D. Bakry, Un critère de non-explosion pour certaines diffusions sur une variété riemannienne compì ete, C. R. Acad. Sci. Paris Sér. I Math, vol.303, pp.23-26, 1986.

D. Bakry and M. ´. Emery, Diffusions hypercontractives, Lecture Notes in Math, pp.177-206, 1123.
DOI : 10.1007/BFb0075847

URL : http://archive.numdam.org/article/SPS_1985__19__177_0.pdf

D. Bakry, I. Gentil, and M. Ledoux, Analysis and geometry of Markov diffusion operators. Grundlehren der mathematischen Wissenschaften
URL : https://hal.archives-ouvertes.fr/hal-00929960

S. G. Bobkov, Probability Measures, The Annals of Probability, vol.27, issue.4, pp.1903-1921, 1999.
DOI : 10.1214/aop/1022677553

S. G. Bobkov, F. Götze, and C. Houdré, On Gaussian and Bernoulli Covariance Representations, Bernoulli, vol.7, issue.3, pp.439-451, 2001.
DOI : 10.2307/3318495

URL : http://projecteuclid.org/download/pdf_1/euclid.bj/1080004759

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 M. Ledoux, Weighted Poincar??-type inequalities for Cauchy and other convex measures, The Annals of Probability, vol.37, issue.2, pp.403-427, 2009.
DOI : 10.1214/08-AOP407

F. Bolley, I. Gentil, and A. Guillin, Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01171361

M. Bonnefont and A. Joulin, Intertwining Relations for One-Dimensional Diffusions and Application to Functional Inequalities, Potential Analysis, vol.17, issue.4, pp.1005-1031, 2014.
DOI : 10.1007/s11118-014-9408-7

URL : https://hal.archives-ouvertes.fr/hal-00812346

M. Bonnefont, A. Joulin, Y. Ma, M. Bonnefont, A. Joulin et al., Spectral gap for spherically symmetric log-concave probability measures, and beyond A note on spectral gap and weighted Poincaré inequalities for some one-dimensional diffusions, 2016.

H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Pr??kopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, Journal of Functional Analysis, vol.22, issue.4, pp.366-389, 1976.
DOI : 10.1016/0022-1236(76)90004-5

E. Carlen, D. Cordero-erausquin, and E. Lieb, Asymmetric covariance estimates of Brascamp???Lieb type and related inequalities for log-concave measures, Annales de l'Institut Henri Poincar??, Probabilit??s et Statistiques, vol.49, issue.1
DOI : 10.1214/11-AIHP462

D. Chafa¨?chafa¨? and A. Joulin, Intertwining and commutation relations for birth???death processes, Bernoulli, vol.19, issue.5A, pp.1855-1879, 2013.
DOI : 10.3150/12-BEJ433

M. F. Chen, Spectral gap and logarithmic Sobolev constant for continuous spin systems, Acta Mathematica Sinica, English Series, vol.144, issue.2, pp.705-736, 2008.
DOI : 10.1007/s10114-007-7293-3

M. F. Chen and F. Y. Wang, Estimation of spectral gap for elliptic operators, Transactions of the American Mathematical Society, vol.349, issue.03, pp.1239-1267, 1997.
DOI : 10.1090/S0002-9947-97-01812-6

D. Cordero-erausquin, Transport inequalities for log-concave measures, quantitative forms and applications

I. Gentil and C. Roberto, Spectral Gaps for Spin Systems: Some Non-convex Phase Examples, Journal of Functional Analysis, vol.180, issue.1, pp.66-84, 2001.
DOI : 10.1006/jfan.2000.3692

G. Hargé, Reinforcement of an inequality due to Brascamp and Lieb, Journal of Functional Analysis, vol.254, issue.2, pp.267-300, 2008.
DOI : 10.1016/j.jfa.2007.07.019

B. Helffer, Remarks on Decay of Correlations and Witten Laplacians Brascamp???Lieb Inequalities and Semiclassical Limit, Journal of Functional Analysis, vol.155, issue.2, pp.571-586, 1998.
DOI : 10.1006/jfan.1997.3239

B. Helffer, Remarks on decay of correlations and Witten Laplacians III. Application to logarithmic Sobolev inequalities, Annales de l'Institut Henri Poincare (B) Probability and Statistics, vol.35, issue.4, pp.483-508, 1999.
DOI : 10.1016/S0246-0203(99)00103-X

B. Helffer, Semiclassical analysis, Witten Laplacians, and statistical mechanics, Series in Partial Differential Equations and Applications, 2002.
DOI : 10.1142/5049

L. Hörmander, L2 estimates and existence theorems for the $\bar \partial $ operator, Acta Mathematica, vol.113, issue.0, pp.89-152, 1965.
DOI : 10.1007/BF02391775

C. Houdré, Remarks on deviation inequalities for functions of infinitely divisible random vectors, The Annals of Probability, vol.30, issue.3, pp.1223-1237, 2002.
DOI : 10.1214/aop/1029867126

E. Kannan, L. Lovász, and M. Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete & Computational Geometry, vol.32, issue.312, pp.541-559, 1995.
DOI : 10.1007/BF02574061

R. Z. Khas-'minskii, Ergodic Properties of Recurrent Diffusion Processes and Stabilization of the Solution to the Cauchy Problem for Parabolic Equations, Theory of Probability & Its Applications, vol.5, issue.2, pp.179-195, 1960.
DOI : 10.1137/1105016

A. V. Kolesnikov and E. Milman, Poincaré and Brunn-Minkowski inequalities on weighted Riemannian manifolds with boundary, 2014.

M. Ledoux, Logarithmic Sobolev Inequalities for Unbounded Spin Systems Revisited, Lecture Notes in Math, pp.167-194, 1755.
DOI : 10.1007/978-3-540-44671-2_13

P. Li, Uniqueness of $L^1$ solutions for the Laplace equation and the heat equation on Riemannian manifolds, Journal of Differential Geometry, vol.20, issue.2, pp.447-457, 1984.
DOI : 10.4310/jdg/1214439287

G. Menz, A Brascamp-Lieb type covariance estimate, Electronic Journal of Probability, vol.19, issue.0, pp.1-15, 2014.
DOI : 10.1214/EJP.v19-2997

G. Menz and F. Otto, Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential, The Annals of Probability, vol.41, issue.3B, pp.2182-2224, 2013.
DOI : 10.1214/11-AOP715

S. P. Meyn and R. L. Tweedie, Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. App. Probab, vol.25, pp.518-548, 1993.

E. Milman, On the role of convexity in isoperimetry, spectral gap and??concentration, Inventiones mathematicae, vol.115, issue.9, pp.1-43, 2009.
DOI : 10.1007/s00222-009-0175-9

V. H. Nguyen, Dimensional variance inequalities of Brascamp???Lieb type and a local approach to dimensional Pr??kopa??s theorem, Journal of Functional Analysis, vol.266, issue.2, pp.931-955, 2014.
DOI : 10.1016/j.jfa.2013.11.003

R. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, Journal of Functional Analysis, vol.52, issue.1, pp.48-79, 1983.
DOI : 10.1016/0022-1236(83)90090-3

L. Veysseire, A harmonic mean bound for the spectral gap of the Laplacian on Riemannian manifolds, Comptes Rendus Mathematique, vol.348, issue.23-24, pp.1319-1322, 2010.
DOI : 10.1016/j.crma.2010.10.015

F. Y. Wang, Modified Curvatures on Manifolds with Boundary and Applications, Potential Analysis, vol.15, issue.3, pp.699-714, 2014.
DOI : 10.1007/s11118-014-9389-6

(. M. Arnaudon, . Umr, and . Cnrs, 1, France E-mail address: mailto:marc.arnaudon(at)math.u-bordeaux1.fr URL: http://www.math.u-bordeaux1.fr 1, France E-mail address: mailto:michel.bonnefont(at)math.u-bordeaux1.fr URL: http://www.math.u-bordeaux1.fr, Joulin) UMR CNRS 5219