The equivalence of the log-Sobolev inequality and a mixing condition for unbounded spin systems on the lattice
Annales de l'I.H.P. Probabilités et statistiques, Volume 37 (2001) no. 2, p. 223-243
@article{AIHPB_2001__37_2_223_0,
     author = {Yoshida, Nobuo},
     title = {The equivalence of the log-Sobolev inequality and a mixing condition for unbounded spin systems on the lattice},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Elsevier},
     volume = {37},
     number = {2},
     year = {2001},
     pages = {223-243},
     zbl = {0992.60089},
     mrnumber = {1819124},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2001__37_2_223_0}
}
Yoshida, Nobuo. The equivalence of the log-Sobolev inequality and a mixing condition for unbounded spin systems on the lattice. Annales de l'I.H.P. Probabilités et statistiques, Volume 37 (2001) no. 2, pp. 223-243. http://www.numdam.org/item/AIHPB_2001__37_2_223_0/

[1] S Aida, T Masuda, I Shigekawa, Logarithmic Sobolev inequalities and exponential integrability, J. Funct. Anal. 126 (1) (1994) 83-101. | MR 1305064 | Zbl 0846.46020

[2] S Albeverio, Yu.G Kondratiev, M Röckner, Dirichlet operators via stochastic analysis, J. Funct. Anal. 128 (1995) 102-138. | MR 1317712 | Zbl 0820.60042

[3] S Albeverio, Yu.G Kondratiev, M Röckner, T.V Tsikalenko, Uniqueness of Gibbs states for quantum lattice systems, Probab. Theory Related Fields 108 (1997) 193-218. | MR 1452556 | Zbl 0883.60094

[4] D Bakry, M Emery, Diffusions hypercontractives, in: Séminaire de Probabilités XIX, Springer Lecture Notes in Math., 1123, 1985, pp. 177-206. | Numdam | MR 889476 | Zbl 0561.60080

[5] J Bellissard, R Høegh-Krohn, Compactness and maximal Gibbs state for random Gibbs fields on the lattice, Comm. Math. Phys. 84 (1982) 297-327. | MR 667405 | Zbl 0495.60057

[6] T Bodineau, B Helffer, Log-Sobolev inequality for unbounded spin systems, J. Funct. Anal. 166 (1999) 168-178. | MR 1704666 | Zbl 0972.82035

[7] T Bodineau, B Helffer, Correlations, spectral gap and log-Sobolev inequality for unbounded spin systems, in: Differential Equations and Mathematical Physics, Birmingham, International Press, 1999, pp. 27-42. | MR 1704666 | Zbl 01780486

[8] H Doss, G Royer, Processus de diffusion associe aux mesures de Gibbs, Z. Wahrsch. verw. Gebiete 46 (1978) 107-124. | MR 512335 | Zbl 0384.60076

[9] R.L Dobrushin, S Shlosman, Constructive criterion for the uniqueness of Gibbs field, in: Fritz J, Jaffe A, Szasz D (Eds.), Statistical Physics and Dynamical Systems, Birkhäuser, 1985. | MR 821306 | Zbl 0569.46042

[10] R.L Dobrushin, S Shlosman, Completely analytical Gibbs fields, in: Fritz J, Jaffe A, Szasz D (Eds.), Statistical Physics and Dynamical Systems, Birkhäuser, 1985. | MR 821307 | Zbl 0569.46043

[11] R.L Dobrushin, S Shlosman, Completely analytical interactions: Constructive description, J. Stat. Phys. 46 (1987) 983-1014. | MR 893129 | Zbl 0683.60080

[12] J.D Deuschel, D.W Stroock, Large Deviations, Academic Press, 1989. | MR 997938 | Zbl 0705.60029

[13] I Gentil, C Roberto, Spectral gaps for spin systems: some non-convex phase examples, preprint, 2000. | MR 1814423

[14] B Helffer, Remarks on the decay of correlations and Witten Laplacians III - Application to logarithmic Sobolev inequalities, Ann. de l'Insti. H. Poincaré (Sect. Probab-Stat) (1998). | Numdam | Zbl 1055.82004

[15] R Holley, D.W Stroock, Logarithmic Sobolev inequality and stochastic Ising models, J. Stat. Phys. 46 (1987) 1159-1194. | MR 893137 | Zbl 0682.60109

[16] M Ledoux, Log-Sobolev inequality for unbounded spin systems revisited, preprint, 1999. | MR 1704666

[17] T.M Liggett, Interacting Particle Systems, Springer Verlag, Berlin, 1985. | MR 776231 | Zbl 0559.60078

[18] T Lindvall, Lectures on the Coupling Method, Wiley, 1992. | MR 1180522 | Zbl 0850.60019

[19] S.L Lu, H.T Yau, Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics, Comm. Math. Phys. 156 (1993) 399-433. | MR 1233852 | Zbl 0779.60078

[20] F Martinelli, E Olivieri, Approach to equilibrium of Glauber dynamics in the one phase region I: Attractive case, Comm. Math. Phys. 161 (1994) 447-486. | MR 1269387 | Zbl 0793.60110

[21] F Martinelli, E Olivieri, Approach to equilibrium of Glauber dynamics in the one phase region II: General case, Comm. Math. Phys. 161 (1994) 487-514. | MR 1269388 | Zbl 0793.60111

[22] D.W Stroock, S.R.S Varadhan, Multidimensional Diffusion Processes, Springer Verlag, Berlin, 1979. | MR 532498 | Zbl 0426.60069

[23] D.W Stroock, B Zegarlinski, The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition, Comm. Math. Phys. 144 (1992) 303-323. | MR 1152374 | Zbl 0745.60104

[24] D.W Stroock, B Zegarlinski, The logarithmic Sobolev inequality for discrete spin systems on the lattice, Comm. Math. Phys. 149 (1992) 175-193. | MR 1182416 | Zbl 0758.60070

[25] Sugiura M., Private communication.

[26] N Yoshida, Sobolev spaces on a Riemannian manifold and their equivalence, J. Math. Kyoto Univ. 33 (1992) 621-654. | MR 1183370 | Zbl 0771.58005

[27] N Yoshida, The log-Sobolev inequality for weakly coupled lattice fields, Probab. Theory Related Fields 115 (1999) 1-40. | MR 1715549 | Zbl 0948.60095

[28] N Yoshida, Application of log-Sobolev inequality to the stochastic dynamics of unbounded spin systems on the lattice, J. Funct. Anal. 173 (2000) 74-102. | MR 1760279 | Zbl 1040.82047

[29] B Zegarlinski, The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice, Comm. Math. Phys. 175 (1996) 401-432. | MR 1370101 | Zbl 0844.46050