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Logarithmic Sobolev, isoperimetry and transport inequalities on graphs

Yu Tao Ma, Ran Wang, Li Ming Wu
2016 Acta Mathematica Sinica. English series  
We provide estimates of the involved constants.  ...  transportation-entropy inequality) for reversible nearest-neighbor Markov processes on connected finite graphs by means of (random) path method.  ...  Acknowledgements The authors are grateful to the anonymous referees for constructive comments and corrections. Logarithmic Sobolev, Isoperimetry and Transport Inequalities on Graphs  ... 
doi:10.1007/s10114-016-5330-9 fatcat:qhzj2rtdw5ez7kal4sf3ol5ogi

Log-Sobolev, isoperimetry and transport inequalities on graphs [article]

Yutao Ma, Ran Wang, Liming Wu
2015 arXiv   pre-print
and transportation-entropy inequalities) for reversible nearest-neighbor Markov processes on a connected finite graph by means of (random) path method.  ...  We provide estimates of the involved constants.  ...  The upper bound in (2.4) gives us a very practical criterion for the logarithmic Sobolev inequality, following the classical idea of Lyapunov function method for stability.  ... 
arXiv:1505.04552v1 fatcat:plkovrl7xzhgrdruhwsnw6phwe

Logarithmic Harnack Inequalities

F. R. K. Chung, S.-T. Yau
1996 Mathematical Research Letters  
The problem of bounding log-Sobolev constants tends to be harder than estimating eigenvalues. Logarithmic Harnack inequalities provide a direct approach for estimating the log-Sobolev constant.  ...  We will establish Logarithmic Harnack inequalities which can be used to derive lower bounds for log-Sobolev constants.  ...  Many methods for bounding log-Sobolev constants for graphs can be extended to the log-Sobolev constant for certain subgraphs as well.  ... 
doi:10.4310/mrl.1996.v3.n6.a8 fatcat:s6lp2csc5necdcqisjoh2vzmzq

Page 5069 of Mathematical Reviews Vol. , Issue 97H [page]

1997 Mathematical Reviews  
The main result, the logarithmic Sobolev inequality, states that there exists a constant C > 0 (depending on G and the metric) such that for all bounded cylinder functions f on 7(G), g € Z(G) and T>0,  ...  Under this setting, it is shown that several previously developed methods may be applied to give a logarithmic Sobolev inequality without potential term.  ... 

Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution

Yutao Ma, Zhengliang Zhang
2014 Electronic Communications in Probability  
In this paper, we consider the circular Cauchy distribution µx on the unit circle S with index 0 ≤ |x| < 1 and we study the spectral gap and the optimal logarithmic Sobolev constant for µx, denoted respectively  ...  first inequality is true since the optimal logarithmic Sobolev constant for the Bernoulli distribution with parameter 1/2 is 1.  ...  We will denote by C LS (µ x ) the optimal logarithmic Sobolev constant of µ x .  ... 
doi:10.1214/ecp.v19-3071 fatcat:xt6jymc27jalbbxt3fc3hbizxa

Page 3262 of Mathematical Reviews Vol. , Issue 93f [page]

1993 Mathematical Reviews  
The constant Cr can be chosen in- dependent of 7, for T small. The author’s proof also involves the method of tubular neighborhoods used by Gross. {For the entire collection see MR 92c:81004.}  ...  The paper is a discussion concerning infinite-dimensional Sobolev inequalities and logarithmic Sobolev inequalities. For the finite- dimensional case, see two other papers [L. Gross, Amer. J.  ... 

On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group [article]

Michel Bonnefont, Ronan Herry
2018 arXiv   pre-print
In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group.  ...  The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk.  ...  Acknowledgements D.C. would like to thank Leonard Gross for his encouragement to explore this problem and Fabrice Baudoin for his hospitality during a visit to Purdue University in Fall 2008.  ... 
arXiv:1607.02741v2 fatcat:7ky7kcxsyfd7xpfbuidb37iupq

On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group

Michel Bonnefont, Djalil Chafaï, Ronan Herry
2020 Annales de la Faculté des Sciences de Toulouse  
Acknowledgements D.C. would like to thank Leonard Gross for his encouragement to explore this problem and Fabrice Baudoin for his hospitality during a visit to Purdue University in Fall 2008.  ...  We follow the method developed by Leonard Gross in [12] for the Gaussian and in [13] for the path space on elliptic Lie groups.  ...  Li inequality For β = 0, Hong-Quan Li has obtained in [17] (see also [1, 10] for a Poincaré inequality) the following logarithmic Sobolev inequality: there exists a constant C LSI > 0 such that for  ... 
doi:10.5802/afst.1633 fatcat:27ae3lsjhbcmzintmw237pxnqa

Martingale Representation and a Simple Proof of Logarithmic Sobolev Inequalities on Path Spaces

Mireille Capitaine, Elton Hsu, Michel Ledoux
1997 Electronic Communications in Probability  
By an appropriate integration by parts formula the proof also yields in the same way a logarithmic Sobolev inequality for the path space equipped with a general diffusion measure as long as the torsion  ...  Fang in his proof of the spectral gap inequality for the Ornstein-Uhlenbeck operator on the path space can yield in a very simple way the logarithmic Sobolev inequality on the same space.  ...  Driver for his help at the early stage of this work and for his suggestion of further developing this approach in the case of a connection with torsion.  ... 
doi:10.1214/ecp.v2-986 fatcat:ygkx2nvjgnfd5bpzz3z25imh7a

Page 559 of Mathematical Reviews Vol. , Issue 2000a [page]

2000 Mathematical Reviews  
The perturbation of the spectral gap and the logarithmic Sobolev constant under a linear transform are given (Theorem 5).  ...  A new proof for computing the logarithmic Sobolev constant in a basic case is also presented (Theorem 7).” 2000a:60141 60527 60J45 He, Ping (J-KANASN; Kanazawa) The structure of excursions of Markov chains  ... 

On the variational interpretation of local logarithmic Sobolev inequalities [article]

Gauthier Clerc
2021 arXiv   pre-print
inequalities such as the Logarithmic Sobolev inequality.  ...  In this short note we close this gap by explaining how Otto calculus applied to the Schrödinger problem yields a variations interpretation of the local logarithmic Sobolev inequalities, that could possibly  ...  Acknowledgements This reflexion was supported by the French ANR-17-CE40-0030 EFI project.  ... 
arXiv:2011.05207v3 fatcat:efgn7trdaje7vakbiyos76ihvq

Page 4756 of Mathematical Reviews Vol. , Issue 2004f [page]

2004 Mathematical Reviews  
These are worked out for some special cases.” 2004f:60017 60C05 Roberto, Cyril (F-TOUL3; Toulouse) A path method for the logarithmic Sobolev constant. (English summary) Combin. Probab.  ...  The author develops a path combinatoric method to obtain new bounds on the logarithmic Sobolev constant on a countable set. He generalizes Sinclair’s bounds [A. Sinclair, Combin. Probab.  ... 

Concentration of measure and logarithmic Sobolev inequalities [chapter]

Michel Ledoux
1999 Lecture notes in mathematics  
We then discuss some recent logarithmic Sobolev inequality for Wiener measure on the paths of a Riemannian manifold.  ...  We only aim to show here unity of the method, deriving this large deviation estimate from logarithmic Sobolev inequalities for heat kernel measures and Wiener measures on path spaces developed recently  ... 
doi:10.1007/bfb0096511 fatcat:hq5vnzqzx5gdlilffh3qa5njom

Logarithmic Sobolev inequalities for harmonic measures on spheres

Franck Barthe, Yutao Ma, Zhengliang Zhang
2014 Journal des Mathématiques Pures et Appliquées  
We study the corresponding optimal constants of logarithmic Sobolev and Poincaré inequalities, denoted respectively by C LS  ...  In this paper, we consider the family of harmonic measures µ n x (indexed by x ∈ R n with |x| < 1) on the unit sphere S n−1 in R n , for n ≥ 3.  ...  We denote by C LS (µ) the best constant in the logarithmic Sobolev inequality.  ... 
doi:10.1016/j.matpur.2013.11.008 fatcat:iuvni4v4ajhopbe2njwqqiwbye

Page 4410 of Mathematical Reviews Vol. , Issue 97G [page]

1997 Mathematical Reviews  
This method is based on use of the hypercontractivity property of the Markov semigroup or equivalently of the logarithmic Sobolev inequality for the corresponding generator.  ...  {For the entire collection see MR 97a:00031.} 97g:60104 60J65 60G17 Brzezniak, Zdzislaw (4-HULL,; Hull) On Sobolev and Besov spaces regularity of Brownian paths.  ... 
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