Logarithmic Sobolev inequalities for unbounded spin systems revisited

Michel Ledoux

Séminaire de probabilités de Strasbourg (2001)

  • Volume: 35, page 167-194

How to cite

top

Ledoux, Michel. "Logarithmic Sobolev inequalities for unbounded spin systems revisited." Séminaire de probabilités de Strasbourg 35 (2001): 167-194. <http://eudml.org/doc/114059>.

@article{Ledoux2001,
author = {Ledoux, Michel},
journal = {Séminaire de probabilités de Strasbourg},
keywords = {Poincaré inequality; logarithm Sobolev inequality; spin system; semigroup analysis; Hamiltonian},
language = {eng},
pages = {167-194},
publisher = {Springer - Lecture Notes in Mathematics},
title = {Logarithmic Sobolev inequalities for unbounded spin systems revisited},
url = {http://eudml.org/doc/114059},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Ledoux, Michel
TI - Logarithmic Sobolev inequalities for unbounded spin systems revisited
JO - Séminaire de probabilités de Strasbourg
PY - 2001
PB - Springer - Lecture Notes in Mathematics
VL - 35
SP - 167
EP - 194
LA - eng
KW - Poincaré inequality; logarithm Sobolev inequality; spin system; semigroup analysis; Hamiltonian
UR - http://eudml.org/doc/114059
ER -

References

top
  1. [Ai] S. Aida. Uniform positivity improving property, Sobolev inequalities and spectral gaps. J. Funct. Anal.158, 152-185 (1998). Zbl0914.47041MR1641566
  2. [A-M-S] S. Aida, T. Masuda, I. Shigekawa. Logarithmic Sobolev inequalities and exponential integrability. J. Funct. Anal.126, 83-101 (1994). Zbl0846.46020MR1305064
  3. [An] C. Ané ET AL. Sur les inégalités de Sobolev logarithmiques (2000). Panoramas et Synthèses, S.M.F., to appear. Zbl0982.46026MR1845806
  4. [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. Zbl0856.47026MR1307413
  5. [Ba2] D. Bakry. On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. New trends in.Stochastic Analysis. 43-75 (1997). World Scientific. MR1654503
  6. [Ba-E] D. Bakry, M. Emery. Diffusions hypercontractives. Séminaire de Probabilités XIX. Lecture Notes in Math.1123, 177-206 (1985). Springer-Verlag. Zbl0561.60080MR889476
  7. [Ba-L] D. Bakry, M. Ledoux. Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator. Invent. math.123, 259-281 (1996). Zbl0855.58011MR1374200
  8. [Bo1] S. Bobkov. An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probability25, 206-214 (1997). Zbl0883.60031MR1428506
  9. [Bo2] S. Bobkov. Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probability27, 1903-1921 (1999). Zbl0964.60013MR1742893
  10. [B-G] S. Bobkov, F. Götze. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities (1998). J. Funct. Anal.163, 1-28 (1999). Zbl0924.46027MR1682772
  11. [B-H1] TH. Bodineau, B. Helffer. On Log-Sobolev inequalities for unbounded spin systems. J. Funct. Anal.166, 168-178 (1999). Zbl0972.82035MR1704666
  12. [B-H2] TH. Bodineau, B. Helffer. Correlations, spectral gaps and Log-Sobolev inequalities for unbounded spins systems. Differential Equations and Mathematical Physics. Birmingham 1999, 27-42. International Press (1999). Zbl1161.82306MR1764741
  13. [B-L] H.J. Brascamp, E.H. Lieb. On extensions of the Brunn-Minkovski and Prékopa-Leindler theorems, including inequalities for log-concave functions, and with an application to the diffusion equation. J. Funct. Anal.22, 366-389 (1976). Zbl0334.26009MR450480
  14. [Fo] P. Fougères. Hypercontractivité et isopérimétrie gaussienne. Applications aux systèmes de spins (1999). Ann. Inst. H. Poincaré, to appear. Zbl0983.60097MR1792659
  15. [G-H-L] S. Gallot, D. Hulin, J. Lafontaine. Riemannian Geometry. Second Edition. Springer (1990). Zbl0716.53001MR1083149
  16. [G-R] I. Gentil, C. Roberto. Spectral gaps for spin system: some non-convex phase examples (2000). J. Funct. Anal., to appear. Zbl0992.60091MR1814423
  17. [Gr] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math.97, 1061-1083 (1975). Zbl0318.46049MR420249
  18. [G-Z] A. Guionnet, B. Zegarlinski. Lectures on logarithmic Sobolev inequalities (2000). Zbl1125.60111
  19. [He1] B. Helffer. Remarks on decay of correlations and Witten Laplacians - Brascamp-Lieb inequalities and semi-classical analysis. J. Funct. Anal. (1999). Zbl0921.35141MR1624506
  20. [He2] B. Helffer. Remarks on decay of correlations and Witten Laplacians III- Application to logarithmic Sobolev inequalites. Ann. Inst. H. Poincaré35, 483-508 (1999). Zbl1055.82004MR1702239
  21. [He3] B. Helffer. Semiclassical analysis and statistical mechanics. Notes (1999). 
  22. [He-S] B. Helffer, J. Sjöstrand. On the correlation for the Kac like models in the convex case. J. Statist. Phys.74, 349-369 (1994). Zbl0946.35508MR1257821
  23. [H-S] R. Holley, D. Stroock. Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys.46, 1159-1194 (1987). Zbl0682.60109MR893137
  24. [Le1] M. Ledoux. Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités XXXIII. Lecture Notes in Math.1709, 120-216 (1999). Springer. Zbl0957.60016MR1767995
  25. [Le2] M. Ledoux. The geometry of Markov diffusion generators (1998). Ann. Fac. Sci. Toulouse, to appear. Zbl0980.60097MR1813804
  26. [L-Y] S.L. Lu, H.T. Yau. Spectral gap and logarithmic Sobolev inequalities for Kawasaki and Glauber dynamics. Comm. Math. Phys.156, 399-433 (1993). Zbl0779.60078MR1233852
  27. [M-O1] F. Martinelli, E. Olivieri. Approach to equilibrium of Glauber dynamics in the one phase region I. The attractive case. Comm. Math. Phys.161, 447-486 (1994). Zbl0793.60110MR1269387
  28. [M-O2] F. Martinelli, E. Olivieri. Approach to equilibrium of Glauber dynamics in the one phase region II. The general case. Comm. Math. Phys.161, 487-514 (1994). Zbl0793.60111MR1269388
  29. [Ro] G. Royer. Une initiation aux inégalités de Sobolev logarithmiques. Cours Spécialisés. Soc. Math. de France (1999). Zbl0927.60006MR1704288
  30. [S-Z1] D. Stroock, B. Zegarlinski. The logarithmic Sobolev inequality for continuous spin systems on a lattice. J. Funct. Anal.104, 299-326 (1992). Zbl0794.46025MR1153990
  31. [S-Z2] D. Stroock, B. Zegarlinski. The logarithmic Sobolev inequality for discrete spin systems on a lattice. Comm. Math. Phys.149, 175-193 (1992). Zbl0758.60070MR1182416
  32. [S-Z3] D. Stroock, B. Zegarlinski. The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition. Comm. Math. Phys.144, 303-323 (1992). Zbl0745.60104MR1152374
  33. [S-Z4] D. Stroock, B. Zegarlinski. On the ergodic properties of Glauber dynamics. J. Stat. Phys.81, 1007-1019 (1995). Zbl1081.60562MR1361304
  34. [Wa] F.-Y. Wang. Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theor. Relat. Fields109, 417-424 (1997). Zbl0887.35012MR1481127
  35. [Yo1] N. Yoshida. The log-Sobolev inequality for weakly coupled lattice fields. Probab. Theor. Relat. Field115, 1-40 (1999). Zbl0948.60095MR1715549
  36. [Yo2] N. Yoshida. Application of log-Sobolev inequality to the stochastic dynamics of unbounded spin systems on the lattice. J. Funct. Anal.173, 74-102 (2000). Zbl1040.82047MR1760279
  37. [Yo3] N. Yoshida. The equivalence of the log-Sobolev inequality and a mixing condition for unbounded spin systems on the lattice (1999). Ann. Inst. H. Poincaré, to appear. Zbl0992.60089MR1819124
  38. [Ze1] B. Zegarlinski. The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice. Comm. Math. Phys.175, 401-432 (1996). Zbl0844.46050MR1370101
  39. [Ze2] B. Zegarlinski. Isoperimetry for Gibbs measures (1999).http://www-sv.cict.fr/lsp/Ledoux/ 

Citations in EuDML Documents

top
  1. Pietro Caputo, Paolo Dai Pra, Gustavo Posta, Convex entropy decay via the Bochner–Bakry–Emery approach
  2. Lorenzo Bertini, Nicoletta Cancrini, Filippo Cesi, The spectral gap for a Glauber-type dynamics in a continuous gas
  3. Liming Wu, Estimate of spectral gap for continuous gas
  4. Natalie Grunewald, Felix Otto, Cédric Villani, Maria G. Westdickenberg, A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit
  5. Pierre-André Zitt, Functional inequalities and uniqueness of the Gibbs measure — from log-Sobolev to Poincaré

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.