Analogues of the Poincaré-Chernoff inequality and the logarithmic Sobolev inequality for processes with independent increments

2007 ◽  
Vol 62 (6) ◽  
pp. 1191-1193
Author(s):  
A T Abakirova
Keyword(s):  
Sobolev Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Chernoff Inequality ◽  
Independent Increments ◽  
Processes With Independent Increments

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 Global sensitivity analysis for stochastic processes with independent increments

2020 ◽  
Vol 62 ◽  
pp. 103098
Author(s):  
Emeline Gayrard ◽  
Cédric Chauvière ◽  
Hacène Djellout ◽  
Pierre Bonnet ◽  
Don-Pierre Zappa
Keyword(s):  
Sensitivity Analysis ◽  
Stochastic Processes ◽  
Global Sensitivity Analysis ◽  
Global Sensitivity ◽  
Independent Increments ◽  
Processes With Independent Increments
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