R. Adamczak, Logarithmic Sobolev Inequalities and Concentration of Measure for Convex Functions and Polynomial Chaoses, Bulletin of the Polish Academy of Sciences Mathematics, vol.53, issue.2, pp.221-238, 2005.
DOI : 10.4064/ba53-2-10

N. Alon and J. H. Spencer, The probabilistic method, Wiley-Interscience Series in Discrete Mathematics and Optimization, 2008.

L. Ambrosio, N. Fusco, and D. Pallara, Free Discontinuity Problems and Special Functions with Bounded Variation, 2000.
DOI : 10.1007/978-3-0348-8974-2_2

L. Ambrosio, N. Gigli, and G. Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Inventiones mathematicae, vol.196, issue.1???2, pp.289-391, 2014.
DOI : 10.1007/s00222-013-0456-1
URL : https://hal.archives-ouvertes.fr/hal-00769378

D. P. Bertsekas and S. E. Shreve, Stochastic optimal control The discrete time case, Mathematics in Science and Engineering, vol.139, 1978.

S. G. Bobkov, An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space, The Annals of Probability, vol.25, issue.1, pp.206-214, 1997.
DOI : 10.1214/aop/1024404285

S. G. Bobkov, I. Gentil, and M. Ledoux, Hypercontractivity of Hamilton???Jacobi equations, Journal de Math??matiques Pures et Appliqu??es, vol.80, issue.7, pp.669-696, 2001.
DOI : 10.1016/S0021-7824(01)01208-9

S. G. Bobkov and F. Götze, Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities, Journal of Functional Analysis, vol.163, issue.1, pp.1-28, 1999.
DOI : 10.1006/jfan.1998.3326

S. G. Bobkov, C. Houdré, and P. Tetali, The subgaussian constant and concentration inequalities, Israel Journal of Mathematics, vol.17, issue.no. 1-3, pp.255-283, 2006.
DOI : 10.1007/BF02773835

S. G. Bobkov and M. Ledoux, Poincar??'s inequalities and Talagrand's concentration phenomenon for the exponential distribution, Probability Theory and Related Fields, vol.107, issue.3, pp.383-400, 1997.
DOI : 10.1007/s004400050090

S. G. Bobkov and M. Ledoux, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities, Geometric and Functional Analysis, vol.10, issue.5, pp.1028-1052, 2000.
DOI : 10.1007/PL00001645

S. G. Bobkov and P. Tetali, Modified Logarithmic Sobolev Inequalities in Discrete Settings, Journal of Theoretical Probability, vol.11, issue.4, pp.289-336, 2006.
DOI : 10.1007/s10959-006-0016-3

A. Bonciocat and K. Sturm, Mass transportation and rough curvature bounds for discrete spaces, Journal of Functional Analysis, vol.256, issue.9, pp.2944-2966, 2009.
DOI : 10.1016/j.jfa.2009.01.029

C. Borell, The Brunn-Minkowski inequality in Gauss space, Inventiones Mathematicae, vol.3, issue.2, pp.207-216, 1975.
DOI : 10.1007/BF01425510

S. Boucheron, G. Lugosi, and P. Massart, Concentration Inequalities, nonasymptotic theory of independence
DOI : 10.1007/978-1-4757-2440-0
URL : https://hal.archives-ouvertes.fr/hal-00751496

A. Dembo, Information inequalities and concentration of measure, The Annals of Probability, vol.25, issue.2, pp.927-939, 1997.
DOI : 10.1214/aop/1024404424

J. Deuschel and D. W. Stroock, Large deviations, Pure and Applied Mathematics, vol.137, 1989.
DOI : 10.1090/chel/342

H. Djellout, A. Guillin, and L. Wu, Transportation cost-information inequalities and applications to random dynamical systems and diffusions, Ann. Probab, vol.32, issue.3B, pp.2702-2732, 2004.

D. P. Dubhashi and A. Panconesi, Concentration of measure for the analysis of randomized algorithms, 2009.
DOI : 10.1017/CBO9780511581274

M. Erbar and J. Maas, Ricci Curvature of Finite Markov Chains via Convexity of the Entropy, Archive for Rational Mechanics and Analysis, vol.196, issue.1, pp.997-1038, 2012.
DOI : 10.1007/s00205-012-0554-z

P. Federbush, Partially Alternate Derivation of a Result of Nelson, Journal of Mathematical Physics, vol.10, issue.1, pp.50-52, 1969.
DOI : 10.1063/1.1664760

N. Gozlan, A characterization of dimension free concentration in terms of transportation inequalities, The Annals of Probability, vol.37, issue.6, pp.2480-2498, 2009.
DOI : 10.1214/09-AOP470
URL : https://hal.archives-ouvertes.fr/hal-00274548

N. Gozlan and C. Léonard, Transport inequalities. A survey, Markov Process, pp.635-736, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00515419

N. Gozlan, C. Roberto, and P. Samson, From concentration to logarithmic Sobolev and Poincar?? inequalities, Journal of Functional Analysis, vol.260, issue.5, pp.1491-1522, 2011.
DOI : 10.1016/j.jfa.2010.11.010

N. Gozlan, C. Roberto, and P. Samson, A new characterization of Talagrand???s transport-entropy inequalities and applications, The Annals of Probability, vol.39, issue.3, pp.857-880, 2011.
DOI : 10.1214/10-AOP570

N. Gozlan, C. Roberto, and P. Samson, Characterization of Talagrand???s transport-entropy inequalities in metric spaces, The Annals of Probability, vol.41, issue.5, pp.3112-3139, 2013.
DOI : 10.1214/12-AOP757

N. Gozlan, C. Roberto, and P. Samson, Hamilton Jacobi equations on metric spaces and transport entropy inequalities, Revista Matem??tica Iberoamericana, vol.30, issue.1, pp.133-163, 2014.
DOI : 10.4171/RMI/772
URL : https://hal.archives-ouvertes.fr/hal-00795829

N. Gozlan, C. Roberto, P. Samson, Y. Shu, and P. Tetali, Characterization of a class of weak transportentropy inequalities on the line, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01199023

N. Gozlan, C. Roberto, P. Samson, and P. Tetali, Displacement convexity of entropy and related inequalities on graphs, Probability Theory and Related Fields, vol.196, issue.1, pp.47-94, 2014.
DOI : 10.1007/s00440-013-0523-y
URL : https://hal.archives-ouvertes.fr/hal-00719848

N. Gozlan, C. Roberto, P. Samson, and P. Tetali, Transport-entropy inequalities for some discrete measures, 2016.

M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol.152, 1999.

M. Gromov and V. D. Milman, A Topological Application of the Isoperimetric Inequality, American Journal of Mathematics, vol.105, issue.4, pp.843-854, 1983.
DOI : 10.2307/2374298

L. Gross, Logarithmic Sobolev Inequalities, American Journal of Mathematics, vol.97, issue.4, pp.1061-1083, 1975.
DOI : 10.2307/2373688

E. Hillion, Concavity of entropy along binomial convolutions, Electronic Communications in Probability, vol.17, issue.0, 2012.
DOI : 10.1214/ECP.v17-1707
URL : https://hal.archives-ouvertes.fr/hal-01296809

M. Ledoux, On Talagrand's deviation inequalities for product measures, ESAIM: Probability and Statistics, vol.1, pp.63-87, 1995.
DOI : 10.1051/ps:1997103

M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol.89, 2001.
DOI : 10.1090/surv/089

C. Léonard, A saddle-point approach to the Monge-Kantorovich optimal transport problem, ESAIM: Control, Optimisation and Calculus of Variations, vol.17, issue.3, pp.682-704, 2011.
DOI : 10.1051/cocv/2010013

J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Annals of Mathematics, vol.169, issue.3, pp.903-991, 2009.
DOI : 10.4007/annals.2009.169.903

K. Marton, A simple proof of the blowing-up lemma (Corresp.), IEEE Transactions on Information Theory, vol.32, issue.3, pp.445-446, 1986.
DOI : 10.1109/TIT.1986.1057176

K. Marton, Bounding $\bar{d}$-distance by informational divergence: a method to prove measure concentration, The Annals of Probability, vol.24, issue.2, pp.857-866, 1996.
DOI : 10.1214/aop/1039639365

K. Marton, A measure concentration inequality for contracting markov chains, Geometric and Functional Analysis, vol.81, issue.158, pp.556-571, 1996.
DOI : 10.1007/BF02249263

B. Maurey, Some deviation inequalities, Geometric and Functional Analysis, vol.104, issue.2, pp.188-197, 1991.
DOI : 10.1007/BF01896377

T. Mikami, A simple proof of duality theorem for Monge-Kantorovich problem, Kodai Mathematical Journal, vol.29, issue.1, pp.1-4, 2006.
DOI : 10.2996/kmj/1143122381

V. D. Milman, A new proof of A. Dvoretzky's theorem on cross-sections of convex bodies, Funkcional. Anal. i Prilo?en, vol.5, issue.4, pp.28-37, 1971.

Y. Ollivier, Ricci curvature of Markov chains on metric spaces, Journal of Functional Analysis, vol.256, issue.3, pp.810-864, 2009.
DOI : 10.1016/j.jfa.2008.11.001

Y. Ollivier and C. Villani, A Curved Brunn--Minkowski Inequality on the Discrete Hypercube, Or: What Is the Ricci Curvature of the Discrete Hypercube?, SIAM Journal on Discrete Mathematics, vol.26, issue.3, pp.983-996, 2012.
DOI : 10.1137/11085966X
URL : https://hal.archives-ouvertes.fr/hal-00974738

F. Otto and C. Villani, Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality, Journal of Functional Analysis, vol.173, issue.2, pp.361-400, 2000.
DOI : 10.1006/jfan.1999.3557

K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, 1967.

D. Paulin, The convex distance inequality for dependent random variables, with applications to the stochastic travelling salesman and other problems, Electron, J. Probab, vol.19, issue.68, p.34, 2014.

L. Saloff-coste, Lectures on finite Markov chains, Lecture Notes in Math, vol.302, issue.S??rieI, pp.301-413, 1996.
DOI : 10.1103/PhysRevLett.58.86

P. Samson, Concentration of measure inequalities for Markov chains and $\Phi$-mixing processes, The Annals of Probability, vol.28, issue.1, pp.416-461, 2000.
DOI : 10.1214/aop/1019160125

P. Samson, Concentration inequalities for convex functions on product spaces, Stochastic inequalities and applications, Progr. Probab, vol.56, pp.33-52, 2003.

P. Samson, Infimum-convolution description of concentration properties of product probability measures, with applications, Annales de l'Institut Henri Poincare (B) Probability and Statistics, vol.43, issue.3, pp.321-338, 2007.
DOI : 10.1016/j.anihpb.2006.05.003
URL : https://hal.archives-ouvertes.fr/hal-00693095

Y. Shu, Hamilton jacobi equation on graphs and applications, In preparation, 2014.

A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Information and Control, vol.2, issue.2, pp.101-112, 1959.
DOI : 10.1016/S0019-9958(59)90348-1

V. Strassen, The Existence of Probability Measures with Given Marginals, The Annals of Mathematical Statistics, vol.36, issue.2, pp.423-439, 1965.
DOI : 10.1214/aoms/1177700153

K. Sturm, On the geometry of metric measure spaces. I, Acta Math, pp.65-131, 2006.

V. N. Sudakov, B. S. Cirel, and ?. Son, Extremal properties of half-spaces for spherically invariant measures, Journal of Soviet Mathematics, vol.270, issue.No. 2, pp.14-24, 1974.
DOI : 10.1007/BF01086099

M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Publications math??matiques de l'IH??S, vol.19, issue.158, pp.73-205, 1995.
DOI : 10.1007/BF02699376

M. Talagrand, New concentration inequalities in product spaces, Inventiones Mathematicae, vol.126, issue.3, pp.505-563, 1996.
DOI : 10.1007/s002220050108

M. Talagrand, Transportation cost for Gaussian and other product measures, Geometric and Functional Analysis, vol.27, issue.3, pp.587-600, 1996.
DOI : 10.1007/BF02249265

X. Tan and N. Touzi, Optimal transportation under controlled stochastic dynamics, The Annals of Probability, vol.41, issue.5, pp.3201-3240, 2013.
DOI : 10.1214/12-AOP797
URL : https://hal.archives-ouvertes.fr/hal-01069125

C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol.58, 2003.
DOI : 10.1090/gsm/058

C. Villani, Optimal transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of, Mathematical Sciences], vol.338, 2009.