Modified logarithmic Sobolev inequalities for canonical ensembles
Fathi, Max1 
1 LPMA, University Paris 6, France.
ESAIM: Probability and Statistics, Tome 19 (2015), pp. 544-559

In this paper, we prove modified logarithmic Sobolev inequalities for canonical ensembles with superquadratic single-site potential. These inequalities were introduced by Bobkov and Ledoux, and are closely related to concentration of measure and transport-entropy inequalities. Our method is an adaptation of the iterated two-scale approach that was developed by Menz and Otto to prove the usual logarithmic Sobolev inequality in this context. As a consequence, we obtain convergence in Wasserstein distance W p for Kawasaki dynamics on the Ginzburg−Landau’s model.

Reçu le :
DOI : 10.1051/ps/2015004
Classification : 60K35, 82B21
Keywords: Modified logarithmic Sobolev inequalities, spin system, coarse-graining

Fathi, Max 1

1 LPMA, University Paris 6, France. 
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Fathi, Max. Modified logarithmic Sobolev inequalities for canonical ensembles. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 544-559. doi: 10.1051/ps/2015004

S.G. Bobkov and M. Ledoux, From Brunn-Minkowski to Brascamp−Lieb and to logarithmic Sobolev inequalities. Geom. Func. Anal. 10 (2000) 1028–1052. | MR | Zbl

S.G. Bobkov and B. Zegarlinski, Entropy Bounds and Isoperimetry. Memoirs of the AMS (2005). | MR | Zbl

S.G. Bobkov, I. Gentil and L. Ledoux, Hypercontractivity of Hamilton−Jacobi equations. J. Math. Pures Appl. 80 (2001) 669–696. | MR | Zbl

M. Fathi, A two-scale approach to the hydrodynamic limit, part II: local Gibbs behavior. ALEA 80 (2013) 625–651. | MR | Zbl

N. Gozlan, A characterization of dimension-free concentration in terms of transportation inequalities. Ann. Probab. 37 (2009) 2480–2498. | MR | Zbl

N. Gozlan, C. Roberto and P. Samson, Characterization of Talagrand’s transport-entropy inequality in metric spaces. Ann. Probab. 41 (2013) 3112–3139. | MR | Zbl

L. Gross, Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975) 1061–1083. | MR | Zbl

N. Grunewald, F. Otto, C. Villani and M.G. Westdickenberg, A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit. Ann. Inst. Henri Poincaré Probab. Statist. 45 (2009) 302–351. | MR | Zbl

M. Ledoux, Logarithmic Sobolev inequalities for spin systems revisited (1999). Available at http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.4917 | MR

M. Ledoux, The Concentration of Measure Phenomenon. Vol. 89 of Math. Surv. Monogr. AMS. Providence, Rhode Island (2001). | MR | Zbl

K. Marton, A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal. 6 (1996) 556–571 | MR | Zbl

G. Menz and F. Otto, Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. Ann. Probab. 41 (2013) 2182–2224. | MR | Zbl

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 243 (2007) 121–157. | Zbl

C. Villani, Optimal Transport, Old and New. Vol. 338 of Grund. Math. Wissenschaften. Springer-Verlag (2009). | MR | Zbl

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