Modified log-Sobolev inequalities, Beckner inequalities and moment estimates

Abstract

We prove that in the context of general Markov semigroups Beckner inequalities with constants separated from zero as $p\to 1^{+}$ are equivalent to the modified log Sobolev inequality (previously only one implication was known to hold in this generality). Further, by adapting an argument by Boucheron et al. we derive Sobolev type moment estimates which hold under these functional inequalities. We illustrate our results with applications to concentration of measure estimates (also of higher order, beyond the case of Lipschitz functions) for various stochastic models, including random permutations, zero-range processes, strong Rayleigh measures, exponential random graphs, and geometric functionals on the Poisson path space.

56 pages, 1 figure; presentation in Sec. 4.7 changed

References in corpus (5)

• Moment inequalities for functions of independent random variables
• Weighted Poincaré-type inequalities for Cauchy and other convex measures
• Concentration of Multilinear Functions of the Ising Model with Applications to Network Data
• Restricted hypercontractivity on the Poisson space
• A family of Beckner inequalities under various curvature-dimension conditions

Cited by in corpus (3)

• Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees
• Large deviations and a phase transition in the Block Spin Potts models
• Concentration on Poisson spaces via modified $\Phi$-Sobolev inequalities