Hyper-contractivity and entropy decay in discrete time

Abstract

Consider a measure-preserving transition kernel $T$ on an arbitrary probability space $(\mathbb{X},cA,\pi )$. In this level of generality, we prove that a one-step hyper-contractivity estimate of the form $\Vert T{\Vert }_{p\to q}\le 1$ with $p<q$ implies a one-step entropy contraction estimate of the form $\mathrm{H}(\mu T|\pi )\le \theta \mathrm{H}(\mu |\pi )$, with $\theta =p/q$. Neither reversibility, nor any sort of regularity is required. This static implication is simultaneously simpler and stronger than the celebrated dynamic relation between exponential hyper-contractivity and exponential entropy decay along continuous-time Markov semi-groups.

7 pages, comments welcome!