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LSI for Kawasaki dynamics with weak interaction. (English) Zbl 1266.82038

The aim of the article is to establish a new logarithmic Sobolev inequality (LSI) with a view to understanding the long-time limit of a random dynamical system of (unbounded, continuous) spins on a lattice. The strategy is now quite well known: it consists in using an LSI for the probability measure of the spin system, which allows one to control a so-called entropy functional and to show convergence to equilibrium. But obtaining an LSI is a difficult step, and many improvements to the standard approach are needed here to treat the case of the Kawasaki spin dynamics. The article is well-written, with many details and explanations, but it is not completely self-contained (it is a summary of some portion of the author’s PhD thesis).

MSC:

82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Bakry, D., Émery, M.: Diffusions hypercontractives. Sem. Probab. XIX, Lecture Notes in Math, 1123. Berlin-Heidelberg-New York: Springer-Verlag, 1985, pp. 177–206
[2] Bodineau T., Helffer B.: The log-Sobolev inequality for unbounded spin systems. J. Funct. Anal. 166(1), 168–178 (1999) · Zbl 0972.82035 · doi:10.1006/jfan.1999.3419
[3] Bodineau, T., Helffer, B.: Correlations, spectral gap and log-Sobolev inequalities for unbounded spins systems. In: Differential equations and mathematical physics (Birmingham, AL, 1999), Volume 16 of AMS/IP Stud. Adv. Math., Providence, RI: Amer. Math. Soc., 2000, pp. 51–66 · Zbl 1161.82306
[4] Cancrini N., Martinelli F., Roberto C.: The logarithmic Sobolev constant of Kawasaki dynamics under a mixing condition revisited. Ann. Inst. H. Poincaré Probab. Statist. 38(4), 385–436 (2002) · Zbl 1174.82310 · doi:10.1016/S0246-0203(01)01096-2
[5] Caputo P.: Uniform Poincaré inequalities for unbounded conservative spin systems: the non-interacting case. Stochastic Process. Appl. 106(2), 223–244 (2003) · Zbl 1075.60581 · doi:10.1016/S0304-4149(03)00044-9
[6] Chafaï D.: Glauber versus Kawasaki for spectral gap and logarithmic Sobolev inequalities of some unbounded conservative spin systems. Markov Process. Rel. Fields 9(3), 341–362 (2003) · Zbl 1040.60081
[7] Feller, W.: An introduction to probability theory and its applications. Vol II. 2nd ed. Wiley Series in Probability and Mathematical Statistics. New York: John Wiley and Sons, Inc., 1971 · Zbl 0219.60003
[8] Gross L.: Logarithmic Sobolev inequalities. Amer. J. Math. 97, 1061–1083 (1975) · Zbl 0318.46049 · doi:10.2307/2373688
[9] Grunewald N., Otto F., Villani C., Westdickenberg M.: A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit. Ann. Inst. H. Poincaré Probab. Statist. 45(2), 302–351 (2009) · Zbl 1179.60068 · doi:10.1214/07-AIHP200
[10] Guionnet, A., Zegarlinski, B.: Lectures on logarithmic Sobolev inequalities. In: Séminaire de Probabilités, XXXVI, Volume 1801 of Lecture Notes in Math, Berlin: Springer, 2003, pp. 1–134 · Zbl 1125.60111
[11] Guo M., Papanicolau G., Varadhan S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys. 118, 31–59 (1988) · Zbl 0652.60107 · doi:10.1007/BF01218476
[12] Helffer B.: Remarks on decay of correlations and Witten Laplacians. III. Application to logarithmic Sobolev inequalities. Ann. Inst. H. Poincaré Probab. Statist. 35(4), 483–508 (1999) · Zbl 1055.82004 · doi:10.1016/S0246-0203(99)00103-X
[13] Holley R., Stroock D.: Logarithmic Sobolev inequalities and stochastic Ising models. J. Stat. Phys. 46, 1159–1194 (1987) · Zbl 0682.60109 · doi:10.1007/BF01011161
[14] Kipnis, C., Landim, C.: Scaling limits of interacting particle systems. Grundlehren der Mathematischen Wissenschaften. 320. Berlin: Springer., 1999 · Zbl 0927.60002
[15] Kosygina E.: The behavior of the specific entropy in the hydrodynamic scaling limit. Ann. Probab. 29(3), 1086–1110 (2001) · Zbl 1018.60096 · doi:10.1214/aop/1015345597
[16] Landim C., Panizo G., Yau H.T.: Spectral gap and logarithmic Sobolev inequality for unbounded conservative spin systems. Ann. Inst. H. Poincaré Probab. Statist. 38(5), 739–777 (2002) · Zbl 1022.60087 · doi:10.1016/S0246-0203(02)01108-1
[17] M. Ledoux. Logarithmic Sobolev inequalities for unbounded spin systems revisted. Sem. Probab. XXXV, Lecture Notes in Math. 1755. Berlin-Heidelberg-New York: Springer-Verlag, 2001, pp. 167–194 · Zbl 0979.60096
[18] Lu S.L., Lu S.L., Lu S.L.: Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Commun. Math. Phys. 156(2), 399–433 (1993) · Zbl 0779.60078 · doi:10.1007/BF02098489
[19] Menz, G.: Equilibrium dynamics of continuous unbounded spin systems. Dissertation, University of Bonn (2011). urn:nbn:de:hbz:5N-25331
[20] Menz, G., Otto, F.: A new covariance estimate. In preparation, 2011
[21] Menz, G., Otto, F.: Uniform logarithmic sobolev inequalities for conservative spin systems with super-quadratic single-site potential. MPI-MIS preprint, Leipzig, 2011 · Zbl 1282.60096
[22] Otto F., Reznikoff M.: A new criterion for the logarithmic Sobolev inequality and two applications. J. Funct. Anal. 243(1), 121–157 (2007) · Zbl 1109.60013 · doi:10.1016/j.jfa.2006.10.002
[23] Otto F., Villani C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361–400 (2000) · Zbl 0985.58019 · doi:10.1006/jfan.1999.3557
[24] Royer, G.: Une initiation aux inégalités de Sobolev logarithmiques. Cours Spéc., Soc. Math. France, 1999 · Zbl 0927.60006
[25] Shiryaev, A.N.: Probability. New York: Springer-Verlag, 2nd edition, 1996 · Zbl 0909.01009
[26] Stroock D., Zegarlinski B.: The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition. Commun. Math. Phys. 144, 303–323 (1992) · Zbl 0745.60104 · doi:10.1007/BF02101094
[27] Stroock D., Zegarlinski B.: The logarithmic Sobolev inequality for discrete spin systems on the lattice. Commun. Math. Phys. 149, 175–193 (1992) · Zbl 0758.60070 · doi:10.1007/BF02096629
[28] Stroock D., Zegarlinski B.: On the ergodic properties of glauber dynamics. J. Stat. Phys. 81, 1007–1019 (1995) · Zbl 1081.60562 · doi:10.1007/BF02179301
[29] Yau H.-T.: Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22, 63–80 (1991) · Zbl 0725.60120 · doi:10.1007/BF00400379
[30] Yau H.-T.: Logarithmic Sobolev inequality for lattice gases with mixing conditions. Commun. Math. Phys. 181(2), 367–408 (1996) · Zbl 0864.60079 · doi:10.1007/BF02101009
[31] Yoshida N.: Application of log-Sobolev inequality to the stochastic dynamics of unbounded spin systems on the lattice. J. Funct. Anal. 173, 74–102 (2000) · Zbl 1040.82047 · doi:10.1006/jfan.1999.3558
[32] Yoshida N.: The equivalence of the log-Sobolev inequality and a mixing condition for unbounded spin systems on the lattice. Ann. Inst. H. Poincaré Probab. Statist. 37(2), 223–243 (2001) · Zbl 0992.60089 · doi:10.1016/S0246-0203(00)01066-9
[33] Zegarlinski B.: The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice. Commun. Math. Phys. 175, 401–432 (1996) · Zbl 0844.46050 · doi:10.1007/BF02102414
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