# 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.

## Key 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.