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 Electronic Journal of Probability > Vol. 15(2010) > Paper 13 open journal systems 


Functional inequalities for heavy tailed distributions and application to isoperimetry

Patrick Cattiaux, Institut Mathématique de Toulouse
Nathael Gozlan, Université Marne La Vallée
Arnaud Guillin, Université Blaise Pascal
Cyril Roberto, Université Marne La Vallée


Abstract
This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincaré and weak Cheeger, weighted Poincaré and weighted Cheeger inequalities and their dual forms. Proofs are short and we cover very large situations. For product measures on $R^n$ we obtain the optimal dimension dependence using the mass transportation method. Then we derive (optimal) isoperimetric inequalities. Finally we deal with spherically symmetric measures. We recover and improve many previous result


Full text: PDF

Pages: 346-385

Published on: April 9, 2010


Bibliography
  1. Ané, Cécile; Blachère, Sébastien; Chafaï, Djalil; Fougères, Pierre; Gentil, Ivan; Malrieu, Florent; Roberto, Cyril; Scheffer, Grégory. Sur les inégalités de Sobolev logarithmiques. (French) [Logarithmic Sobolev inequalities] With a preface by Dominique Bakry and Michel Ledoux. Panoramas et Synthèses [Panoramas and Syntheses], 10. Société Mathématique de France, Paris, 2000. xvi+217 pp. ISBN: 2-85629-105-8 MR1845806
  2. 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 (95m:47075)
  3. 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 (2009d:60039)
  4. 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
  5. Bakry, D.; Ledoux, M. Lévy-Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator. Invent. Math. 123 (1996), no. 2, 259--281. MR1374200 (97c:58162)
  6. Barlow, Richard E.; Marshall, Albert W.; Proschan, Frank. Properties of probability distributions with monotone hazard rate. Ann. Math. Statist. 34 1963 375--389. MR0171328 (30 #1559)
  7. Barthe, Franck. Levels of concentration between exponential and Gaussian. Ann. Fac. Sci. Toulouse Math. (6) 10 (2001), no. 3, 393--404. MR1923685 (2003f:60038)
  8. Barthe, F. Isoperimetric inequalities, probability measures and convex geometry. European Congress of Mathematics, 811--826, Eur. Math. Soc., Zürich, 2005. MR2185783 (2006h:60030)
  9. 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 (2006h:60031)
  10. 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 (2008k:47097)
  11. Barthe, F.; Cattiaux, P.; Roberto, C. Isoperimetry between exponential and Gaussian. Electron. J. Probab. 12 (2007), no. 44, 1212--1237 (electronic). MR2346509 (2008j:60046)
  12. Barthe, F.; Roberto, C. Sobolev inequalities for probability measures on the real line.Dedicated to Professor Aleksander Peƚczyński on the occasion of his 70th birthday (Polish). Studia Math. 159 (2003), no. 3, 481--497. MR2052235 (2006c:60019)
  13. Barthe, F.; Roberto, C. Modified logarithmic Sobolev inequalities on $Bbb R$. Potential Anal. 29 (2008), no. 2, 167--193. MR2430612 (2010e:26013)
  14. Blanchet, Adrien; Bonforte, Matteo; Dolbeault, Jean; Grillo, Gabriele; Vázquez, Juan Luis. Asymptotics of the fast diffusion equation via entropy estimates. Arch. Ration. Mech. Anal. 191 (2009), no. 2, 347--385. MR2481073
  15. Bobkov, S. G. Isoperimetric inequalities for distributions of exponential type. Ann. Probab. 22 (1994), no. 2, 978--994. MR1288139 (95h:60025)
  16. Bobkov, S. A functional form of the isoperimetric inequality for the Gaussian measure. J. Funct. Anal. 135 (1996), no. 1, 39--49. MR1367623 (96m:60091)
  17. Bobkov, S. G. Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 (1999), no. 4, 1903--1921. MR1742893 (2001h:60026)
  18. 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 (2005i:60028)
  19. Bobkov, Sergey G. Large deviations and isoperimetry over convex probability measures with heavy tails. Electron. J. Probab. 12 (2007), 1072--1100 (electronic). MR2336600 (2008g:60066)
  20. Bobkov, S. G.; Houdré, C. Isoperimetric constants for product probability measures. Ann. Probab. 25 (1997), no. 1, 184--205. MR1428505 (98g:60032)
  21. 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 (98b:46038)
  22. Bobkov, Sergey G.; Houdré, Christian. Weak dimension-free concentration of measure. Bernoulli 6 (2000), no. 4, 621--632. MR1777687 (2001i:28005)
  23. Bobkov, Sergey G.; Ledoux, Michel. Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab. 37 (2009), no. 2, 403--427. MR2510011
  24. Bobkov, S. G.; Zegarlinski, B. Entropy bounds and isoperimetry. Mem. Amer. Math. Soc. 176 (2005), no. 829, x+69 pp. MR2146071 (2006c:46027)
  25. Bobkov, S. G.; Zegarlinski, B. Distribution with slow tails and ergodicity of markov semigroups in infinite dimensions. Preprint 2007.
  26. Borell, Christer. The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 (1975), no. 2, 207--216. MR0399402 (53 #3246)
  27. Borell, C. Convex set functions in $d$-space. Period. Math. Hungar. 6 (1975), no. 2, 111--136. MR0404559 (53 #8359)
  28. Cattiaux, Patrick. A pathwise approach of some classical inequalities. Potential Anal. 20 (2004), no. 4, 361--394. MR2032116 (2004k:60217)
  29. Cattiaux, Patrick; Guillin, Arnaud. Trends to equilibrium in total variation distance. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 1, 117--145. MR2500231
  30. 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
  31. 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 (90e:35123)
  32. 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 (2005k:35214)
  33. Dolbeault, Jean; Gentil, Ivan; Guillin, Arnaud; Wang, Feng-Yu. $Lsp q$-functional inequalities and weighted porous media equations. Potential Anal. 28 (2008), no. 1, 35--59. MR2366398 (2009d:35171)
  34. 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
  35. Gozlan, Nathael. Characterization of Talagrand's like transportation-cost inequalities on the real line. J. Funct. Anal. 250 (2007), no. 2, 400--425. MR2352486 (2009c:60039)
  36. Gozlan, Nathael. Poincar'e inequalities and dimension free concentration of measure. To appear in Ann. Inst. H. Poinc., (2010)
  37. 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 (95h:47061)
  38. Guionnet, A.; Zegarlinski, B. Lectures on logarithmic Sobolev inequalities. Séminaire de Probabilités, XXXVI, 1--134, Lecture Notes in Math., 1801, Springer, Berlin, 2003. MR1971582 (2004b:60226)
  39. Hairer, Martin; Mattingly, Jonathan C. Slow energy dissipation in anharmonic oscillator chains. Comm. Pure Appl. Math. 62 (2009), no. 8, 999--1032. MR2531551
  40. 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 (96e:52018)
  41. Ledoux, Michel. Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités, XXXIII, 120--216, Lecture Notes in Math., 1709, Springer, Berlin, 1999. MR1767995 (2002j:60002)
  42. 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 (2003k:28019)
  43. 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 (2007f:58049)
  44. 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 (95j:60103)
  45. 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 (94g:60136)
  46. 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 (94g:60137)
  47. Milman, Emanuel. On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177 (2009), no. 1, 1--43. MR2507637
  48. 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 (47 #418)
  49. Röckner, Michael; Wang, Feng-Yu. Weak Poincaré inequalities and $Lsp 2$-convergence rates of Markov semigroups. J. Funct. Anal. 185 (2001), no. 2, 564--603. MR1856277 (2002j:47075)
  50. Ros, Antonio. The isoperimetric problem. Global theory of minimal surfaces, 175--209, Clay Math. Proc., 2, Amer. Math. Soc., Providence, RI, 2005. MR2167260 (2006e:53023)
  51. Royer, Gilles. An initiation to logarithmic Sobolev inequalities. Translated from the 1999 French original by Donald Babbitt. SMF/AMS Texts and Monographs, 14. American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2007. viii+119 pp. ISBN: 978-0-8218-4401-4; 0-8218-4401-6 MR2352327
  52. 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 (51 #1932)
  53. 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 (93d:60095)
  54. Talagrand, Michel. The supremum of some canonical processes. Amer. J. Math. 116 (1994), no. 2, 283--325. MR1269606 (95g:60052)
  55. 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 (95b:35101)
  56. Veretennikov, A. Yu. On polynomial mixing bounds for stochastic differential equations. Stochastic Process. Appl. 70 (1997), no. 1, 115--127. MR1472961 (99k:60158)
  57. Wang, Feng-Yu. Functional inequalities, Markov processes and Spectral theory. Science Press, Beijing, 2005.
  58. 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 (2010a:60276)
















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Electronic Journal of Probability. ISSN: 1083-6489