Concentration of measure and logarithmic Sobolev inequalities
Ledoux, Michel
Séminaire de probabilités de Strasbourg, Tome 33 (1999), p. 120-216 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU) 
@article{SPS_1999__33__120_0,
     author = {Ledoux, Michel},
     title = {Concentration of measure and logarithmic Sobolev inequalities},
     journal = {S\'eminaire de probabilit\'es de Strasbourg},
     volume = {33},
     year = {1999},
     pages = {120-216},
     mrnumber = {1767995},
     zbl = {0957.60016},
     language = {en},
     url = {http://dml.mathdoc.fr/item/SPS_1999__33__120_0}
}

Ledoux, Michel. Concentration of measure and logarithmic Sobolev inequalities. Séminaire de probabilités de Strasbourg, Tome 33 (1999) pp. 120-216. http://gdmltest.u-ga.fr/item/SPS_1999__33__120_0/

[Ai] S. Aida. Uniform positivity improving property, Sobolev inequalities and spectral gaps. J. Funct. Anal. 158, 152-185 (1998). | MR 1641566 | Zbl 0914.47041

[A-M-S] S. Aida, T. Masuda, I. Shigekawa. Logarithmic Sobolev inequalities and exponential integrability. J. Funct. Anal. 126, 83-101 (1994). | MR 1305064 | Zbl 0846.46020

[A-S] S. Aida, D. Stroock. Moment estimates derived from Poincaré and logarithmic Sobolev inequalities. Math. Res. Lett. 1, 75-86 (1994). | MR 1258492 | Zbl 0862.60064

[Al] N. Alon. Eigenvalues and expanders. J. Combin. Theory, Ser. B, 38, 78-88 (1987). | MR 875835

[A-L] C. Ané, M. Ledoux. On logarithmic Sobolev inequalities for continuous time random walks on graphs. Preprint (1998). | MR 1757600

[Ba1] D. Bakry. L'hypercontractivité et son utilisation en théorie des semigroupes. Ecole d'Eté de Probabilités de St-Flour. Lecture Notes in Math. 1581, 1-114 (1994). Springer-Verlag. | MR 1307413 | Zbl 0856.47026

[Ba2] D. Bakry. On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. New trends in Stochastic Analysis. 43-75 (1997). World Scientific. | MR 1654503

[Ba-E] D. Bakry, M. Emery. Diffusions hypercontractives. Séminaire de Probabilites XIX. Lecture Notes in Math. 1123, 177-206 (1985). Springer-Verlag. | Numdam | MR 889476 | Zbl 0561.60080

[Ba-L] D. Bakry, M. Ledoux. Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator. Invent. math. 123, 259-281 (1996). | MR 1374200 | Zbl 0855.58011

[B-L-Q] D. Bakry, M. Ledoux, Z. Qian. Preprint (1997).

[Be] W. Beckner. Personal communication (1998).

[BA-L] G. Ben Arous, M. Ledoux. Schilder's large deviation principle without topology. Asymptotic problems in probability theory: Wiener functionals and asymptotics. Pitman Research Notes in Math. Series 284, 107-121 (1993). Longman. | MR 1354164 | Zbl 0801.60019

[B-M1] L. Birgé, P. Massart. From model selection to adaptive estimation. Festschrift for Lucien LeCam: Research papers in Probability and Statistics (D. Pollard, E. Torgersen and G. Yang. eds.) 55-87 (1997). Springer-Verlag. | MR 1462939 | Zbl 0920.62042

[B-M2] L. Birgé, P. Massart. Minimum contrast estimators on sieves: exponential bounds and rates of convergence (1998). Bernoulli, to appear. | MR 1653272 | Zbl 0954.62033

[B-B-M] A. Barren, L. Birgé, P. Massart. Risk bounds for model selection via penalization (1998). Probab. Theory Relat. Fields, to appear. | MR 1679028 | Zbl 0946.62036

[Bob1] S. Bobkov. and On Gross' Talagrand's inequalities on the discrete cube. Vestnik of Syktyvkar University, Ser. 1, 1, 12-19 (1995) (in Russian). | MR 1717705 | Zbl 1011.60500

[Bob2] S. Bobkov. Some extremal properties of Bernoulli distribution. Probability Theor. Appl. 41, 877-884 (1996). | MR 1687168 | Zbl 0895.60012

[Bob3] S. Bobkov. A functional form of the isoperimetric inequality for the Gaussian measure. J. Funct. Anal. 135, 39-49 (1996). | MR 1367623 | Zbl 0838.60013

[Bob4] S. Bobkov. An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probability 25, 206-214 (1997). | MR 1428506 | Zbl 0883.60031

[Bob5] S. Bobkov. Isoperimetric and analytic inequalities for log-concave probability measures (1998). Ann. Probability, to appear. | MR 1742893 | Zbl 0964.60013

[B-G] S. Bobkov, F. Götze. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities (1997). J. Funct. Anal., to appear. | MR 1682772 | Zbl 0924.46027

[B-H] S. Bobkov, C. Houdré. Isoperimetric constants for product probability measures. Ann. Probability 25, 184-205 (1997). | MR 1428505 | Zbl 0878.60013

[B-L1] S. Bobkov, M. Ledoux. Poincaré's inequalities and Talagrand's concentration phenomenon for the exponential measure. Probab. Theory Relat. Fields 107, 383-400 (1997). | MR 1440138 | Zbl 0878.60014

[B-L2] S. Bobkov, M. Ledoux. On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156, 347-365 (1998). | MR 1636948 | Zbl 0920.60002

[Bon] A. Bonami. Etude des coefficients de Fourier des fonctions de LP(G). Ann. Inst. Fourier 20, 335-402 (1970). | Numdam | MR 283496 | Zbl 0195.42501

[Bor] C. Borell. The Brunn-Minkowski inequality in Gauss space. Invent. math. 30, 207-216 (1975). | MR 399402 | Zbl 0311.60007

[Br] R. Brooks. On the spectrum of non-compact manifolds with finite volume. Math. Z. 187, 425-437 (1984). | MR 757481 | Zbl 0552.58034

[Cha1] I. Chavel. Eigenvalues in Riemannian geometry. Academic Press (1984). | MR 768584 | Zbl 0551.53001

[Cha2] I. Chavel. Riemannian geometry - A modern introduction. Cambridge Univ. Press (1993). | MR 1271141 | Zbl 0810.53001

[Che] S.-Y. Cheng. Eigenvalue comparison theorems and its geometric applications. Math. Z. 143, 289-297 (1975). | MR 378001 | Zbl 0329.53035

[Da] E.B. Davies. Heat kernel and spectral theory. Cambridge Univ. Press (1989). | MR 990239

[D-S] E.B. Davies, B. Simon. Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59, 335-395 (1984). | MR 766493 | Zbl 0568.47034

[De] A. Dembo. Information inequalities and concentration of measure. Ann. Probability 25, 927-939 (1997). | MR 1434131 | Zbl 0880.60018

[D-Z] A. Dembo, O. Zeitouni. Transportation approach to some concentration inequalities in product spaces. Elect. Comm. in Probab. 1, 83-90 (1996). | MR 1423908 | Zbl 0916.28003

[De-S] J.-D. Deuschel, D. Stroock. Large deviations. Academic Press (1989). | MR 997938 | Zbl 0705.60029

[D-SC] P. Diaconis, L. Saloff-Coste. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Prob. 6, 695-750 (1996). | MR 1410112 | Zbl 0867.60043

[Eh] A. Ehrhard. Symétrisation dans l'espace de Gauss. Math. Scand. 53, 281-301 (1983). | MR 745081 | Zbl 0542.60003

[G-M] M. Gromov, V.D. Milman. A topological application of the isoperimetric inequality. Amer. J. Math. 105, 843-854 (1983). | MR 708367 | Zbl 0522.53039

[Gr1] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math. 97, 1061-1083 (1975). | MR 420249 | Zbl 0318.46049

[Gr2] L. Gross. Logarithmic Sobolev inequalities and contractive properties of semigroups. Dirichlet Forms, Varenna 1992. Lect. Notes in Math. 1563, 54-88 (1993). Springer-Verlag. | MR 1292277 | Zbl 0812.47037

[G-R] L. Gross, O. Rothaus. Herbst inequalities for supercontractive semigroups. Preprint (1997). | MR 1648283

[H-Y] Y. Higuchi, N. Yoshida. Analytic conditions and phase transition for Ising models. Lecture Notes in Japanese (1995).

[H-S] R. Holley, D. Stroock. Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys. 46, 1159-1194 (1987). | MR 893137 | Zbl 0682.60109

[H-T] C. Houdré, P. Tetali. Concentration of measure for products of Markov kernels via functional inequalities. Preprint (1997). | MR 1827807

[Hs1] E.P. Hsu. Logarithmic Sobolev inequalities on path spaces over Riemannian manifolds. Commun. Math. Phys. 189, 9-16 (1997). | MR 1478528 | Zbl 0892.58083

[Hs2] E.P. Hsu. Analysis on Path and Loop Spaces (1996). To appear in IAS/Park City Mathematics Series, Vol. 5, edited by E. P. Hsu and S. R. S. Varadhan, American Mathematical Society and Institute for Advanced Study (1997). | MR 1678311 | Zbl 1069.60500

[J-S] W.B. Johnson, G. Schechtman. Remarks on Talagrand's deviation inequality for Rademacher functions. Longhorn Notes, Texas (1987). | Zbl 0753.60024

[Kl] C.A.J. Klaassen. On an inequality of Chernoff. Ann. Probability 13, 966-974 (1985). | MR 799431 | Zbl 0576.60015

[K-S] A. Korzeniowski, D. Stroock. An example in the theory of hypercontractive semigroups. Proc. Amer. Math. Soc. 94, 87-90 (1985). | MR 781062 | Zbl 0577.47043

[Kw-S] S. Kwapien, J. Szulga. Hypercontraction methods in moment inequalities for series of independent random variables in normed spaces. Ann. Probability 19, 369-379 (1991). | MR 1085342 | Zbl 0718.60044

[K-L-O] S. Kwapien, R. Latala, K. Oleszkiewicz. Comparison of moments of sums of independent random variables and differential inequalities. J. Funct. Anal. 136, 258-268 (1996). | MR 1375162 | Zbl 0864.60014

[Le1] M. Ledoux. Isopérimétrie et inégalités de Sobolev logarithmiques gaussiennes. C. R. Acad. Sci. Paris, 306, 79-92 (1988). | MR 929114 | Zbl 0646.60006

[Le2] M. Ledoux. Remarks on logarithmic Sobolev constants, exponential integrability and bounds on the diameter. J. Math. Kyoto Univ. 35, 211-220 (1995). | MR 1346225 | Zbl 0836.60074

[Le3] M. Ledoux. Isoperimetry and Gaussian Analysis. Ecole d'Eté de Probabilités de St-Flour 1994. Lecture Notes in Math. 1648, 165-294 (1996). Springer-Verlag. | MR 1600888 | Zbl 0874.60005

[Le4] M. Ledoux. On Talagrand's deviation inequalities for product measures. ESAIM Prob. & Stat. 1, 63-87 (1996). | Numdam | Zbl 0869.60013

[L-T] M. Ledoux, M. Talagrand. Probability in Banach spaces (Isoperimetry and processes). Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag (1991). | MR 1102015 | Zbl 0748.60004

[L-Y] T.Y. Lee, H.-T. Yau. Logarithmic Sobolev inequality fo some models of random walks. Preprint (1998). | MR 1675008

[Li] P. Li. A lower bound for the first eigenvalue of the Laplacian on a compact manifold. Indiana Univ. Math. J. 28, 1013-1019 (1979). | MR 551166 | Zbl 0429.35054

[Ly] T. Lyons. Random thoughts on reversible potential theory. Summer School in Potentiel Theory, Joensuu 1990. Publications in Sciences 26, 71-114 University of Joensuu. | MR 1175217 | Zbl 0757.31007

[MD] C. Mcdiarmid. On the method of bounded differences. Surveys in Combinatorics. London Math. Soc. Lecture Notes 141, 148-188 (1989). Cambridge Univ. Press. | MR 1036755 | Zbl 0712.05012

[Mar1] K. Marton. Bounding d-distance by information divergence: a method to prove measure concentration. Ann. Probability 24, 857-866 (1996). | MR 1404531 | Zbl 0865.60017

[Mar2] K. Marton. A measure concentration inequality for contracting Markov chains. Geometric and Funct. Anal. 6, 556-571 (1997). | MR 1392329 | Zbl 0856.60072

[Mar3] K. Marton. Measure concentration for a class of random processes. Probab. Theory Relat. Fields 110, 427-439 (1998). | MR 1616492 | Zbl 0927.60050

[Mar4] K. Marton. On a measure concentration of Talagrand for dependent random variables. Preprint (1998). | MR 1392329

[Mas] P. Massart. About the constants in Talagrand's deviation inequalities for empirical processes (1998). Ann. Probability, to appear. | MR 1782276 | Zbl 01905939

[Mau1] B. Maurey. Constructions de suites symétriques. C. R. Acad. Sci. Paris 288, 679-681 (1979). | MR 533901 | Zbl 0398.46019

[Mau2] B. Maurey. Some deviations inequalities. Geometric and Funct. Anal. 1, 188-197 (1991). | MR 1097258 | Zbl 0756.60018

[Mi] V.D. Milman. Dvoretzky theorem - Thirty years later. Geometric and Funct. Anal. 2, 455-479 (1992) . | MR 1191569 | Zbl 0787.46016

[M-S] V.D. Milman, G. Schechtman. Asymptotic theory of finite dimensional normed spaces. Lecture Notes in Math. 1200 (1986). Springer-Verlag. | MR 856576 | Zbl 0606.46013

[M-W] C. Muller, F. Weissler. Hypercontractivity of the heat semigroup for ultraspherical polynomials and on the n-sphere. J. Funct. Anal. 48, 252-283 (1982). | MR 674060 | Zbl 0506.46022

[O-V] F. Otto, C. Villani. Generalization of an inequality by Talagrand, viewed as a consequence of the logarithmic Sobolev inequality. Preprint (1998). | MR 1760620

[Pi] M.S. Pinsker. Information and information stability of random variables and processes. Holden-Day, San Franscico (1964). | MR 213190 | Zbl 0125.09202

[Ro1] O. Rothaus. Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities. J. Funct. Anal. 42, 358-367 (1981). | MR 620581 | Zbl 0471.58027

[Ro2] O. Rothaus. Hypercontractivity and the Bakry-Emery criterion for compact Lie groups. J. Funct. Anal. 65, 358-367 (1986). | MR 826433 | Zbl 0589.58036

[Ro3] O. Rothaus. Logarithmic Sobolev inequalities and the growth of Lp norms (1996). | Zbl 0896.46021

[SC1] L. Saloff-Coste. Convergence to equilibrium and logarithmic Sobolev constant on manifolds with Ricci curvature bounded below. Colloquium Math. 67, 109-121 (1994). | MR 1292948 | Zbl 0816.53027

[SC2] L. Saloff-Coste. Lectures on finite Markov chains. Ecole d'Eté de Probabilités de St-Flour 1996. Lecture Notes in Math. 1665, 301-413 (1997). Springer-Verlag. | MR 1490046 | Zbl 0885.60061

[Sa] P.-M. Samson. Concentration of measure inequalities for Markov chains and Φ-mixing processes. Preprint (1998). | MR 1756011

[Sc] M. Schmuckenschläger. Martingales, Poincaré type inequalities and deviations inequalities. J. Funct. Anal. 155, 303-323 (1998). | MR 1624545 | Zbl 0924.58111

[St] D. Stroock. Logarithmic Sobolev inequalities for Gibbs states. Dirichlet forms, Varenna 1992. Lecture Notes in Math. 1563, 194-228 (1993). | MR 1292280 | Zbl 0801.60056

[S-Z] D. Stroock, B. Zegarlinski. The logarithmic Sobolev inequality for continuous spin systems on a lattice. J. Funct. Anal. 104, 299-326 (1992). | MR 1153990 | Zbl 0794.46025

[S-T] V.N. Sudakov, B.S. Tsirel'Son. Extremal properties of half-spaces for spherically invariant measures. J. Soviet. Math. 9, 9-18 (1978); translated from Zap. Nauch. Sem. L.O.M.I. 41, 14-24 (1974). | MR 365680 | Zbl 0395.28007

[Tak] M. Takeda. On a martingale method for symmetric diffusion process and its applications. Osaka J. Math. 26, 605-623 (1989). | MR 1021434 | Zbl 0717.60090

[Ta1] M. Talagrand. An isoperimetric theorem on the cube and the Khintchine-Kahane inequalities. Proc. Amer. Math. Soc. 104, 905-909 (1988). | MR 964871 | Zbl 0691.60015

[Ta2] M. Talagrand. Isoperimetry and integrability of the sum of independent Banach space valued random variables. Ann. Probability 17, 1546-1570 (1989). | MR 1048946 | Zbl 0692.60016

[Ta3] M. Talagrand. A new isoperimetric inequality for product measure. and the concentration of measure phenomenon. Israel Seminar (GAFA), Lecture Notes in Math. 1469, 91-124 (1991). Springer-Verlag. | MR 1122615 | Zbl 0818.46047

[Ta4] M. Talagrand. Some isoperimetric inequalities and their applications. Proc. of the International Congress of Mathematicians, Kyoto 1990, vol. II, 1011-1024 (1992). Springer-Verlag. | MR 1159286 | Zbl 0791.60004

[Ta5] M. Talagrand. Sharper bounds for Gaussian and empirical processes. Ann. Probability 22, 28-76 (1994). | MR 1258865 | Zbl 0798.60051

[Ta6] M. Talagrand. Concentration of measure and isoperimetric inequalities in product spaces. Publications Mathématiques de l'I.H.E.S. 81, 73-205 (1995). | Numdam | MR 1361756 | Zbl 0864.60013

[Ta7] M. Talagrand. A new look at independence. Ann. Probability, 24, 1-34 (1996). | MR 1387624 | Zbl 0858.60019

[Ta8] M. Talagrand. New concentration inequalities in product spaces. Invent. math. 126, 505-563 (1996). | MR 1419006 | Zbl 0893.60001

[Ta9] M. Talagrand. Transportation cost for Gaussian and other product measures. Geometric and Funct. Anal. 6, 587-600 (1996). | MR 1392331 | Zbl 0859.46030

[Wan] F.-Y. Wang. Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Proab. Theory Relat. Fields 109, 417-424 (1997). | MR 1481127 | Zbl 0887.35012

[Wat] G.N. Watson. A treatise on the theory of Bessel functions. Cambridge Univ. Press (1944). | MR 10746 | Zbl 0063.08184

[Z-Y] J.Q. Zhong, H.C. Yang. On the estimate of the first eigenvalue of a compact Riemanian manifold. Sci. Sinica Ser. A 27 (12), 1265-1273 (1984). | MR 794292 | Zbl 0561.53046