A new criterion for the logarithmic Sobolev inequality and two applications

Otto, Félix; Reznikoff, Maria G.

doi:10.1016/j.jfa.2006.10.002

Cited by 49 publications

(67 citation statements)

References 17 publications

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“…The statement follows from a direct application of the criterion for LSI of Otto & Reznikoff (see Theorem A.4 in the appendix or Theorem 1 in [OR07]). We check the first requirement of Theorem A.4.…”

Section: Microscopic Lsimentioning

confidence: 99%

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LSI for Kawasaki Dynamics with Weak Interaction

Menz

2011

Commun. Math. Phys.

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We consider a large lattice system of unbounded continuous spins that are governed by a Ginzburg-Landau type potential and a weak quadratic interaction. We derive the logarithmic Sobolev inequality (LSI) for Kawasaki dynamics uniform in the boundary data. The scaling of the LSI constant is optimal in the system size and our argument is independent of the geometric structure of the system. The proof consists of an application of the two-scale approach of Grunewald, Otto, Westdickenberg & Villani. Several ideas are needed to solve new technical difficulties due to the interaction. Let us mention the application of a new covariance estimate, a conditioning technique, and a generalization of the local Cramér theorem.MSC: Primary 60K35; secondary 60J25; 82B21

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Section: Microscopic Lsimentioning

confidence: 99%

“…More recently, Otto & Reznikoff [OR07] deduced a criterion that is capable to deal with certain non-convex Hamiltonians in high dimensions. exp(−H(x)) dx be a probability measure on a direct product of Euclidean spaces X = X 1 × · · · × X M .…”

Section: A Basic Facts About the Lsimentioning

confidence: 99%

LSI for Kawasaki Dynamics with Weak Interaction

Menz

2011

Commun. Math. Phys.

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“…In the more standard Euclidean model, this problem has been extensively studied in the case q = 2 , for example in [8], [14], [18], [21] and more recently in [23], as well as in many of the afore mentioned papers. The case q < 2 was looked at in [7].…”

Section: The Logarithmic Sobolev Inequality For Gibbs Measuresmentioning

confidence: 99%

“…They have proved extremely useful as a tool in the control of the rate of convergence to equilibrium of spin systems, and were extensively studied (see for example [8], [14], [18], [23], [27], [29], [31]). Up until recently, however, most of the attention has been focused on the case of elliptic generators, for which there are some very powerful methods for proving such inequalities.…”

Section: Introductionmentioning

confidence: 99%

Logarithmic Sobolev Inequalities for Infinite Dimensional Hörmander Type Generators on the Heisenberg Group

Inglis

Papageorgiou

2009

Potential Anal

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“…The aim of this work is to give an answer, which is a generalization to non-linear coarse-graining operators of results in [6,9].…”

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confidence: 99%

A general two-scale criteria for logarithmic Sobolev inequalities

Lelièvre

2009

Journal of Functional Analysis

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We present a general criteria to prove that a probability measure satisfies a logarithmic Sobolev inequality, knowing that some of its marginals and associated conditional laws satisfy a logarithmic Sobolev inequality. This is a generalization of a result by N. Grunewald et al. [N. Grunewald, F. Otto, C. Villani, M.G. Westdickenberg, A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit, Ann. Inst. H. Poincaré Probab. Statist., in press].The motivation behind this work is molecular dynamics (in the canonical statistical ensemble), and more precisely, (i) the analysis of numerical methods for the computation of free energy differences [8] (see Remark 1 below) and (ii) the derivation of effective dynamics on coarse-grained variables [7]. In both cases, it appears that estimates based on entropies for measures related to the Boltzmann-Gibbs measure is a useful tool. One important question is the following: what is the link between the logarithmic Sobolev inequality (LSI) constant of the Boltzmann-Gibbs measure for the original variables (microscopic level) and the LSI constant of the Boltzmann-

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A new criterion for the logarithmic Sobolev inequality and two applications

Cited by 49 publications

References 17 publications

LSI for Kawasaki Dynamics with Weak Interaction

LSI for Kawasaki Dynamics with Weak Interaction

Logarithmic Sobolev Inequalities for Infinite Dimensional Hörmander Type Generators on the Heisenberg Group

A general two-scale criteria for logarithmic Sobolev inequalities

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