D. Bakry, P. Cattiaux, and A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincar??, Journal of Functional Analysis, vol.254, issue.3, pp.727-759, 2008.
DOI : 10.1016/j.jfa.2007.11.002

D. Bakry and M. ´. Emery, Diffusions hypercontractives, Sém. de probabilités XIX, Lecture Notes in Math, vol.84, p.1123, 1983.
DOI : 10.1007/bfb0075847
URL : http://archive.numdam.org/article/SPS_1985__19__177_0.pdf

D. Benedetto, E. Caglioti, J. A. Carillo, and M. Pulvirenti, A non Maxwellian steady distribution for one-dimensional granular media, J. Stat. Physics, vol.916, issue.5, pp.979-990, 1998.

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

F. Bolley, Separability and completeness for the Wasserstein distance, Sém. de probabilités XLI, Lecture Notes in Math, 1934.

F. Bolley, A. Guillin, and C. Villani, Quantitative Concentration Inequalities for Empirical Measures on Non-compact Spaces, Probability Theory and Related Fields, vol.206, issue.1, pp.3-4, 2007.
DOI : 10.1007/s00440-006-0004-7
URL : https://hal.archives-ouvertes.fr/hal-00453883

F. Bouchut and J. Dolbeault, On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov- Poisson-Fokker-Planck system with Coulombic and Newtonian potentials, Diff. Int. Equations, vol.8, issue.3, pp.487-514, 1995.

J. A. Carrillo, R. J. Mccann, and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Revista Matem??tica Iberoamericana, vol.19, issue.3, pp.971-1018, 2003.
DOI : 10.4171/RMI/376

J. A. Carrillo and G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma, vol.6, issue.7, pp.75-198, 2007.

P. Cattiaux, A. Guillin, and F. Malrieu, Probabilistic approach for granular media equations in the non-uniformly convex case, Probability Theory and Related Fields, vol.13, issue.2, pp.19-40, 2008.
DOI : 10.1007/s00440-007-0056-3
URL : https://hal.archives-ouvertes.fr/hal-00021591

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.

F. Hérau, Short and long time behavior of the Fokker???Planck equation in a confining potential and applications, Journal of Functional Analysis, vol.244, issue.1, pp.95-118, 2007.
DOI : 10.1016/j.jfa.2006.11.013

F. Hérau and F. Nier, Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential, Journ??es ??quations aux d??riv??es partielles, vol.2, issue.171, pp.151-218, 2004.
DOI : 10.5802/jedp.606

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

F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE's, Stochastic Process, Appl, vol.95, issue.1, pp.109-132, 2001.

S. Meléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, Lecture Notes in Math, vol.28, issue.2, pp.42-95, 1995.
DOI : 10.1007/BF01055714

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

R. Esposito, Y. Guo, and R. Marra, Stability of the Front under a Vlasov???Fokker???Planck Dynamics, Archive for Rational Mechanics and Analysis, vol.58, issue.2, 2009.
DOI : 10.1007/s00205-008-0184-7

D. Talay, Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme, Mark, Proc. Rel. Fields, pp.163-198, 2002.

C. Villani, Hypocoercivity, Memoirs of the Amer, Math. Society Grund. der Math. Wissenschaften, vol.26, issue.338, 2009.

L. Wu, Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems, Stochastic Processes and their Applications, vol.91, issue.2, pp.205-238, 2001.
DOI : 10.1016/S0304-4149(00)00061-2

F. Bolley, mailto:bolley(AT)ceremade.dauphine(DOT)fr

A. Guillin, mailto:guillin(AT)math.univ-bpclermont(DOT)fr UMR CNRS 6620, Laboratoire de Mathématiques Universite Blaise Pascal, avenue des Landais, F-63177 Aubiere Cedex Florent Malrieu, mailto:florent.malrieu(AT)univ-rennes1(DOT)fr UMR CNRS 6625, IRMAR