Lectures on Logarithmic Sobolev Inequalities
Séminaire de probabilités de Strasbourg, Tome 36 (2002), pp. 1-134.
@article{SPS_2002__36__1_0,
     author = {Guionnet, A. and Zegarlinski, B.},
     title = {Lectures on {Logarithmic} {Sobolev} {Inequalities}},
     journal = {S\'eminaire de probabilit\'es de Strasbourg},
     pages = {1--134},
     publisher = {Springer - Lecture Notes in Mathematics},
     volume = {36},
     year = {2002},
     mrnumber = {1971582},
     zbl = {02046374},
     language = {en},
     url = {http://www.numdam.org/item/SPS_2002__36__1_0/}
}
TY  - JOUR
AU  - Guionnet, A.
AU  - Zegarlinski, B.
TI  - Lectures on Logarithmic Sobolev Inequalities
JO  - Séminaire de probabilités de Strasbourg
PY  - 2002
SP  - 1
EP  - 134
VL  - 36
PB  - Springer - Lecture Notes in Mathematics
UR  - http://www.numdam.org/item/SPS_2002__36__1_0/
LA  - en
ID  - SPS_2002__36__1_0
ER  - 
%0 Journal Article
%A Guionnet, A.
%A Zegarlinski, B.
%T Lectures on Logarithmic Sobolev Inequalities
%J Séminaire de probabilités de Strasbourg
%D 2002
%P 1-134
%V 36
%I Springer - Lecture Notes in Mathematics
%U http://www.numdam.org/item/SPS_2002__36__1_0/
%G en
%F SPS_2002__36__1_0
Guionnet, A.; Zegarlinski, B. Lectures on Logarithmic Sobolev Inequalities. Séminaire de probabilités de Strasbourg, Tome 36 (2002), pp. 1-134. http://www.numdam.org/item/SPS_2002__36__1_0/

[1] Aida, S.; Stroock, D.W.; Moment estimates derived from Poincaré and logarithmic Sobolev inequalities. Math.Res.Lett. 1, No. 1, 75-86 (1994) | MR | Zbl

[2] Aizenmann, M.; Holley, R.; Rapid Convergence to Equilibrium of Stochastic Ising Models in the Dobrushin-Shlosman Regime, Percolation Theory and Ergodic Theory of Infinite Particle Systems, ed. Kesten H, Springer-Verlag, 1-11 (1987) | MR | Zbl

[3] Albeverio, S.; Kondratiev Yu.G. ; Röckner, M.; Dirichlet operators and Gibbs measures, Collection: On Klauder's path: a field trip, World Sci. Publishing, River Edge, NJ, 1-10, (1994) | MR | Zbl

[4] Albeverio, S.; Röckner, M.; Dirichlet Form Methods for Uniqueness of Martingale Problems and Applications Collection: Stochastic analysis (Ithaca, NY, 1993), 513-528 Proc. Sympos. Pure Math. 57, Amer. Math. Soc., Providence, RI (1995) | MR | Zbl

[5] Bakry, D.; L'Hypercontractivité et son Utilisation en Théorie des Semi-Groupes, Lectures on Probability Theory, Ecole d'Eté de Probabilités de Saint-Flour XXII - 1992, Ed. P. Bernard, LNM 1581, 1-114 (1994) | MR | Zbl

[6] Bakry, D.; Emery, M.; Diffusions Hypercontractives, Séminaire de Probabilités XIX (1983/84 Proceedings), LNM 1123 (Eds J. Azéma and M. Yor), 177-206 (1985) | Numdam | MR | Zbl

[7] Bakry, D. and Ledoux, M.; Lévy-Gromov Isoperimetric Inequality for an Infinite Dimensional Diffusion Generator, Invent. Math. 123, 259-281 (1996). | MR | Zbl

[8] Beckner, W.; Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. 138, 213 - 242 (1993) | MR | Zbl

[9] Bertini, L.; Zegarlinski, B.; Coercive inequalities for Gibbs measure. J. Funct. Anal. 162, no 2, 257-286 (1999) | MR | Zbl

[10] Bobkov, S.G.; An Isoperimetric Inequality on the Discrete Cube, And an Elementary Proof of the Isoperimetric Inequality in Gauss Space, Ann. Prob. 25, 206 - 214 (1997) | MR | Zbl

[11] Bobkov, S.G.; Gotze, F. ; Exponential integrability and transportation cost related to log-Sobolev inequalities. J. Funct. Anal. 163, no 1, 1-28 (1999) | MR | Zbl

[12] Bourbaki, N.; Théories Spectrales, Hermann (1967) | Zbl

[13] Carlen, E.A.; Stroock, D.W.; An application of the Bakry-Emery criterion to infinite dimensional diffusions, Sém. de Probabilités XX, Azéma J. and Yor M. (eds.) LNM 1204, 341-348 (1986) | Numdam | MR | Zbl

[14] Cesi, F.; Maes, C.; Martinelli, F.; Relaxation of disordered magnets in the Griffiths regime. Commun. Math. Phys. 188, no 1, 135-173 (1997) | MR | Zbl

[15] Cesi, F.; Maes, C.; Martinelli, F.; Relaxation to equilibrium for two dimensional disordered Ising systems in the Griffiths phase. Commun. Math. Phys. 189, No 2, 323-335 (1997) | MR | Zbl

[16] Chen, M.; Wang, F.; Estimates of logarithmic Sobolev constant: An improvement of Bakry-Emery criterion. J. Funct. Anal. 144, no.2, 287-300 (1997) | MR | Zbl

[17] Chung, F.K.; Grigor'Yan, A.; Yau, S.T. ; Eigenvalues and a diameter for manifolds and graphs. Tsing Hua lectures on geometry and analysis Taiwan (1990-91), International press, 79-105 (1997) | MR | Zbl

[18] Davies, E.B.; Heat kernels and spectral theory, Cambridge University Press (1989) | MR | Zbl

[19] Davies, E.B.; Gross L.; Simon, B.; Hypercontractivity : A bibliographical review, Proceedings of the Hoegh-Krohn Memorial Conference, Eds. S. Albeverio, J.E. Fenstad, H. Holden and T. Lindstrom, 370-389. | MR | Zbl

[20] Davies, E.B.; Simon, B.; Ultracontractivity and the Heat Kernel for Schrödinger Operators and Dirichlet Laplacians, J. Func. Anal. 59, 335-395 (1984) | MR | Zbl

[21] Deuschel, J.-D.; Algebraic L2 Decay of Attractive Critical Processes on the Lattice, Ann. Prob. 22, 264-283 (1994) | MR | Zbl

[22] Deuschel, J.-D. ; Stroock, D.W.; Large deviations, Academic Press Inc. (1989) | MR | Zbl

[23] Deuschel, J.-D.; Stroock, D.W.; Hypercontractivity and Spectral Gap of Symmetric Diffusions with applications to the Stochastic Ising Model, J. Func. Anal. 92, 30-48 (1990) | MR | Zbl

[24] Diaconis, P.; Saloff-Coste, L.; Nash inequalities for finite Markov chains, J. Theor. Probab. 9, no 2, 459-510 (1996) | MR | Zbl

[25] Diaconis, P.; Saloff-Coste, L.; Logarithmic Sobolev inequalities for finite Markov chains, Ann. Appl. Probab. 6, no 3, 695-750 (1996) | MR | Zbl

[26] Dobrushin, R.; The description of Random Fields by Means of Conditional Probabilities and Conditions of its Regularity, Theor. Prob. its Appl. 13, 197-224 (1968) | Zbl

[27] Dobrushin, R.; Prescribing a system of random variables by conditionnal distribution, Theor. prob. Appl. 15, 458-486 (1970) | Zbl

[28] Dobrushin, R.; Markov Processes with a Large Number of Locally Interacting Components Problems, Inf. Trans. 7, 149-164 and 235-241 (1971)

[29] Dobrushin, R.; Kassalygo, L.A.; Uniqueness of a Gibbs Fields with Random Potential - An Elementary Approach, Theory. Probab. Appl. 31, 572-589 (1986) | MR | Zbl

[30] Dobrushin, R.; Kotecky, R.; Shlosman, S.; Wulff construction. A global shape from local interaction, AMS Translations of mathematical monographs (1992) | MR | Zbl

[31] Dobrushin, R.; Shlosman, S.; Constructive criterion for the uniqueness of Gibbs field. Statistical Physics and Dynamical Systems, Rigorous Results Eds. Fritz, Jaffe and Szasz, 347-370, Birkhäuser (1985) | MR | Zbl

[32] Dobrushin, R.; Shlosman, S.; Completely analytical Gibbs fields, Statistical Physics and Dynamical Systems, Rigorous Results Eds. Fritz, Jaffe and Szasz, 371-403, Birkhäuser (1985) | MR | Zbl

[33] Dobrushin, R.; Shlosman, S.; Completely analytical interactions: constructive description, J. Stat. Phys. 46, 983-1014 (1987) | MR | Zbl

[34] Von Dreyfus, H. ; Klein, A.; Perez, J.F.; Taming Griffiths singularities : Infinite differentiability of quenched correlation functions, Comm. Math. Phys. 170, 21-39 (1995) | MR | Zbl

[35] Driver, B.; Lohrenz, T.; Logarithmic Sobolev inequalities for pinned loop groups, J. Funct. Anal. 140, no 2, 381-448 (1996) | MR | Zbl

[36] Dunlop, F.; Correlation Inequalities for multicomponent Rotators, Commun. Math. Phys., 49, 247-256 (1976) | MR

[37] Ethier, S.N.; Kurtz, T.G.; Markov Processes, characterization and convergence, J. Wiley and Sons (1985) | MR | Zbl

[38] Van Enter, A.C.D. ; Zegarlinski, B. ; A Remark on Differentiability of the Pressure Functional, Rev. Math. Phys. 17, 959-977 (1995) | MR | Zbl

[39] Federbush, I.; A partially alternative derivation of a result of Nelson, J. Math. Phys 10, 50-52 (1969) | Zbl

[40] Fisher, D.; Huse, D.; Dynamics of droplet fluctuations in pure and random Ising systems, Phys. Rev. 35, 6841 (1987)

[41] Föllmer, H.; A covariance estimate for Gibbs measures, J. Func. Anal. 46, 387-395 (1982) | MR | Zbl

[42] Fougères, P.; Hypercontractivité et isopérimétrie gaussienne. Applications aux systèmes de spins., Ann. Inst. Henri Poincaré, Probabilités et Statistiques 36, no 5, 647-689 (2000) | Numdam | MR | Zbl

[43] Fröhlich, J.; Imbrie, J.Z.; Improved Perturbation Expansion for Disordered Systems: Beating Griffiths Singularities. Commun. Math. Phys. 96 145-180 (1984) | MR | Zbl

[44] Fukushima, M.; Oshima, Y.; Takeda, M.; Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics 19, Walter de Gruyter and Co., Berlin, (1994) | MR | Zbl

[45] Georgii, H.O.; Gibbs measures and phase transitions, Walter de Gruyter (1988) | MR | Zbl

[46] Gielis, G. ; PhD Thesis, Leuven (1995)

[47] Gielis, G.; Maes, C.; The uniqueness regime of Gibbs fields with unbounded disorder, J. Statist. Phys. 81, no. 3-4, 829-835 (1995) | MR | Zbl

[48] Glimm, J.; Jaffe, A.; Quantum Physics: A Functional Integral Point of View, Springer-Verlag 1981, 1987 | Zbl

[49] Goldstein, J.A.; Semigroups of linear operators and Applications, Oxford Science Publications (1970)

[50] Goldstein, J.A.; Ruiz Goldstein, G.; Semigroups of linear and non linear operators and Applications, Proceedings of the Curucao conference, August 1992. Kluwer Academic publishers. | MR | Zbl

[51] Gray, L.; Griffeath, D.; On the Uniqueness of Certain Interacting Particle Systems, Z. Wahr. v. Geb. 35, 75-86 (1976) | MR | Zbl

[52] Gross, L.; Logarithmic Sobolev inequalities, Amer. J. Math. 97 , 1061-1083 (1976) | MR | Zbl

[53] Guionnet, A.; Zegarlinski, B.; Decay to Equilibrium in Random Spin Systems on a Lattice. Commun. Math. Phys. 181, no 3, 703-732 (1996) | MR | Zbl

[54] Guionnet, A.; Zegarlinski, B.; Decay to Equilibrium in Random Spin Systems on a Lattice. Journal of Stat. 86, 899-904 (1997) | MR | Zbl

[55] Hebey, E.; Sobolev spaces on Riemannian manifolds, LNM 1635, Springer-Verlag (1996) | MR | Zbl

[56] Herbst, I.; On Canonical Quantum Field Theories, J. Math. Phys. 17, 1210-1221 (1976) | MR

[57] Higuchi, Y.; Yoshida, N.; Slow relaxation of stochastic Ising models with random and non random boundary conditions, New trends in stochastic analysis, ed. K. Elworthy, S. Kusuoka, I. Shigekawa, 153-167 (1997) | MR

[58] Holley, R.; The one-dimensional stochastic X-Y model. Collection: Random walks, Brownian motion, and interacting particle systems, Progr. Probab. 28, Birkhäuser, Boston MA, 295-307 (1991) | MR | Zbl

[59] Holley, R.; Stroock, D.W.; Logarithmic Sobolev inequalities and stochastic Ising models.J. Stat. Phys. 46, 1159-1194 (1987) | MR | Zbl

[60] Holley, R.; Stroock, D.W., Uniform and L2 Convergence in One Dimensional Stochastic Ising models, Commun. Math. Phys. 123, 85-93 (1989) | MR | Zbl

[61] Karatzas, I.; Shreve, S.E.; Brownian motion and stochastic calculus. Second edition. Graduate texts in Mathematics. Springer-Verlag. | MR | Zbl

[62] Kunita, H.; Absolute continuity of Markov processes and generators Nagoya Math. J. 36, 1-26 (1969) | MR | Zbl

[63] Kesten, H.; Aspect of first passage percolation. Ecole d'été de St-Flour, LNM 1180, 125-264 (1986) | MR | Zbl

[64] Laroche, E.; Hypercontractivité pour des systèmes de spins de portée infinie, Prob. Theo. Rel. Fields 101, No. 1, 89-132 (1995) | MR | Zbl

[65] Latala, R. and Oleszkiewicz, K.; Between Sobolev and Poincare, in Geometric Aspects of Functional Analysis, pp. 147-168, Lecture Notes in Math. 1745, Springer, Berlin 2000 | MR | Zbl

[66] Ledoux, M.; Concentration of measure and logarithmic Sobolev inequalities, Sém. de Proba. 33 (1997) Lecture Notes in Mathematics 1709, 120-216 Springer (1999) | Numdam | MR | Zbl

[67] Liggett, T.M.; Infinite Particle Systems, Springer-Verlag, Grundlehren Series 276, New York (1985) | Zbl

[68] Liggett, T.M.; L2 Rates of Convergence for Attractive Reversible Nearest Particle Systems: The Critical Case, Ann. Probab. 19, 935-959 (1991) | MR | Zbl

[69] Lu, S.-L. ; Yau, H.-T.; Spectral Gap and Logarithmic Sobolev Inequality for Kawasaki and Glauber Dynamics, Commun. Math. Phys. 156, 399-433 (1993) | MR | Zbl

[70] Ma, Z.M. ; , Röckner, M. ; Introduction to the theory of (nonsymmetric) Dirichlet forms. Universitext. Springer-Verlag, Berlin (1992) | MR | Zbl

[71] Maes, C.; Shlosman, S.; Ergodicity of probabilistic cellular automata: a constructive criterion. Comm. Math. Phys. 135, no. 2, 233-251 (1991) | MR | Zbl

[72] Maes, C.; Shlosman, S.; When is an interacting particle system ergodic?, Comm. Math. Phys. 151, no. 3, 447-466 (1993) | MR | Zbl

[73] Martinelli, F.; On the two dimensionnal dynamical Ising model in the phase coexistence region. Journal Stat. Phys. 76, 1179 (1994) | MR | Zbl

[74] Martinelli, F.; Lectures on Glauber dynamics for discrete spin models Lectures on probability theory and statistics LNM 1717, 93-191 (St-Flour, 1997) Springer-Verlag (1999) | MR | Zbl

[75] Martinelli, F.; Olivieri, E.; Approach to Equilibrium of Glauber Dynamics in the One Phase Region: I. The Attractive case/ II. The General Case. Commun. Math. Phys. 161, 447-486 / 487-514 (1994) | MR | Zbl

[76] Minlos, R.A.; Invariant subspaces of the stochastic Ising high temperature dynamics, Markov Process and Rel. Fields 2, no. 2, 263-284 (1996) | MR | Zbl

[77] Olivieri, E.; Picco, P.; Cluster Expansion for D-dimensional Lattice Systems and Finite Volume Factorization Properties, J. Stat. Phys. 59, 221 (1990) | MR | Zbl

[78] Olkiewicz, R.; Zegarlinski, B.; Hypercontractivity in Non-commutative Lp spaces, J. Funct. Anal. 161 , 246-285 (1999) | MR | Zbl

[79] Prakash, C.; High-temperature differentiability of lattice Gibbs states by Dobrushin uniqueness techniques, J. Stat. Phys. 31, 169-228 (1983) | MR

[80] Preston, C.; Random fields, LNM 534 , Springer 1976 | MR | Zbl

[81] Rao, M.M. and Ren, Z.D.; Theory of Orlicz spaces , New York : Marcel Dekker, 1991 | MR | Zbl

[82] Revuz, D.; Yor, M. ; Continuous martingales and Brownian motion. Springer New York (1991) | MR | Zbl

[83] Röckner, M.; Dirichlet forms on infinite-dimensional state space and applications. Collection: Stochastic analysis and related topics (Silivri, 1990), 131-185 Progr. Probab. 31, Birkhäuser, Boston, MA (1992) | MR | Zbl

[84] Reed, J.; Simon, B.; Methods of Modern Mathematical Physics , Academic Press 1975

[85] Rosen, J.; Sobolev Inequalities for Weight Spaces and Supercontractivity, Trans. A.M.S. 222, 367-376 (1976) | MR | Zbl

[86] Roth, J.P. ; Opérateurs dissipatifs et semi-groupes dans les espaces de fonctions continues. Ann. Inst. Fourier 26 , 1-97 (1976) | Numdam | MR | Zbl

[87] Rothaus, O.S. ; Logarithmic Sobolev Inequalities and the Spectrum of Schrödinger Operators, J. Func. Anal. 42, 110-378 (1981) | MR | Zbl

[88] G. Royer; Une initiation aux inégalités de Sobolev logarithmiques, Société Mathématique de France, Cours spécialisés (1999) | MR | Zbl

[89] W. Rudin; Real and Complex Analysis Mc Graw-Hill International | Zbl

[90] Ruelle D.; Statistical Mechanics: Rigorous Results. W.A. Benjamin Inc. (1969) | MR | Zbl

[91] Saloff Coste, L.; Lecture notes on finite Markov Chains, LNM 1665, 301-413 Springer (1997) | MR | Zbl

[92] Schonmann, R.; Schlosman, S.; Complete analyticity for 2D Ising model completed Comm. Math. Phys 170, 453 (1995) | MR | Zbl

[93] Simon B.; The P(φ)2 Euclidian (Quantum) Field Theory, Princeton Univ. Press 1974

[94] Simon, B.; The Statistical Mechanics of Lattice Gasses, Princeton Univ. Press (1993) | MR | Zbl

[95] Simon, B.; A remark on Nelson's best hypercontractive estimates, Proc. A.M.S. 55, 376-378 (1976) | MR | Zbl

[96] Sinai, Y.G.; Theory of Phase Transitions : Rigorous Results, Pergamon Press, Oxford (1982) | MR | Zbl

[97] Stroock, D.W.; Logarithmic Sobolev Inequalities for Gibbs states, Dirichlet Forms, Varenna 1992, LNM 1563, 194-228 (Springer-Verlag 993) eds G. Dell'Antonio, U. Mosco. (1993) | MR | Zbl

[98] Stroock D.W.; Zegarlinski B.; The Logarithmic Sobolev Inequality for Continuous Spin Systems on a Lattice. J. Funct. Anal. 104, 299-326 (1992) | MR | Zbl

[99] Stroock D.W.; Zegarlinski B.; The Equivalence of the Logarithmic Sobolev Inequality and the Dobrushin-Shlosman Mixing Condition. Commun. Math. Phys. 144, 303-323 (1992) | MR | Zbl

[100] Stroock D.W.; Zegarlinski B.; The Logarithmic Sobolev inequality for Discrete Spin Systems on a Lattice. Commun. Math. Phys. 149, 175-193 (1992) | MR | Zbl

[101] Stroock D.W.; Zegarlinski B.; On the ergodic properties of Glauber dynamics. J. Stat. Phys. 81, 1007-1019 (1995) | MR | Zbl

[102] Stroock D.W.; Varadhan, S.R.S.; Diffusion processes with continuous coefficients, I and II. Comm. Pure Applied Math. 22, 345-400, 479-530 (1969) | Zbl

[103] Sullivan W.G.; A unified Existence and Ergodic Theorem for Markov Evolution of Random Fields, Z. Wahr. Verv. Geb. 31, 47-56 (1974) | MR | Zbl

[104] Thomas L.E.; Bound on the Mass Gap for Finite Volume Stochastic Ising Models at Low Temperature. Commun. Math. Phys. 126, 1-11 (1989) | MR | Zbl

[105] Wang F.Y.; Estimates of logarithmic Sobolev constant for finite-volume continuous spin systems. J. Stat. Phys. 84, No. 1-2, 277-293 (1996) | MR | Zbl

[106] Yau H.T.; Logarithmic Sobolev inequality for lattice gases with mixing conditions. Comm. Math. Phys. 181, no. 2, 367-408 (1996) | MR | Zbl

[107] Yau H.T.; Logarithmic Sobolev inequality for generalized simple exclusion processes. Probab. Theory Related Fields 109, no. 4, 507-538 (1997) | MR | Zbl

[108] Yoshida, N.; Relaxed criteria of the Dobrushin-Shlosman mixing condition. J. Stat. Phys. 87, no. 1-2, 293-309 (1997) | MR | Zbl

[109] Zegarlinski B.; On log-Sobolev Inequalities for Infinite Lattice Systems, Lett. Math. Phys. 20, 173-182 (1990) | MR | Zbl

[110] Zegarlinski B.; Log-Sobolev Inequalities for Infinite One Dimensional Lattice Systems, Commun. Math. Phys. 133, 147-162 (1990) | MR | Zbl

[111] Zegarlinski B.; Dobrushin Uniqueness Theorem and Logarithmic Sobolev Inequalities, J. Funct. Anal. 105, 77-111 (1992) | MR | Zbl

[112] Zegarlinski B.; Hypercontractive Semigroups and Applications, Bochum Lecture Notes 1992, unpublished

[113] Zegarlinski B.; Strong Decay to Equilibrium in One Dimensional Random Spin Systems. J. Stat. Phys. 77, 717-732 (1994) | MR | Zbl

[114] Zegarlinski B.; Ergodicity of Markov Semigroups. Proc. of the Conference : Stochastic Partial Differential Equations, Edinburgh 1994, Ed. A. Etheridge, London Math Soc. Lect. Notes 216, Cambridge University Press (1995) | MR | Zbl

[115] Zegarlinski B.; The Strong Decay to Equilibrium for the Stochastic Dynamics of an Unbounded Spin System on a Lattice, Commun. Math. Phys. 175, 401-432 (1996) | MR | Zbl

[116] Zegarlinski B.; Isoperimetry for Gibbs measures, Ann. of Probab. (2001) | MR | Zbl

[117] Zegarlinski B.; Entropy bounds for Gibbs Measures with non-Gaussian tails, J. Funct. Anal. to appear | MR | Zbl

[118] Zegarlinski B.; Analysis of classical and quantum interacting particle systems, in Trento School 2000 Lecture Notes, World Scientific to appear | MR