On quadratic transportation cost inequalities

Abstract In this paper we study quadratic transportation cost inequalities. To this end we introduce new families of inequalities (for quadratic transportation cost and for relative entropy) that are shown to be equivalent to the Poincare inequality. This allows us to give some examples of measures satisfying T 2 but not the logarithmic-Sobolev inequality.

[1]  M. Rao,et al.  Theory of Orlicz spaces , 1991 .

[2]  C. McDiarmid.  Concentration , 1862, The Dental register.

[3]  F. Barthe,et al.  Sobolev inequalities for probability measures on the real line , 2003 .

[4]  P. Cattiaux.  Hypercontractivity for perturbed diffusion semigroups , 2005, math/0510258.

[5]  F. Barthe,et al.  Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry , 2004, math/0407219.

[6]  M. Ledoux.  Concentration of measure and logarithmic Sobolev inequalities , 1999 .

[7]  S. Bobkov,et al.  Hypercontractivity of Hamilton-Jacobi equations , 2001 .

[8]  K. Marton.  Bounding $\bar{d}$-distance by informational divergence: a method to prove measure concentration , 1996 .

[9]  Feng-Yu Wang.  Probability distance inequalities on Riemannian manifolds and path spaces , 2004 .

[10]  Dario Cordero-Erausquin,et al.  Some Applications of Mass Transport to Gaussian-Type Inequalities , 2002 .

[11]  A. Guillin,et al.  Transportation cost-information inequalities and applications to random dynamical systems and diffusions , 2004, math/0410172.

[12]  P. Cattiaux.  A Pathwise Approach of Some Classical Inequalities , 2004 .

[13]  C. Villani,et al.  Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities , 2005 .

[14]  P. Cattiaux,et al.  Minimization of the Kullback information for some Markov processes , 1996 .

[15]  P. Meyer,et al.  Sur les inegalites de Sobolev logarithmiques. I , 1982 .

[16]  Feng-Yu Wang.  A Generalization of Poincaré and Log-Sobolev Inequalities , 2005 .

[17]  Djalil CHAFAÏ.  On Φ-entropies and Φ-Sobolev inequalities , 2002 .

[18]  S. Bobkov,et al.  Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution , 1997 .

[19]  S. Bobkov,et al.  Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities , 1999 .

[20]  Feng-Yu Wang,et al.  Logarithmic Sobolev inequalities on noncompact Riemannian manifolds , 1997 .

[21]  C. Villani.  Topics in Optimal Transportation , 2003 .

[22]  R. Latala,et al.  Between Sobolev and Poincaré , 2000, math/0003043.

[23]  C. Villani,et al.  Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality , 2000 .

[24]  Djalil CHAFAÏ.  Entropies, convexity, and functional inequalities : On Phi-entropies and Phi-Sobolev inequalities , 2004 .

[25]  G. Royer,et al.  Une initiation aux inégalités de Sobolev logarithmiques , 1999 .

[26]  M. Talagrand.  Transportation cost for Gaussian and other product measures , 1996 .

[27]  LOGARITHMIC SOBOLEV INEQUALITIES: CONDITIONS AND COUNTEREXAMPLES , 2001 .