Correlation and Brascamp-Lieb inequalities for Markov semigroups

TL;DR

This paper develops a unifying Markov semigroup framework for Brascamp-Lieb inequalities across different settings and explains the combinatorial reasons behind unexpected exponents in these inequalities.

Contribution

It introduces a general framework based on Markov generators and elucidates the combinatorial origins of unusual exponents in Brascamp-Lieb inequalities.

Findings

Unified Markov generator framework for inequalities

Explanation of unexpected exponents in inequalities

Extension beyond previous Euclidean and spherical settings

Abstract

This paper builds upon several recent works, where semigroup proofs of Brascamp-Lieb inequalities are provided in various settings (Euclidean space, spheres and symmetric groups). Our aim is twofold. Firstly, we provide a general, unifying, framework based on Markov generators, in order to cover a variety of examples of interest going beyond previous investigations. Secondly, we put forward the combinatorial reasons for which unexpected exponents occur in these inequalities.