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 for Kawasaki dynamics on the Ginzburg−Landau’s model.
DOI : 10.1051/ps/2015004
Mots clés : Modified logarithmic Sobolev inequalities, spin system, coarse-graining
@article{PS_2015__19__544_0, author = {Fathi, Max}, title = {Modified logarithmic {Sobolev} inequalities for canonical ensembles}, journal = {ESAIM: Probability and Statistics}, pages = {544--559}, publisher = {EDP-Sciences}, volume = {19}, year = {2015}, doi = {10.1051/ps/2015004}, mrnumber = {3433425}, zbl = {1336.60187}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps/2015004/} }
TY - JOUR AU - Fathi, Max TI - Modified logarithmic Sobolev inequalities for canonical ensembles JO - ESAIM: Probability and Statistics PY - 2015 SP - 544 EP - 559 VL - 19 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2015004/ DO - 10.1051/ps/2015004 LA - en ID - PS_2015__19__544_0 ER -
Fathi, Max. Modified logarithmic Sobolev inequalities for canonical ensembles. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 544-559. doi : 10.1051/ps/2015004. http://www.numdam.org/articles/10.1051/ps/2015004/
From Brunn-Minkowski to Brascamp−Lieb and to logarithmic Sobolev inequalities. Geom. Func. Anal. 10 (2000) 1028–1052. | MR | Zbl
and ,S.G. Bobkov and B. Zegarlinski, Entropy Bounds and Isoperimetry. Memoirs of the AMS (2005). | MR | Zbl
Hypercontractivity of Hamilton−Jacobi equations. J. Math. Pures Appl. 80 (2001) 669–696. | MR | Zbl
, and ,A two-scale approach to the hydrodynamic limit, part II: local Gibbs behavior. ALEA 80 (2013) 625–651. | MR | Zbl
,A characterization of dimension-free concentration in terms of transportation inequalities. Ann. Probab. 37 (2009) 2480–2498. | MR | Zbl
,Characterization of Talagrand’s transport-entropy inequality in metric spaces. Ann. Probab. 41 (2013) 3112–3139. | MR | Zbl
, and ,Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975) 1061–1083. | MR | Zbl
,A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit. Ann. Inst. Henri Poincaré Probab. Statist. 45 (2009) 302–351. | MR | Zbl
, , and ,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
A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal. 6 (1996) 556–571 | MR | Zbl
,Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. Ann. Probab. 41 (2013) 2182–2224. | MR | Zbl
and ,Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 243 (2007) 121–157. | Zbl
and ,C. Villani, Optimal Transport, Old and New. Vol. 338 of Grund. Math. Wissenschaften. Springer-Verlag (2009). | MR | Zbl
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