Shifted Composition I: Harnack and Reverse Transport Inequalities

TL;DR

This paper introduces the shifted composition rule, a new information-theoretic principle, to establish reverse transport inequalities for diffusions, linking geometric analysis, differential privacy, and sampling techniques.

Contribution

It formulates the shifted composition rule and applies it to derive reverse transport inequalities, offering new coupling methods and functional inequalities for stochastic processes.

Findings

Proves reverse transport inequalities for diffusions.

Develops an alternative coupling method based on optimal transport.

Establishes reverse Harnack inequalities for stochastic processes.

Abstract

We formulate a new information-theoretic principle--the shifted composition rule--which bounds the divergence (e.g., Kullback-Leibler or R\'enyi) between the laws of two stochastic processes via the introduction of auxiliary shifts. In this paper, we apply this principle to prove reverse transport inequalities for diffusions which, by duality, imply F.-Y. Wang's celebrated dimension-free Harnack inequalities. Our approach bridges continuous-time coupling methods from geometric analysis with the discrete-time shifted divergence technique from differential privacy and sampling. It also naturally gives rise to (1) an alternative continuous-time coupling method based on optimal transport, which bypasses Girsanov transformations, (2) functional inequalities for discrete-time processes, and (3) a family of "reverse" Harnack inequalities.