M. Agueh and &. Carlier, Barycenters in the Wasserstein space, In: SIAM J. Math. Anal, vol.43, pp.36-1410, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00637399

S. Aida and &. Stroock, Moment estimates derived from Poincaré and logarithmic Sobolev inequalities, Math. Res. Lett, vol.1, issue.1, pp.1073-2780, 1994.

L. Ambrosio-&-r and . Ghezzi, Sobolev and bounded variation functions on metric measure spaces, Geometry, analysis and dynamics on sub-Riemannian manifolds, vol.II, pp.211-273, 2016.

L. Ambrosio, N. Gigli, and &. Savaré, Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds, Ann. Probab, vol.43, pp.91-1798, 2015.

L. Ambrosio, N. Gigli, and &. Savaré, Gradient flows in metric spaces and in the space of probability measures, Second. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, pp.978-981, 2008.

C. Ané, Panoramas et Synthèses [Panoramas and Syntheses, vol.10, pp.2-85629, 2000.

S. Artstein, K. M. Ball, F. Barthe, and &. Naor, On the rate of convergence in the entropic central limit theorem, Probab. Theory Related Fields, vol.129, pp.178-8051, 2004.

K. Atkinson and &. W. Han, Spherical harmonics and approximations on the unit sphere: an introduction, vol.2044, 2012.

E. Azmoodeh, S. Campese, and &. Poly, Fourth Moment Theorems for Markov diffusion generators, J. Funct. Anal, vol.266, pp.2341-2359, 2014.

L. Bachelier, Théorie de la spéculation, Ann. Sci. École Norm. Sup, issue.3, pp.12-9593, 1900.

L. Bachelier, Théorie mathématique du jeu, Ann. Sci. École Norm. Sup, issue.3, pp.12-9593, 1901.

D. Bakry and &. Émery, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math, vol.1123, pp.177-206, 1985.

D. Bakry, I. Gentil, and &. Ledoux, Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften, vol.348
URL : https://hal.archives-ouvertes.fr/hal-00929960

C. Springer, , 2014.

A. Barron and &. O. Johnson, Fisher information inequalities and the central limit theorem, Probab. Theory Related Fields, vol.129, pp.391-409, 2004.

A. C. Berry, The accuracy of the Gaussian approximation to the sum of independent variates, In: Trans. Amer. Math. Soc, vol.49, pp.122-136, 1941.

P. Billingsley, Probability and measure, Wiley Series in Probability and Mathematical Statistics, 1995.

S. G. Bobkov and &. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal, vol.163, pp.22-1236, 1999.

S. G. Bobkov and &. Ledoux, On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures, J. Funct. Anal, vol.156, pp.347-365, 1998.

S. Bobkov and &. Ledoux, Poincaré's inequalities and Talagrand's concentration phenomenon for the exponential distribution, Probab. Theory Related Fields, vol.107, pp.383-400, 1997.

S. G. Bobkov, I. Gentil, and &. Ledoux, Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl, vol.80, issue.9, pp.1208-1217, 2001.

S. G. Bobkov and &. Tetali, Modified logarithmic Sobolev inequalities in discrete settings, J. Theoret. Probab, vol.19, issue.2, pp.289-336, 2006.

S. Bochner, Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse, In: Math. Ann, vol.108, 1933.

N. Bouleau and &. F. Hirsch, Dirichlet forms and analysis on Wiener space, vol.14, 1991.

N. Bourbaki, Intégration, Chapitre 9 Intégration sur les espaces topologiques séparés. Eléments de mathématiques, 2006.

N. Bourbaki, Éléments de mathématique, Algèbre. Chapitre, vol.1, issue.3, 2007.

S. Bourguin and &. Peccati, Portmanteau inequalities on the Poisson space: mixed regimes and multidimensional clustering, In: Electron. J. Probab, vol.19, issue.66, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00731861

S. Bourguin and &. Peccati, The Malliavin-Stein method on the Poisson space, vol.7, pp.185-228, 2016.

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math, vol.44, pp.375-417, 1991.

P. Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup, vol.15, issue.4, pp.12-9593, 1982.

S. Campese, I. Nourdin, G. Peccati, and &. Poly, Multivariate Gaussian approximations on Markov chaoses, Paper No. 48, 9, vol.21, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01219698

D. Chafaï, Entropies, convexity, and functional inequalities: on ?-entropies and ?-Sobolev inequalities, vol.44, pp.325-363, 2004.

I. Chavel, Pure and Applied Mathematics. Including a chapter, With an appendix by Jozef Dodziuk, vol.115, 1984.

J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal, vol.9, pp.428-517, 1999.

F. R. Chung, A. Grigor'yan, and &. Yau, Upper bounds for eigenvalues of the discrete and continuous Laplace operators, Adv. Math, vol.117, pp.165-178, 1996.

C. Dellacherie and &. Meyer, Probabilités et potentiel. Chapitres I à IV. Édition entièrement refondue, Actualités Scientifiques et Industrielles, No. 1372. Hermann, p.291, 1975.

C. Dellacherie and &. Meyer, Probabilités et potentiel. Chapitres XVII à XXIV, Processus de Markov (fin), Compléments de calcul stochastique, p.429, 1992.

P. Dirac, The quantum theory of the emission and absorption of radiation, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol.114, pp.243-265, 1927.

C. Döbler and &. Krokowski, On the fourth moment condition for Rademacher chaos, 2017.

C. Döbler and &. Peccati, Fourth moment theorems on the Poisson space: analytic statements via product formulae, 2018.

C. Döbler and &. Peccati, The fourth moment theorem on the Poisson space, Ann. Probab, vol.46, pp.91-1798, 2018.

C. Döbler, A. Vidotto, and &. Zheng, Fourth moment theorems on the Poisson space in any dimension, Paper No. 36, vol.23, 2018.

R. M. Dudley, The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Functional Analysis, vol.1, pp.290-330, 1967.

N. &. Dunford, G. Schwartz-;-robert, and . Bartle, Linear operators. Part II. Wiley Classics Library. Spectral theory. Selfadjoint operators in Hilbert space, 1988.

M. Erbar and &. Huesmann, Curvature bounds for configuration spaces, Calc. Var. Partial Differential Equations, vol.54, 2015.

C. Esseen, On the Liapounoff limit of error in the theory of probability, Ark. Mat. Astr. Fys. 28A, vol.9, pp.4-2080, 1942.

F. Bruno, Sullo sviluppo delle funzioni, Annali di Scienze Matematiche e Fisiche 6 (1855), pp.479-480

T. S. Ferguson, Mathematical statistics: A decision theoretic approach, Probability and Mathematical Statistics, vol.1, p.396, 1967.

V. Fock, Konfiguationsraum und zweite Quantelung, Z. Physik, vol.75, issue.9, pp.622-647, 1932.

J. Friedman and &. Tillich, Laplacian eigenvalues and distances between subsets of a manifold, J. Differential Geom, vol.56, pp.22-040, 2000.

K. Funano, Estimates of eigenvalues of the Laplacian by a reduced number of subsets, Israel J. Math, vol.217, pp.413-433, 2017.

K. Funano and &. Shioya, Concentration, Ricci curvature, and eigenvalues of Laplacian, Geom. Funct. Anal, vol.23, pp.888-936, 2013.

W. Gangbo and &. Swiech, Optimal maps for the multidimensional MongeKantorovich problem, Comm. Pure Appl. Math, vol.51, pp.23-45, 1998.

N. Gozlan-&-r and . Herry, Multiple sets exponential concentration and higher order eigenvalues, 2018.

N. Gozlan, Characterization of Talagrand's like transportation-cost inequalities on the real line, J. Funct. Anal, vol.250, pp.400-425, 2007.

N. Gozlan and &. Léonard, Transport inequalities. A survey, Markov Process. Related Fields, vol.16, pp.1024-2953, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00515419

N. Gozlan, C. Roberto, and &. Samson, From dimension free concentration to the Poincaré inequality, Calc. Var. Partial Differential Equations, vol.52, pp.944-2669, 2015.

N. Gozlan, C. Roberto, and &. Samson, Hamilton Jacobi equations on metric spaces and transport entropy inequalities, In: Rev. Mat. Iberoam, vol.30, pp.133-163, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00795829

N. Gozlan, C. Roberto, P. Samson, and &. Tetali, Kantorovich duality for general transport costs and applications, J. Funct. Anal, vol.273, pp.22-1236, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01098114

M. D. Gromov-&-v and . Milman, A topological application of the isoperimetric inequality, Amer. J. Math, vol.105, pp.843-854, 1983.

M. Gromov, Metric structures for Riemannian and non-Riemannian spaces. English. Modern Birkhäuser Classics. Based on the 1981 French original, 2007.

L. Gross, Logarithmic Sobolev inequalities, In: Amer. J. Math, vol.97, pp.2-9327, 1975.

M. Hairer, Advanced stochastic calculus. Lecture notes, 2016.

D. Harnett and &. Nualart, Central limit theorem for a Stratonovich integral with Malliavin calculus, Ann. Probab, vol.41, pp.91-1798, 2013.

J. Heinonen, Lectures on Lipschitz analysis, vol.100, pp.951-990, 2005.

W. Hoeffding, The strong law of large numbers for U -statistics, In: Mimeo. Series, vol.302, pp.1-10, 1961.

L. Hörmander, Hypoelliptic second order differential equations, Acta Math, vol.119, pp.147-171, 1967.

K. Itô, Multiple Wiener integral, J. Math. Soc. Japan, vol.3, pp.25-5645, 1951.

K. Itô, Spectral type of the shift transformation of differential processes with stationary increments, In: Trans. Amer. Math. Soc, vol.81, pp.2-9947, 1956.

K. Itô, Stochastic integral, Proc. Imp. Acad. Tokyo, vol.20, pp.519-524, 1944.

J. N. Jacod-&-a and . Shiryaev, Limit theorems for stochastic processes, Grundlehren der Mathematischen Wissenschaften, vol.288

. Springer-verlag, , p.661, 2003.

O. Kallenberg, Foundations of modern probability. Second. Probability and its Applications, 2002.

M. Kirszbraun, Uber die zusammenziehende und Lipschitzsche Transformationen, Fundamenta Mathematicae, vol.22, pp.77-108, 1934.

A. N. Kolmogorov, Curves in a Hilbert space invariant with respect to a oneparameter group of motions, vol.26, pp.6-9, 1940.

A. N. Kolmogorov-;-v, V. A. Arnold, P. L. Skvortsov, and . Ulyanov, Mathematics and mechanics, With commentaries by, Mathematics and its Applications (Soviet Series), vol.25, pp.90-277, 1991.

R. Lachièze-rey and &. G. Peccati, Fine Gaussian fluctuations on the Poisson space II: rescaled kernels, marked processes and geometric U -statistics, In: Stochastic Process. Appl. 123, vol.12, pp.304-4149, 2013.

R. Lachièze-rey and &. G. Peccati, Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs, In: Electron. J. Probab, vol.18, issue.32, 2013.

P. De-laplace, Théorie analytique des probabilités, Mme Ve Courcier, vol.1812

P. De-laplace, Introduction: Essai philosophique sur les probabilités, Livre I: Du calcul des fonctions génératrices, vol.I, pp.2-87647, 1995.

P. De-laplace, Livre II: Théorie générale des probabilités, Reprint of the 1820 third edition (Book II) and of the 1816, 1818, 1820 and 1825 originals (Supplements). Éditions Jacques Gabay, vol.II, 1995.

G. Last, Stochastic analysis for Poisson processes, Stochastic analysis for Poisson point processes, vol.7, pp.1-36, 2016.

G. Last, G. Peccati, and &. Schulte, Normal approximation on Poisson spaces: Mehler's formula, second order Poincaré inequalities and stabilization, In: Probab. Theory Related Fields, vol.165, pp.178-8051, 2016.
DOI : 10.1007/s00440-015-0643-7
URL : http://arxiv.org/pdf/1401.7568

G. Last and &. Penrose, Institute of Mathematical Statistics Textbooks, Lectures on the Poisson process, vol.7, 2018.

G. D. Last-&-m and . Penrose, Poisson process Fock space representation, chaos expansion and covariance inequalities, Probab. Theory Related Fields, vol.150, pp.3-4, 2011.

M. Ledoux, Chaos of a Markov operator and the fourth moment condition, Ann. Probab, vol.40, pp.91-1798, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00934894

M. Ledoux, The concentration of measure phenomenon, vol.89, pp.0-8218, 2001.

M. Ledoux, The geometry of Markov diffusion generators, Ann. Fac. Sci. Toulouse Math, vol.9, issue.6, pp.240-2963, 2000.

M. Ledoux, I. Nourdin, and &. Peccati, Stein's method, logarithmic Sobolev and transport inequalities, Geom. Funct. Anal. 25, vol.1, pp.256-306, 2015.
DOI : 10.1007/s00039-015-0312-0
URL : http://arxiv.org/pdf/1403.5855

A. Lichnerowicz, Géométrie des groupes de transformations, Travaux et Recherches Mathématiques, III. Dunod, p.193, 1958.

S. Liu, An optimal dimension-free upper bound for eigenvalue ratios, 2014.

J. Lott and &. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math, vol.3, issue.2, pp.3-486, 2009.
DOI : 10.4007/annals.2009.169.903
URL : http://annals.math.princeton.edu/wp-content/uploads/annals-v169-n3-p04.pdf

Y. Ma, S. Shen, X. Wang, and &. Wu, Transportation inequalities: from Poisson to Gibbs measures, Bernoulli, vol.17, pp.155-169, 2011.
DOI : 10.3150/00-bej268
URL : https://doi.org/10.3150/00-bej268

P. Malliavin, Stochastic calculus of variation and hypoelliptic operators, Proceedings of the International Symposium on Stochastic Differential Equations, pp.195-263, 1976.

K. Marton, A simple proof of the blowing-up lemma, IEEE Trans. Inform. Theory, vol.32, pp.445-446, 1986.

K. Marton, Bounding d-distance by informational divergence: a method to prove measure concentration, In: Ann. Probab, vol.24, pp.91-1798, 1996.
DOI : 10.1214/aop/1039639365
URL : https://doi.org/10.1214/aop/1039639365

K. Marton, A measure concentration inequality for contracting Markov chains, In: Geom. Funct. Anal, vol.6, pp.556-571, 1996.
DOI : 10.1007/bf02249263

B. Maurey, Some deviation inequalities, Geom. Funct. Anal, vol.1, pp.188-197, 1991.
DOI : 10.1007/bf01896377
URL : http://arxiv.org/pdf/math/9201216

R. J. Mccann, A convexity principle for interacting gases, Adv. Math, vol.128, pp.153-179, 1997.

E. J. Mcshane, Extension of range of functions, Bull. Amer. Math. Soc, vol.40, pp.837-842, 1934.

P. Meyer, Séminaire de probabilités, 1967.

E. Milman, Spectral Estimates, Contractions and Hypercontractivity". In: ArXiv e-prints, 2015.
DOI : 10.4171/jst/210
URL : http://arxiv.org/pdf/1508.00606

E. Milman, Isoperimetric and concentration inequalities: equivalence under curvature lower bound, Duke Math. J, vol.154, pp.207-239, 2010.
DOI : 10.1215/00127094-2010-038
URL : http://arxiv.org/pdf/0902.1560

E. Milman, On the role of convexity in isoperimetry, spectral gap and concentration, In: Invent. Math, vol.177, 2009.

E. Milman, On the role of convexity in isoperimetry, spectral gap and concentration, In: Invent. Math, vol.177, 2009.

A. De-moivre, Approximatio ad summam terminorum binomii (a+b)n in seriem expansi, p.7

A. De-moivre, The doctrine of chances or, a method of calculating the probabilities of events in play, Cass Library of Science Classics, issue.1, p.257, 1967.

E. Nelson, The free Markoff field, J. Functional Analysis, vol.12, pp.211-227, 1973.

I. Nourdin, Malliavin-Stein approach, 2018.

I. Nourdin, Selected aspects of fractional Brownian motion, Bocconi & Springer Series, vol.4, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01314412

I. Nourdin and &. Nualart, Central limit theorems for multiple Skorokhod integrals, J. Theoret. Probab, vol.23, pp.39-64, 2010.
DOI : 10.1007/s10959-009-0258-y
URL : https://hal.archives-ouvertes.fr/hal-00469195

I. Nourdin, D. Nualart, and &. Peccati, Quantitative stable limit theorems on the Wiener space, Ann. Probab, vol.44, pp.91-1798, 2016.
URL : https://hal.archives-ouvertes.fr/hal-00823380

I. Nourdin and &. Peccati, Cambridge Tracts in Mathematics. From Stein's method to universality, vol.192, 2012.

I. Nourdin and &. Peccati, Stein's method on Wiener chaos, Probab. Theory Related Fields, vol.145, pp.75-118, 2009.

I. Nourdin-&-j.-rosí and . Nski, Asymptotic independence of multiple Wiener-Itô integrals and the resulting limit laws, Ann. Probab, vol.42, pp.497-526, 2014.

D. Nualart and &. Ortiz-latorre, Central limit theorems for multiple stochastic integrals and Malliavin calculus, Stochastic Process, vol.118, pp.614-628, 2008.

D. Nualart, CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, vol.110, 2009.

D. Nualart and &. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab, vol.33, pp.91-1798, 2005.

H. Ogura, Orthogonal functionals of the Poisson process, IEEE Trans. Information Theory IT, vol.18, pp.18-9448, 1972.

F. Otto and &. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal, vol.173, pp.22-1236, 2000.

F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, vol.26, pp.101-174, 2001.

G. Peccati, The Chen-Stein method for Poisson functionals, ArXiv e-prints, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00654235

G. Peccati, J. L. Solé, M. S. Taqqu, and &. Utzet, Stein's method and normal approximation of Poisson functionals, vol.38, pp.91-1798, 2010.

, Wiener-Itô chaos expansions and stochastic geometry, Bocconi & Springer Series. Malliavin calculus, vol.7

. Springer, , pp.978-981, 2016.

G. S. Peccati-&-m and . Taqqu, Central limit theorems for double Poisson integrals, In: Bernoulli, vol.14, pp.1350-7265, 2008.

G. S. Peccati-&-m and . Taqqu, Bocconi & Springer Series. A survey with computer implementation, Supplementary material available online, vol.1, 2011.

G. Peccati and &. Thäle, Gamma limits and U -statistics on the Poisson space, ALEA Lat. Am. J. Probab. Math. Stat, vol.10, pp.1980-0436, 2013.

G. Peccati and &. Yor, Four limit theorems for quadratic functionals of Brownian motion and Brownian bridge, Asymptotic methods in stochastics, vol.44, pp.75-87, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00104941

G. Peccati and &. Zheng, Multi-dimensional Gaussian fluctuations on the Poisson space, Electron. J. Probab, vol.15, issue.48, pp.1083-6489, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00472225

G. Perelman, The entropy formula for the Ricci flow and its geometric applications, ArXiv Mathematics e-prints, 2002.

V. S. Pipiras-&-m and . Taqqu, Integration questions related to fractional Brownian motion, Probab. Theory Related Fields, vol.118, pp.251-291, 2000.

S. Poisson, Recherches sur la probabilité des jugements en matière criminelle et en matière civile, p.415

L. Pratelli and &. Rigo, Total variation bounds for Gaussian functionals, In: Stochastic Process. Appl

M. Reitzner, Poisson point processes: large deviation inequalities for the convex distance, In: Electron. Commun. Probab, vol.18, issue.96, 2013.

M. Reitzner and &. Schulte, Central limit theorems for U -statistics of Poisson point processes, Ann. Probab, vol.41, pp.3879-3909, 2013.

A. Rényi, On mixing sequences of sets, Acta Math. Acad. Sci. Hungar, vol.9, pp.215-228, 1958.

A. Rényi, On stable sequences of events, In: Sankhy¯ a Ser. A, vol.25, pp.581-572, 1963.

G. C. Rota-&-t and . Wallstrom, Stochastic integrals: a combinatorial approach, Ann. Probab, vol.25, pp.91-1798, 1997.

M. Schmuckenschläger, Martingales, Poincaré type inequalities, and deviation inequalities, J. Funct. Anal, vol.155, pp.303-323, 1998.

M. Schulte, A central limit theorem for the Poisson-Voronoi approximation, Adv. in Appl. Math, vol.49, pp.285-306, 2012.

M. Sion, On general minimax theorems, In: Pacific J. Math, vol.8, pp.171-176, 1958.

A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, In: Information and Control, vol.2, issue.59, pp.90348-90349, 1959.

K. Sturm, On the geometry of metric measure spaces. I". In: Acta Math, vol.196, pp.1-5962, 2006.

K. Sturm, On the geometry of metric measure spaces. II, Acta Math, vol.196, issue.1, pp.1-5962, 2006.

D. Surgailis, On multiple Poisson stochastic integrals and associated Markov semigroups, Probab. Math. Statist. 3, vol.2, pp.208-4147, 1984.

M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal, vol.6, pp.587-600, 1996.

M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, In: Inst. Hautes Études Sci. Publ. Math, vol.81, pp.73-8301, 1995.

C. Villani, Grundlehren der Mathematischen Wissenschaften, vol.338

. Springer-verlag, , 2009.

F. Wang, Functional inequalities for empty essential spectrum, J. Funct. Anal, vol.170, pp.22-1236, 2000.

H. Whitney, Analytic extensions of differentiable functions defined in closed sets, In: Trans. Amer. Math. Soc, vol.36, issue.1, pp.63-89, 1934.

N. Wiener, Generalized harmonic analysis, Acta Math, vol.55, pp.117-258, 1930.

N. Wiener, The Homogeneous Chaos, Amer. J. Math, vol.60, pp.2-9327, 1938.

N. Wiener and &. Wintner, The discrete chaos, In: Amer. J. Math, vol.65, pp.279-298, 1943.

L. Wu, A new modified logarithmic Sobolev inequality for Poisson point processes and several applications, Probab. Theory Related Fields, vol.118, pp.178-8051, 2000.

G. Zheng, A Peccati-Tudor type theorem for Rademacher chaoses, 2017.