A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 2, pp. 302-351.

Nous étudions un système sur réseau à variable de spin continue. Dans la première partie, nous établissons deux résultats abstraits : des conditions suffisantes pour une inégalité de Sobolev logarithmique avec constante indépendante de la dimension (Théorème 3), et des conditions suffisantes pour la convergence vers la limite hydrodynamique (Theorème 8). Dans la seconde partie, nous utilisons ces résultats abstraits pour traiter un exemple spécifique, à savoir la dynamique de Kawasaki avec un potentiel de type Ginzburg-Landau.

We consider the coarse-graining of a lattice system with continuous spin variable. In the first part, two abstract results are established: sufficient conditions for a logarithmic Sobolev inequality with constants independent of the dimension (Theorem 3) and sufficient conditions for convergence to the hydrodynamic limit (Theorem 8). In the second part, we use the abstract results to treat a specific example, namely the Kawasaki dynamics with Ginzburg-Landau-type potential.

DOI : 10.1214/07-AIHP200
Classification : 60K35, 60J25
Mots-clés : logarithmic Sobolev inequality, hydrodynamic limit, spin system, Kawasaki dynamics, canonical ensemble, Coarse-graining 
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     title = {A two-scale approach to logarithmic {Sobolev} inequalities and the hydrodynamic limit},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
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Grunewald, Natalie; Otto, Felix; Villani, Cédric; Westdickenberg, Maria G. A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 2, pp. 302-351. doi : 10.1214/07-AIHP200. http://archive.numdam.org/articles/10.1214/07-AIHP200/

[1] D. Bakry and M. Émery. Diffusions hypercontractives. In Sem. Probab. XIX. Lecture Notes in Mathematics 1123 177-206. Springer, Berlin, 1985. | Numdam | MR | Zbl

[2] G. Blower and F. Bolley. Concentration of measure on product spaces with applications to Markov processes. Studia Math. 175 (2006) 47-72. | MR | Zbl

[3] Th. Bodineau and B. Helffer. On log Sobolev inequalities for unbounded spin systems. J. Funct. Anal. 166 (1999) 168-178. | MR | Zbl

[4] D. Chafaï. Glauber versus Kawasaki for spectral gap and logarithmic Sobolev inequalities of some unbounded conservative spin systems. Markov Process. Related Fields 9 (2001) 341-362. | MR | Zbl

[5] P. Caputo. Uniform Poincaré inequalities for unbounded conservative spin systems: The non-interacting case. Stochastic Process. Appl. 106 (2003) 223-244. | MR | Zbl

[6] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edn. Appl. Math. 38. Springer, New York, 1998. | MR | Zbl

[7] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley, New York, 1971. | MR | Zbl

[8] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975) 1061-1083. | MR | Zbl

[9] A. Guionnet and B. Zegarlinski. Lectures on logarithmic Sobolev inequalities. In Sminaire de Probabilits, XXXVI. Lecture Notes in Mathematics 1801 1-134. Springer, Berlin, 2003. | Numdam | MR | Zbl

[10] M. Z. Guo, G. C. Papanicolaou and S. R. S. Varadhan. Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys. 118 (1988) 31-59. | MR | Zbl

[11] R. Holley and D. Stroock. Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys. 46 (1987) 1159-1194. | MR | Zbl

[12] C. Kipnis and C. Landim. Scaling Limits of Interacting Particle Systems. Springer, Berlin, 1999. | MR | Zbl

[13] E. Kosygina. The behavior of the specific entropy in the hydrodynamic scaling limit. Ann. Probab. 29 (2001) 1086-1110. | MR | Zbl

[14] C. Landim, G. Panizo and H. T. Yau. Spectral gap and logarithmic Sobolev inequality for unbounded conservative spin systems. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 739-777. | Numdam | MR | Zbl

[15] M. Ledoux. The Concentration of Measure Phenomenon, Math. Surveys and Monographs 89. Amer. Math. Soc., Providence, RI, 2001. | MR | Zbl

[16] M. Ledoux. Logarithmic Sobolev inequalities for unbounded spin systems revisted. In Sem. Probab. XXXV. Lecture Notes in Mathematics 1755 167-194. Springer, Berlin. 2001. | Numdam | MR | Zbl

[17] S. L. Lu. Hydrodynamic scaling with deterministic initial configurations. Ann. Probab. 23 (1995) 1831-1852. | MR | Zbl

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

[19] F. Otto and M. G. Reznikoff. A new criterion for the logarithmic Sobolev inequality and two applications. J. Funct. Anal. 243 (2007) 121-157. | MR | Zbl

[20] F. Otto and C. Villani. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000) 361-400. | MR | Zbl

[21] G. Royer. Une Initiation aux Inégalités de Sobolev Logarithmiques. Cours Spécialisés. Soc. Math. de France, Paris, 1999. | MR | Zbl

[22] G. Toscani and C. Villani. On the trend to equilibrium for some dissipative systems with slowly increasing a-priori bounds. J. Stat. Phys. 98 (2000) 1279-1309. | MR | Zbl

[23] C. Villani. Topics in Optimal Transportation. Graduate Texts in Mathematics 58. Amer. Math. Soc., Providence, RI, 2003. | MR | Zbl

[24] H.-T. Yau. Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22 (1991) 63-80. | MR | Zbl

[25] H.-T. Yau. Logarithmic Sobolev inequality for lattice gases with mixing conditions. Commun. Math. Phys. 181 (1996) 367-408. | MR | Zbl

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