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.

arXiv comment56 pages, 1 figure; presentation in Sec. 4.7 changed

References in corpus (23)

• Estimating and understanding exponential random graph models
• Exponential and moment inequalities for U-statistics
• Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities
• Estimates of moments and tails of Gaussian chaoses
• Moment inequalities for functions of independent random variables
• Note on sampling without replacing from a finite collection of matrices
• Moment inequalities for U-statistics
• Weighted Poincaré-type inequalities for Cauchy and other convex measures
• A very simple proof of the LSI for high temperature spin systems
• On the variational problem for upper tails in sparse random graphs
• Higher order concentration for functions of weakly dependent random variables
• Intertwining and commutation relations for birth-death processes
• Logarithmic Sobolev inequalities for finite spin systems and applications
• On concentration inequalities and their applications for Gibbs measures in lattice systems
• Concentration inequalities for bounded functionals via generalized log-Sobolev inequalities
• On the convex Poincaré inequality and weak transportation inequalities
• A note on concentration for polynomials in the Ising model
• Concentration of Multilinear Functions of the Ising Model with Applications to Network Data
• High-probability bounds for the reconstruction error of PCA
• Concentration for Poisson functionals: component counts in random geometric graphs
• Modified log-Sobolev inequalities for strong-Rayleigh measures
• 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