scholarly journals Logarithmic Sobolev Inequality and Exponential Convergence of a Markovian Semigroup in the Zygmund Space

Entropy ◽  
2018 ◽  
Vol 20 (4) ◽  
pp. 220 ◽  
Author(s):  
Ichiro Shigekawa
Keyword(s):  
Sobolev Inequality ◽  
Exponential Convergence ◽  
Logarithmic Sobolev Inequality ◽  
Zygmund Space ◽  
Markovian Semigroup

scholarly journals Functional Inequalities for Two-Level Concentration

2021 ◽  
Author(s):  
Franck Barthe ◽  
Michał Strzelecki
Keyword(s):  
Sobolev Inequality ◽  
Poincaré Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Probability Measures ◽  
Exponential Rate ◽  
Functional Inequalities ◽  
Concentration Inequality ◽  
Poincare Inequality ◽  
Free Concentration ◽  
Concentration Property

AbstractProbability measures satisfying a Poincaré inequality are known to enjoy a dimension-free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincaré inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincaré inequality ensures a stronger dimension-free concentration property, known as two-level concentration. We show that a similar phenomenon occurs for the Latała–Oleszkiewicz inequalities, which were devised to uncover dimension-free concentration with rate between exponential and Gaussian. Motivated by the search for counterexamples to related questions, we also develop analytic techniques to study functional inequalities for probability measures on the line with wild potentials.

Equivalence between logarithmic Sobolev inequality and hypercontractivity in a probability gage space

2020 ◽  
pp. 1-13
Author(s):  
Zhang Lunchuan
Keyword(s):  
Dirichlet Form ◽  
Sobolev Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Markov Semigroup ◽  
Quantum Markov Semigroup

Abstract In this paper, we prove the equivalence between logarithmic Sobolev inequality and hypercontractivity of a class of quantum Markov semigroup and its associated Dirichlet form based on a probability gage space.

scholarly journals Application of Initial Boundary Value Problem on Logarithmic Wave Equation in Dynamics

2018 ◽  
Author(s):  
Shkelqim Hajrulla ◽  
Leonard Bezati ◽  
Fatmir Hoxha
Keyword(s):  
Wave Equation ◽  
Global Existence ◽  
Boundary Value Problem ◽  
Sobolev Inequality ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Logarithmic Sobolev Inequality ◽  
Initial Boundary Value ◽  
The Galerkin Method

We introduce a class of logarithmic wave equation. We study the global existence of week solution for this class of equation. We deal with the initial boundary value problem of this class. Using the Galerkin method and the Gross logarithmic Sobolev inequality we establish the main theorem of existence of week solution for this class of equation arising from Q-Ball Dynamic in particular.

A Free Logarithmic Sobolev Inequality on the Circle

2006 ◽  
Vol 49 (3) ◽  
pp. 389-406 ◽  
Author(s):  
Fumio Hiai ◽  
Dénes Petz ◽  
Yoshimichi Ueda
Keyword(s):  
Fisher Information ◽  
Sobolev Inequality ◽  
Large Deviation ◽  
Logarithmic Sobolev Inequality ◽  
Approximation Procedure ◽  
Matrix Approximation ◽  
Unitary Matrices ◽  
Free Entropy ◽  
Large Deviation Result ◽  
Free Fisher Information

AbstractFree analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.

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