Correlation and Brascamp–Lieb Inequalities for Markov Semigroups

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 two-fold. 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. Related superadditivity of information and entropy inequalities are also studied.