Modified log-Sobolev inequalities, Beckner inequalities and moment estimates

TL;DR

This paper establishes the equivalence between Beckner inequalities and modified log-Sobolev inequalities for Markov semigroups, and derives moment estimates with applications to concentration of measure in various stochastic models.

Contribution

It proves the equivalence of Beckner and modified log-Sobolev inequalities in a general setting and develops Sobolev type moment estimates under these inequalities.

Findings

Equivalence of Beckner and modified log-Sobolev inequalities for Markov semigroups.

Derived Sobolev type moment estimates applicable to concentration inequalities.

Applications demonstrated in models like random permutations, zero-range processes, and exponential random graphs.

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.