Logarithmiques. Panoramas et Synthèses [ Panoramas and Syntheses ] 10 . Société Mathématique de France, Paris. · Zbl 0982.46026
[2] Balogh, Z., Engoulatov, A., Hunziker, L. and Maasalo, O. E. (2009). Functional inequalities and Hamilton-Jacobi equations in geodesic spaces. Preprint. Available at .
[3] Barthe, F. and Kolesnikov, A. V. (2008). Mass transport and variants of the logarithmic Sobolev inequality. J. Geom. Anal. 18 921-979. · Zbl 1170.46031
[4] Barthe, F. and Roberto, C. (2008). Modified logarithmic Sobolev inequalities on \Bbb R. Potential Anal. 29 167-193. · Zbl 1170.26010
[5] Bobkov, S. G., Gentil, I. and Ledoux, M. (2001). Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pures Appl. (9) 80 669-696. · Zbl 1038.35020
[6] Bobkov, S. G. and Götze, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 1-28. · Zbl 0924.46027
[7] Bobkov, S. and Ledoux, M. (1997). Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution. Probab. Theory Related Fields 107 383-400. · Zbl 0878.60014
[8] Cattiaux, P. and Guillin, A. (2006). On quadratic transportation cost inequalities. J. Math. Pures Appl. (9) 86 341-361. · Zbl 1118.58017
[9] Cheeger, J. and Ebin, D. G. (2008). Comparison Theorems in Riemannian Geometry . AMS Chelsea Publishing, Providence, RI. · Zbl 1142.53003
[10] Evans, L. C. (1998). Partial Differential Equations. Graduate Studies in Mathematics 19 . Amer. Math. Soc., Providence, RI. · Zbl 0902.35002
[11] Gentil, I., Guillin, A. and Miclo, L. (2005). Modified logarithmic Sobolev inequalities and transportation inequalities. Probab. Theory Related Fields 133 409-436. · Zbl 1080.26010
[12] Gentil, I., Guillin, A. and Miclo, L. (2007). Modified logarithmic Sobolev inequalities in null curvature. Rev. Mat. Iberoamericana 23 235-258. · Zbl 1123.26022
[13] Gozlan, N. (2007). Characterization of Talagrand’s like transportation-cost inequalities on the real line. J. Funct. Anal. 250 400-425. · Zbl 1135.46022
[14] Gozlan, N. (2009). A characterization of dimension free concentration in terms of transportation inequalities. Ann. Probab. 37 2480-2498. · Zbl 1201.60016
[15] Gozlan, N. and Léonard, C. (2010). Transport inequalities. A survey. Markov Process. Related Fields . To appear. Available at . · Zbl 1229.26029
[16] Gross, L. (1975). Logarithmic Sobolev inequalities. Amer. J. Math. 97 1061-1083. JSTOR: · Zbl 0318.46049
[17] Holley, R. and Stroock, D. (1987). Logarithmic Sobolev inequalities and stochastic Ising models. J. Stat. Phys. 46 1159-1194. · Zbl 0682.60109
[18] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89 . Amer. Math. Soc., Providence, RI. · Zbl 0995.60002
[19] Lott, J. and Villani, C. (2007). Hamilton-Jacobi semigroup on length spaces and applications. J. Math. Pures Appl. (9) 88 219-229. · Zbl 1210.53047
[20] Marton, K. (1986). A simple proof of the blowing-up lemma. IEEE Trans. Inform. Theory 32 445-446. · Zbl 0594.94003
[21] Maurey, B. (1991). Some deviation inequalities. Geom. Funct. Anal. 1 188-197. · Zbl 0756.60018
[22] Milman, E. (2010). Isoperimetric and concentration inequalities. Equivalence under curvature lower bound. Duke Math. J. 154 207-239. · Zbl 1205.53038
[23] Milman, E. (2010). Properties of isoperimetric, functional and transport-entropy inequalities via concentration. Preprint. Available at . · Zbl 1205.53038
[24] Ohta, S.-I. (2009). Gradient flows on Wasserstein spaces over compact Alexandrov spaces. Amer. J. Math. 131 475-516. · Zbl 1170.68033
[25] Otto, F. and Villani, C. (2000). Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 361-400. · Zbl 0985.58019
[26] Savaré, G. (2007). Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds. C. R. Math. Acad. Sci. Paris 345 151-154. · Zbl 1125.53064
[27] Stam, A. J. (1959). Some inequalities satisfied by the quantities of information of Fisher and Shannon. Information and Control 2 101-112. · Zbl 0085.34701
[28] Talagrand, M. (1991). A new isoperimetric inequality and the concentration of measure phenomenon. In Geometric Aspects of Functional Analysis (1989-90). Lecture Notes in Math. 1469 94-124. Springer, Berlin. · Zbl 0818.46047
[29] Talagrand, M. (1996). Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 587-600. · Zbl 0859.46030
[30] Villani, C. (2009). Optimal Transport : Old and New. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 338 . Springer, Berlin. · Zbl 1156.53003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.