References

• C. Ané, S. Blachère, D. Chafa"i, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, and G. Scheffer. Sur les inégalités de Sobolev logarithmiques, volume~10 of Panoramas et Synthèses. Société Mathématique de France, Paris, 2000.
• Bakry, Dominique. L'hypercontractivité et son utilisation en théorie des semigroupes. (French) [Hypercontractivity and its use in semigroup theory] Lectures on probability theory (Saint-Flour, 1992), 1--114, Lecture Notes in Math., 1581, Springer, Berlin, 1994. MR1307413
• Bakry, Dominique; Barthe, Franck; Cattiaux, Patrick; Guillin, Arnaud. A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case. Electron. Commun. Probab. 13 (2008), 60--66. MR2386063
• Bakry, Dominique; Cattiaux, Patrick; Guillin, Arnaud. Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254 (2008), no. 3, 727--759. MR2381160
• Bakry, D.; Ledoux, M. Lévy-Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator. Invent. Math. 123 (1996), no. 2, 259--281. MR1374200
• Barlow, Richard E.; Marshall, Albert W.; Proschan, Frank. Properties of probability distributions with monotone hazard rate. Ann. Math. Statist. 34 1963 375--389. MR0171328
• Barthe, Franck. Levels of concentration between exponential and Gaussian. Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), no. 3, 393--404. MR1923685
• Barthe, F. Isoperimetric inequalities, probability measures and convex geometry. European Congress of Mathematics, 811--826, Eur. Math. Soc., Zürich, 2005. MR2185783
• Barthe, F.; Cattiaux, P.; Roberto, C. Concentration for independent random variables with heavy tails. AMRX Appl. Math. Res. Express 2005, no. 2, 39--60. MR2173316
• Barthe, Franck; Cattiaux, Patrick; Roberto, Cyril. Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoam. 22 (2006), no. 3, 993--1067. MR2320410
• Barthe, F.; Cattiaux, P.; Roberto, C. Isoperimetry between exponential and Gaussian. Electron. J. Probab. 12 (2007), no. 44, 1212--1237 (electronic). MR2346509
• Barthe, F.; Roberto, C. Sobolev inequalities for probability measures on the real line. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday (Polish). Studia Math. 159 (2003), no. 3, 481--497. MR2052235
• Barthe, F.; Roberto, C. Modified logarithmic Sobolev inequalities on $\Bbb R$. Potential Anal. 29 (2008), no. 2, 167--193. MR2430612
• A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo, and J.L. Vázquez. Asymptotics of the fast diffusion equation via entropy estimates. To appear in Archive for Rational Mechanics and Analysis, 2008.
• Bobkov, S. G. Isoperimetric inequalities for distributions of exponential type. Ann. Probab. 22 (1994), no. 2, 978--994. MR1288139
• Bobkov, S. A functional form of the isoperimetric inequality for the Gaussian measure. J. Funct. Anal. 135 (1996), no. 1, 39--49. MR1367623
• Bobkov, S. G. Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 (1999), no. 4, 1903--1921. MR1742893
• Bobkov, S. G. Spectral gap and concentration for some spherically symmetric probability measures. Geometric aspects of functional analysis, 37--43, Lecture Notes in Math., 1807, Springer, Berlin, 2003. MR2083386
• Bobkov, Sergey G. Large deviations and isoperimetry over convex probability measures with heavy tails. Electron. J. Probab. 12 (2007), 1072--1100. MR2336600
• Bobkov, S. G.; Houdré, C. Isoperimetric constants for product probability measures. Ann. Probab. 25 (1997), no. 1, 184--205. MR1428505
• Bobkov, Serguei G.; Houdré, Christian. Some connections between isoperimetric and Sobolev-type inequalities. Mem. Amer. Math. Soc. 129 (1997), no. 616, viii+111 pp. MR1396954
• Bobkov, Sergey G.; Houdré, Christian. Weak dimension-free concentration of measure. Bernoulli 6 (2000), no. 4, 621--632. MR1777687
• S. G. Bobkov and M. Ledoux. Weighted Poincaré-type inequalities for Cauchy and other convex measures. To appear in Annals of Probability., 2007.
• Bobkov, S. G.; Zegarlinski, B. Entropy bounds and isoperimetry. Mem. Amer. Math. Soc. 176 (2005), no. 829, x+69 pp. MR2146071
• Bobkov, Sergey; Zegarlinski, Boguslaw. Distributions with slow tails and ergodicity of Markov semigroups in infinite dimensions. Around the research of Vladimir Maz'ya. I, 13--79, Int. Math. Ser. (N. Y.), 11, Springer, New York, 2010. MR2723813
• Borell, Christer. The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 (1975), no. 2, 207--216. MR0399402
• Borell, C. Convex set functions in $d$-space. Period. Math. Hungar. 6 (1975), no. 2, 111--136. MR0404559
• Cattiaux, Patrick. A pathwise approach of some classical inequalities. Potential Anal. 20 (2004), no. 4, 361--394. MR2032116
• Cattiaux, Patrick; Guillin, Arnaud. Trends to equilibrium in total variation distance. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 1, 117--145. MR2500231
• Cattiaux, Patrick; Guillin, Arnaud; Wang, Feng-Yu; Wu, Liming. Lyapunov conditions for super Poincaré inequalities. J. Funct. Anal. 256 (2009), no. 6, 1821--1841. MR2498560
• Davies, E. B. Heat kernels and spectral theory. Cambridge Tracts in Mathematics, 92. Cambridge University Press, Cambridge, 1989. x+197 pp. ISBN: 0-521-36136-2 MR0990239
• Denzler, Jochen; McCann, Robert J. Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology. Arch. Ration. Mech. Anal. 175 (2005), no. 3, 301--342. MR2126633
• Dolbeault, Jean; Gentil, Ivan; Guillin, Arnaud; Wang, Feng-Yu. $L^ q$-functional inequalities and weighted porous media equations. Potential Anal. 28 (2008), no. 1, 35--59. MR2366398
• Douc, Randal; Fort, Gersende; Guillin, Arnaud. Subgeometric rates of convergence of $f$-ergodic strong Markov processes. Stochastic Process. Appl. 119 (2009), no. 3, 897--923. MR2499863
• Gozlan, Nathael. Characterization of Talagrand's like transportation-cost inequalities on the real line. J. Funct. Anal. 250 (2007), no. 2, 400--425. MR2352486
• Gozlan, Nathael. Poincaré inequalities and dimension free concentration of measure. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 3, 708--739. MR2682264
• Gross, Leonard. Logarithmic Sobolev inequalities and contractivity properties of semigroups. Dirichlet forms (Varenna, 1992), 54--88, Lecture Notes in Math., 1563, Springer, Berlin, 1993. MR1292277
• A. Guionnet and B. Zegarlinski. Lectures on logarithmic Sobolev inequalities. Séminaire de Probabilités XXXVI. Lect. Notes Math., 1801, 2002.
• Hairer, Martin; Mattingly, Jonathan C. Slow energy dissipation in anharmonic oscillator chains. Comm. Pure Appl. Math. 62 (2009), no. 8, 999--1032. MR2531551
• Kannan, R.; Lovász, L.; Simonovits, M. Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 (1995), no. 3-4, 541--559. MR1318794
• Ledoux, Michel. Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités, XXXIII, 120--216, Lecture Notes in Math., 1709, Springer, Berlin, 1999. MR1767995
• Ledoux, Michel. The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI, 2001. x+181 pp. ISBN: 0-8218-2864-9 MR1849347
• Ledoux, Michel. Spectral gap, logarithmic Sobolev constant, and geometric bounds. Surveys in differential geometry. Vol. IX, 219--240, Surv. Differ. Geom., IX, Int. Press, Somerville, MA, 2004. MR2195409
• Meyn, S. P.; Tweedie, R. L. Markov chains and stochastic stability. Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 1993. xvi+ 548 pp. ISBN: 3-540-19832-6 MR1287609
• Meyn, Sean P.; Tweedie, R. L. Stability of Markovian processes. II. Continuous-time processes and sampled chains. Adv. in Appl. Probab. 25 (1993), no. 3, 487--517. MR1234294
• Meyn, Sean P.; Tweedie, R. L. Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes. Adv. in Appl. Probab. 25 (1993), no. 3, 518--548. MR1234295
• Milman, Emanuel. On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177 (2009), no. 1, 1--43. MR2507637
• Muckenhoupt, Benjamin. Hardy's inequality with weights. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I. Studia Math. 44 (1972), 31--38. MR0311856
• Röckner, Michael; Wang, Feng-Yu. Weak Poincaré inequalities and $L^ 2$-convergence rates of Markov semigroups. J. Funct. Anal. 185 (2001), no. 2, 564--603. MR1856277
• Ros, Antonio. The isoperimetric problem. Global theory of minimal surfaces, 175--209, Clay Math. Proc., 2, Amer. Math. Soc., Providence, RI, 2005. MR2167260
• G. Royer. Une initiation aux inégalités de Sobolev logarithmiques. S.M.F., Paris, 1999.
• Sudakov, V. N.; Cirelʹson, B. S. Extremal properties of half-spaces for spherically invariant measures. (Russian) Problems in the theory of probability distributions, II. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 (1974), 14--24, 165. MR0365680
• Talagrand, Michel. A new isoperimetric inequality and the concentration of measure phenomenon. Geometric aspects of functional analysis (1989–90), 94--124, Lecture Notes in Math., 1469, Springer, Berlin, 1991. MR1122615
• Talagrand, Michel. The supremum of some canonical processes. Amer. J. Math. 116 (1994), no. 2, 283--325. MR1269606
• Vázquez, Juan Luis. An introduction to the mathematical theory of the porous medium equation. Shape optimization and free boundaries (Montreal, PQ, 1990), 347--389, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 380, Kluwer Acad. Publ., Dordrecht, 1992. MR1260981
• Veretennikov, A. Yu. On polynomial mixing bounds for stochastic differential equations. Stochastic Process. Appl. 70 (1997), no. 1, 115--127. MR1472961
• F. Y. Wang. Functional inequalities, Markov processes and Spectral theory. Science Press, Beijing, 2005.
• Wang, Feng-Yu. From super Poincaré to weighted log-Sobolev and entropy-cost inequalities. J. Math. Pures Appl. (9) 90 (2008), no. 3, 270--285. MR2446080