Hyper-contractivity and entropy decay in discrete time

TL;DR

This paper proves that in discrete time, hyper-contractivity of a measure-preserving transition kernel implies entropy contraction without requiring reversibility or regularity, strengthening the connection between these concepts.

Contribution

It establishes a static, general implication from hyper-contractivity to entropy decay in discrete-time Markov kernels, without assumptions like reversibility.

Findings

Hyper-contractivity implies entropy contraction in discrete time.

No reversibility or regularity assumptions are needed.

The result is stronger and simpler than continuous-time analogs.

Abstract

Consider a measure-preserving transition kernel $T$ on an arbitrary probability space $(\mathbb X,\mathcal cA,\pi)$. In this level of generality, we prove that a one-step hyper-contractivity estimate of the form $\|T\|_{p\to q}\le 1$ 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.