Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. (English) Zbl 0985.58019

Summary: The authors show that transport inequalities, similar to the one derived by M. Talagrand [Geom. Funct. Anal. 6, 587-600 (1996; Zbl 0859.46030)] for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand’s inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately loc-concave, in a precise sense. All constants are independent of the dimension and optimal in certain cases.
The proofs are based on partial differential equations and an interpolation inequality involving the Wasserstein distance, the entropy functional, and the Fisher information.


58J65 Diffusion processes and stochastic analysis on manifolds
28A35 Measures and integrals in product spaces
60E15 Inequalities; stochastic orderings
60G15 Gaussian processes


Zbl 0859.46030
Full Text: DOI


[1] A. Arnold, P. Markowich, G. Toscani, and, A. Unterreiter, On logarithmic Sobolev inequalities, and the rate of convergence to equilibrium for Fokker–Planck type equations, Comm. Partial Differential Equations, to appear. · Zbl 0982.35113
[2] Arnold, V. I.; Khesin, B. A.: Topological methods in hydrodynamics. (1998) · Zbl 0902.76001
[3] Bakry, D.; Emery, M.: Diffusions hypercontractives. Lecture notes in math. 1123 (1985) · Zbl 0561.60080
[4] Benamou, J. -D.; Brenier, Y.: A numerical method for the optimal time-continuous mass transport problem and related problems. Contemp. math. 226 (1999) · Zbl 0916.65068
[5] Bobkov, S.; Götze, F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. funct. Anal. 163, 1-28 (1999) · Zbl 0924.46027
[6] Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Comm. pure appl. Math. 44, 375-417 (1991) · Zbl 0738.46011
[7] Caffarelli, L. A.: The regularity of mappings with a convex potential. J. amer. Math. soc. 5, 99-104 (1992) · Zbl 0753.35031
[8] Caffarelli, L. A.: Boundary regularity of maps with convex potentials. Comm. pure appl. Math. 45, 1141-1151 (1992) · Zbl 0778.35015
[9] Carlen, E.: Superadditivity of Fisher’s information and logarithmic Sobolev inequalities. J. funct. Anal. 101, 194-211 (1991) · Zbl 0732.60020
[10] Carlen, E.; Soffer, A.: Entropy production by block variable summation and central limit theorems. Comm. math. Phys. 140, 339-371 (1991) · Zbl 0734.60024
[11] Csiszár, I.: Information-type measures of difference of probability distributions and indirect observations. Studia sci. Math. hungar. 2, 299-318 (1967) · Zbl 0157.25802
[12] Cordero-Erausquin, D.: Sur le transport de mesures périodiques. CR acad. Sci. Paris I 329, 199-202 (1999) · Zbl 0942.28015
[13] Evans, L. C.; Gariepy, R. F.: Measure theory and fine properties of functions. (1992) · Zbl 0804.28001
[14] Gangbo, W.: An elementary proof of the polar factorization of vector-valued functions. Arch. rational mech. Anal. 128, 381-399 (1994) · Zbl 0828.57021
[15] Jordan, R.; Kinderlehrer, D.; Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. anal. 29, 1-17 (1998) · Zbl 0915.35120
[16] Holley, R.; Stroock, D.: Logarithmic Sobolev inequalities and stochastic Ising models. J. statist. Phys. 46, 1159-1194 (1987) · Zbl 0682.60109
[17] Kullback, S.: A lower bound for discrimination information in terms of variation. IEEE trans. Inform. 4, 126-127 (1967)
[18] Ledoux, M.: On an integral criterion for hypercontractivity of diffusion semigroups and extremal functions. J. funct. Anal. 105, 444-465 (1992) · Zbl 0759.60079
[19] M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, lectures, Berlin, 1997. · Zbl 0957.60016
[20] M. Ledoux, The geometry of Markov processes, lectures, Zürich, 1998.
[21] Marton, K.: A measure concentration inequality for contracting Markov chains. Geom. funct. Anal. 6, 556-571 (1996) · Zbl 0856.60072
[22] Maurey, B.: Some deviation inequalities. Geom. funct. Anal. 1, 188-197 (1991) · Zbl 0756.60018
[23] Cann, R. J. Mc: Existence and uniqueness of monotone measure-preserving maps. Duke math. J. 80, 309-323 (1995) · Zbl 0873.28009
[24] Mccann, R. J.: A convexity principle for interacting gases. Adv. math. 128, 153-179 (1997) · Zbl 0901.49012
[25] R. J. McCann, Polar factorization on Riemannian manifolds, preprint, 1999.
[26] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, in press. · Zbl 0984.35089
[27] Rachev, S.; Rüschendorf, L.: Mass transportation problems. Probability and its applications (1998) · Zbl 0990.60500
[28] Rothaus, O.: Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities. J. funct. Anal. 42, 102-109 (1981) · Zbl 0471.58027
[29] Saloff-Coste, L.: Convergence to equilibrium and logarithmic Sobolev constant on manifolds with Ricci curvature bounded below. Colloq. math. 67, 109-121 (1994) · Zbl 0816.53027
[30] Talagrand, M.: Transportation cost for Gaussian and other product measures. Geom. funct. Anal. 6, 587-600 (1996) · Zbl 0859.46030
[31] Urbas, J.: On the second boundary value problem for equations of Monge–Ampère type. J. reine angew. Math. 487, 115-124 (1997) · Zbl 0880.35031
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.