메뉴 건너뛰기




Volumn 44, Issue 2, 2004, Pages 325-363

Entropies, convexity, and functional inequalities: On φ-entropies and φ-Sobolev inequalities

Author keywords

[No Author keywords available]

Indexed keywords


EID: 10844281118     PISSN: 0023608X     EISSN: None     Source Type: Journal    
DOI: 10.1215/kjm/1250283556     Document Type: Article
Times cited : (170)

References (56)
  • 2
    • 19544378332 scopus 로고    scopus 로고
    • On logarithmic Sobolev inequalities for continuous time random walks on graphs
    • [AL00]
    • [AL00] C. Ané and M. Ledoux, On logarithmic Sobolev inequalities for continuous time random walks on graphs, Probab. Theory Related Fields 116-4 (2000), 573-602.
    • (2000) Probab. Theory Related Fields , vol.116 , Issue.4 , pp. 573-602
    • Ané, C.1    Ledoux, M.2
  • 3
    • 0002925263 scopus 로고
    • L'hypercontractivité et son utilisation en théorie des semigroupes
    • [Bak94] Saint-Flour, 1992 Springer, Berlin
    • [Bak94] D. Bakry, L'hypercontractivité et son utilisation en théorie des semigroupes, Lectures on probability theory (Saint-Flour, 1992), Springer, Berlin, 1994, pp. 1-114.
    • (1994) Lectures on Probability Theory , pp. 1-114
    • Bakry, D.1
  • 4
    • 0002432396 scopus 로고    scopus 로고
    • On Sobolev and logarithmic Sobolev inequalities for Markov semigroups
    • [Bak97] Charingworth, 1994, World Sci. Publishing, River Edge, NJ
    • [Bak97] _, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, New trends in stochastic analysis (Charingworth, 1994), World Sci. Publishing, River Edge, NJ, 1997, 43-75.
    • (1997) New Trends in Stochastic Analysis , pp. 43-75
  • 5
  • 6
    • 4043138417 scopus 로고    scopus 로고
    • Levels of concentration between exponential and Gaussian
    • [Bar01]
    • [Bar01] F. Barthe, Levels of concentration between exponential and Gaussian, Ann. Fac. Sci. Toulouse Math. (6) 10-3 (2001), 393-404.
    • (2001) Ann. Fac. Sci. Toulouse Math. (6) , vol.10 , Issue.3 , pp. 393-404
    • Barthe, F.1
  • 9
    • 0034553975 scopus 로고    scopus 로고
    • From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities
    • [BL00]
    • [BL00] S. Bobkov and M. Ledoux, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities, Geom. Funct. Anal. 10-5 (2000), 1028-1052.
    • (2000) Geom. Funct. Anal. , vol.10 , Issue.5 , pp. 1028-1052
    • Bobkov, S.1    Ledoux, M.2
  • 10
    • 84966220156 scopus 로고
    • A generalized Poincaré inequality for Gaussian measures
    • [Bec89]
    • [Bec89] W. Beckner, A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc. 105-2 (1989), 397-400.
    • (1989) Proc. Amer. Math. Soc. , vol.105 , Issue.2 , pp. 397-400
    • Beckner, W.1
  • 12
    • 0001445688 scopus 로고
    • Entropy differential metric, distance and divergence measures in probability spaces: A unified approach
    • [BR82a]
    • [BR82a] J. Burbea and C. R. Rao, Entropy differential metric, distance and divergence measures in probability spaces: a unified approach, J. Multivariate Anal. 12-4 (1982), 575-596.
    • (1982) J. Multivariate Anal. , vol.12 , Issue.4 , pp. 575-596
    • Burbea, J.1    Rao, C.R.2
  • 13
    • 0020205292 scopus 로고
    • On the convexity of higher order Jensen differences based on entropy functions
    • [BR82b]
    • [BR82b] _, On the convexity of higher order Jensen differences based on entropy functions, IEEE Trans. Inform. Theory 28-6 (1982), 961-963.
    • (1982) IEEE Trans. Inform. Theory , vol.28 , Issue.6 , pp. 961-963
  • 14
    • 0020126872 scopus 로고
    • On the convexity of some divergence measures based on entropy functions
    • [BR82c]
    • [BR82c] _, On the convexity of some divergence measures based on entropy functions, IEEE Trans. Inform. Theory 28-3 (1982), 489-495.
    • (1982) IEEE Trans. Inform. Theory , vol.28 , Issue.3 , pp. 489-495
  • 15
    • 0348234223 scopus 로고    scopus 로고
    • Sobolev inequalities for probability measures on the real line
    • [BR03]
    • [BR03] F. Barthe and C. Roberto, Sobolev inequalities for probability measures on the real line, Studia Math. 159 (2003), 481-497.
    • (2003) Studia Math. , vol.159 , pp. 481-497
    • Barthe, F.1    Roberto, C.2
  • 16
    • 0022773583 scopus 로고
    • Rate distortion theory with generalized information measures via convex programming duality
    • [BTT86]
    • [BTT86] A. Ben-Tal and M. Teboulle, Rate distortion theory with generalized information measures via convex programming duality, IEEE Trans. Inform. Theory 32-5 (1986), 630-641.
    • (1986) IEEE Trans. Inform. Theory , vol.32 , Issue.5 , pp. 630-641
    • Ben-Tal, A.1    Teboulle, M.2
  • 18
    • 10844286005 scopus 로고    scopus 로고
    • Gaussian maximum of entropy and reversed logarithmic Sobolev inequality
    • [Cha03] Séminaire de Probabilité XXXVI, Springer, Berlin
    • [Cha03] D. Chafaï, Gaussian maximum of entropy and reversed logarithmic Sobolev inequality, Séminaire de Probabilité XXXVI, Lecture Notes in Math. 1801, Springer, Berlin, 2003, 194-200.
    • (2003) Lecture Notes in Math. , vol.1801 , pp. 194-200
    • Chafaï, D.1
  • 19
    • 0039907778 scopus 로고    scopus 로고
    • Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces
    • [CHL97] electronic
    • [CHL97] M. Capitaine, E. P. Hsu and M. Ledoux, Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces, Electron. Comm. Probab. 2 (1997), 71-81 (electronic).
    • (1997) Electron. Comm. Probab. , vol.2 , pp. 71-81
    • Capitaine, M.1    Hsu, E.P.2    Ledoux, M.3
  • 21
    • 0001856347 scopus 로고
    • Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten
    • [Csi63]
    • [Csi63] I. Csiszár, Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten, Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1963), 85-108.
    • (1963) Magyar Tud. Akad. Mat. Kutató Int. Közl. , vol.8 , pp. 85-108
    • Csiszár, I.1
  • 22
    • 0001569844 scopus 로고
    • A class of measures of informativity of observation channels
    • [Csi72] Collection of articles dedicated to the memory of Alfréd Rényi, I.
    • [Csi72] _, A class of measures of informativity of observation channels, Period. Math. Hungar. 2 (1972), 191-213, Collection of articles dedicated to the memory of Alfréd Rényi, I.
    • (1972) Period. Math. Hungar. , vol.2 , pp. 191-213
  • 25
    • 0000145028 scopus 로고
    • Inégalité du type de Poincaré sur l'espace des chemins riemanniens
    • [Fan94]
    • [Fan94] S. Fang, Inégalité du type de Poincaré sur l'espace des chemins riemanniens, C. R. Acad. Sci. Paris Sér. I Math. 318-3 (1994), 257-260.
    • (1994) C. R. Acad. Sci. Paris Sér. I Math. , vol.318 , Issue.3 , pp. 257-260
    • Fang, S.1
  • 27
    • 0000291394 scopus 로고    scopus 로고
    • The log-Sobolev inequality on loop space over a compact Riemannian manifold
    • [GM98a]
    • [GM98a] F.-Z. Gong and Z.-M. Ma, The log-Sobolev inequality on loop space over a compact Riemannian manifold, J. Funct. Anal. 157-2 (1998), 599-623.
    • (1998) J. Funct. Anal. , vol.157 , Issue.2 , pp. 599-623
    • Gong, F.-Z.1    Ma, Z.-M.2
  • 28
    • 18144452001 scopus 로고    scopus 로고
    • Martingale representation and log-Sobolev inequality on loop space
    • [GM98b]
    • [GM98b] _, Martingale representation and log-Sobolev inequality on loop space, C. R. Acad. Sci. Paris Sér. I Math. 326-6 (1998), 749-753.
    • (1998) C. R. Acad. Sci. Paris Sér. I Math. , vol.326 , Issue.6 , pp. 749-753
  • 29
    • 0038445488 scopus 로고    scopus 로고
    • Clark formula and logarithmic Sobolev inequalities for Bernoulli measures
    • [GP03]
    • [GP03] F. Gao and N. Privault, Clark formula and logarithmic Sobolev inequalities for Bernoulli measures, C. R. Math. Acad. Sci. Paris 336-1 (2003), 51-56.
    • (2003) C. R. Math. Acad. Sci. Paris , vol.336 , Issue.1 , pp. 51-56
    • Gao, F.1    Privault, N.2
  • 30
    • 0347466412 scopus 로고    scopus 로고
    • Lectures on logarithmic Sobolev inequalities
    • [GZ03] Séminaire de Probabilité XXXVI, Springer, Berlin
    • [GZ03] A. Guionnet and B. Zegarliński, Lectures on logarithmic Sobolev inequalities, Séminaire de Probabilité XXXVI, Lecture Notes in Math. 1801, Springer, Berlin, 2003, 1-134.
    • (2003) Lecture Notes in Math. , vol.1801 , pp. 1-134
    • Guionnet, A.1    Zegarliński, B.2
  • 31
    • 0004151494 scopus 로고
    • [HJ90] Cambridge University Press, Cambridge, corrected reprint of the 1985 original
    • [HJ90] R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1990, corrected reprint of the 1985 original.
    • (1990) Matrix Analysis
    • Horn, R.A.1    Johnson, C.R.2
  • 32
    • 0001359676 scopus 로고
    • Logarithmic Sobolev inequalities and stochastic Ising models
    • [HS87]
    • [HS87] R. A. Holley and D. W. Stroock, Logarithmic Sobolev inequalities and stochastic Ising models, J. Statist. Phys. 46-5-6 (1987), 1159-1194.
    • (1987) J. Statist. Phys. , vol.46 , Issue.5-6 , pp. 1159-1194
    • Holley, R.A.1    Stroock, D.W.2
  • 33
    • 10844231509 scopus 로고    scopus 로고
    • Stochastic analysis on manifolds
    • [Hsu02] American Mathematical Society, Providence, RI
    • [Hsu02] E. P. Hsu, Stochastic analysis on manifolds, Graduate Studies in Mathematics, vol. 38, American Mathematical Society, Providence, RI, 2002.
    • (2002) Graduate Studies in Mathematics , vol.38
    • Hsu, E.P.1
  • 34
    • 0005512419 scopus 로고    scopus 로고
    • A unified approach to several inequalities for Gaussian and diffusion measures
    • [Hu00] Springer, Berlin
    • [Hu00] Y.-Z. Hu, A unified approach to several inequalities for Gaussian and diffusion measures, Séminaire de Probabilités, XXXIV, Springer, Berlin, 2000, 329-335.
    • (2000) Séminaire de Probabilités , vol.34 , pp. 329-335
    • Hu, Y.-Z.1
  • 35
    • 0038032110 scopus 로고    scopus 로고
    • [HUL01] Grundlehren Text Editions, Springer-Verlag, Berlin, abridged version of Convex analysis and minimization algorithms. I [Springer, Berlin, 1993; MR 95m:90001] and II [ibid.; MR 95m:90002]
    • [HUL01] J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of convex analysis, Grundlehren Text Editions, Springer-Verlag, Berlin, 2001, abridged version of Convex analysis and minimization algorithms. I [Springer, Berlin, 1993; MR 95m:90001] and II [ibid.; MR 95m:90002].
    • (2001) Fundamentals of Convex Analysis
    • Hiriart-Urruty, J.-B.1    Lemaréchal, C.2
  • 37
    • 0141529857 scopus 로고    scopus 로고
    • Entropy and a generalisation of "Poincaré's observation"
    • [Joh03]
    • [Joh03] O. Johnson, Entropy and a generalisation of "Poincaré's observation", Math. Proc. Cambridge Philos. Soc. 135-2 (2003), 375-384.
    • (2003) Math. Proc. Cambridge Philos. Soc. , vol.135 , Issue.2 , pp. 375-384
    • Johnson, O.1
  • 38
    • 0004171561 scopus 로고
    • Brownian motion and stochastic calculus, second ed
    • [KS91] Springer-Verlag, New York
    • [KS91] I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, second ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991.
    • (1991) Graduate Texts in Mathematics , vol.113
    • Karatzas, I.1    Shreve, S.E.2
  • 39
    • 0000458262 scopus 로고    scopus 로고
    • Concentration of measure and logarithmic Sobolev inequalities
    • [Led99] Springer, Berlin
    • [Led99] M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, Séminaire de Probabilités, XXXIII, Springer, Berlin, 1999, 120-216.
    • (1999) Séminaire de Probabilités , vol.33 , pp. 120-216
    • Ledoux, M.1
  • 40
    • 0000614531 scopus 로고    scopus 로고
    • The geometry of Markov diffusion generators
    • [Led00] Probability Theory
    • [Led00] _, The geometry of Markov diffusion generators, Ann. Fac. Sci. Toulouse Math. (6) 9-2 (2000), 305-366, Probability Theory.
    • (2000) Ann. Fac. Sci. Toulouse Math. (6) , vol.9 , Issue.2 , pp. 305-366
  • 41
    • 0038609707 scopus 로고    scopus 로고
    • [Led01] American Mathematical Society, Providence, RI
    • [Led01] _, The concentration of measure phenomenon, American Mathematical Society, Providence, RI, 2001.
    • (2001) The Concentration of Measure Phenomenon
  • 42
    • 84996133877 scopus 로고
    • On Talagrand's deviation inequalities for product measures
    • [Led97] electronic
    • [Led97] _, On Talagrand's deviation inequalities for product measures, ESAIM Probab. Statist. 1 (1995/97), 63-87 (electronic).
    • (1995) ESAIM Probab. Statist. , vol.1 , pp. 63-87
  • 43
    • 84968519073 scopus 로고
    • Some convexity and subadditivity properties of entropy
    • [Lie75]
    • [Lie75] E. H. Lieb, Some convexity and subadditivity properties of entropy, Bull. Amer. Math. Soc. 81 (1975), 1-13.
    • (1975) Bull. Amer. Math. Soc. , vol.81 , pp. 1-13
    • Lieb, E.H.1
  • 44
    • 0038722404 scopus 로고    scopus 로고
    • Between Sobolev and Poincaré
    • [LO00] Geometric aspects of functional analysis, Springer, Berlin
    • [LO00] R. Latała and K. Oleszkiewicz, Between Sobolev and Poincaré, Geometric aspects of functional analysis, Lecture Notes in Math. 1745, Springer, Berlin, 2000, 147-168.
    • (2000) Lecture Notes in Math. , vol.1745 , pp. 147-168
    • Latała, R.1    Oleszkiewicz, K.2
  • 45
    • 10844238565 scopus 로고    scopus 로고
    • [Lug04] Preprint, presented at the Machine Learning Summer School 2003, Australian National University, Canberra and at the Workshop on combinatorics, probability and algorithms 2003, held at the CRM in Montreal, February
    • [Lug04] G. Lugosi, Concentration-of-measure inequalities, Preprint, presented at the Machine Learning Summer School 2003, Australian National University, Canberra and at the Workshop on combinatorics, probability and algorithms 2003, held at the CRM in Montreal, February, 2004.
    • (2004) Concentration-of-measure Inequalities
    • Lugosi, G.1
  • 46
    • 9444248338 scopus 로고    scopus 로고
    • Concentration inequalities and model selection
    • [Mas03] Saint-Flour Summer School XXXIII, preprint, July
    • [Mas03] P. Massart, Concentration Inequalities and Model Selection, Lecture Notes, Saint-Flour Summer School XXXIII, preprint, July, 2003.
    • (2003) Lecture Notes
    • Massart, P.1
  • 47
  • 48
    • 10844228664 scopus 로고    scopus 로고
    • On logarithmic Sobolev inequalities for normal martingales
    • [Pri00]
    • [Pri00] N. Privault, On logarithmic Sobolev inequalities for normal martingales, Ann. Fac. Sci. Toulouse Math. (6) 9-3 (2000), 509-518.
    • (2000) Ann. Fac. Sci. Toulouse Math. (6) , vol.9 , Issue.3 , pp. 509-518
    • Privault, N.1
  • 49
    • 0012168784 scopus 로고    scopus 로고
    • Une initiation aux inégalités de Sobolev logarithmiques
    • [Roy99] Société Mathématique de France, Paris
    • [Roy99] G. Royer, Une initiation aux inégalités de Sobolev logarithmiques, Cours Spécialisés [Specialized Courses] 5, Société Mathématique de France, Paris, 1999.
    • (1999) Cours Spécialisés [Specialized Courses] , vol.5
    • Royer, G.1
  • 51
    • 0000180793 scopus 로고
    • On multiple Poisson stochastic integrals and associated Markov semigroups
    • [Sur84]
    • [Sur84] D. Surgailis, On multiple Poisson stochastic integrals and associated Markov semigroups, Probab. Math. Statist. 3-2 (1984), 217-239.
    • (1984) Probab. Math. Statist. , vol.3 , Issue.2 , pp. 217-239
    • Surgailis, D.1
  • 52
    • 0027283206 scopus 로고
    • Convergence of best φ-entropy estimates
    • [TV93]
    • [TV93] M. Teboulle and I. Vajda, Convergence of best φ-entropy estimates, IEEE Trans. Inform. Theory 39-1 (1993), 297-301.
    • (1993) IEEE Trans. Inform. Theory , vol.39 , Issue.1 , pp. 297-301
    • Teboulle, M.1    Vajda, I.2
  • 53
    • 1542342359 scopus 로고    scopus 로고
    • Topics in optimal transportation
    • [Vil03] American Mathematical Society, Providence, RI
    • [Vil03] C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics 58, American Mathematical Society, Providence, RI, 2003.
    • (2003) Graduate Studies in Mathematics , vol.58
    • Villani, C.1
  • 54
    • 0039938081 scopus 로고    scopus 로고
    • Logarithmic Sobolev inequalities for diffusion processes with application to path space
    • [Wan96]
    • [Wan96] F.-Y. Wang, Logarithmic Sobolev inequalities for diffusion processes with application to path space, Chinese J. Appl. Probab. Statist. 12-3 (1996), 255-264.
    • (1996) Chinese J. Appl. Probab. Statist. , vol.12 , Issue.3 , pp. 255-264
    • Wang, F.-Y.1
  • 55
    • 10844227036 scopus 로고    scopus 로고
    • [Wan02] preprint, October to appear in Potential Analysis
    • [Wan02] _, A Generalized Beckner-Type Inequality, preprint, October 2002, to appear in Potential Analysis.
    • (2002) A Generalized Beckner-Type Inequality
  • 56
    • 0034367243 scopus 로고    scopus 로고
    • A new modified logarithmic Sobolev inequality for Poisson point processes and several applications
    • [Wu00]
    • [Wu00] L. Wu, A new modified logarithmic Sobolev inequality for Poisson point processes and several applications, Probab. Theory Related Fields 118-3 (2000), 427-438.
    • (2000) Probab. Theory Related Fields , vol.118 , Issue.3 , pp. 427-438
    • Wu, L.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.