An entropic interpolation proof of the HWI inequality

TL;DR

This paper introduces a new proof of the HWI inequality using entropic interpolations, which are smoother than displacement interpolations, providing a more regular and intuitive approach aligned with Otto-Villani heuristics.

Contribution

The paper offers a novel pathwise proof of the HWI inequality based on entropic interpolations, contrasting with traditional displacement-based methods.

Findings

Entropic interpolations are regular in space and time.

The proof aligns with Otto-Villani heuristics.

Provides a more intuitive understanding of the HWI inequality.

Abstract

We present a pathwise proof of the HWI inequality which is based on en-tropic interpolations rather than displacement ones. Unlike the latter, entropic interpolations are regular both in space and time. Consequently, our approach is closer to the Otto-Villani heuristics, presented in the first part of the article [23], than the original rigorous proof presented in the second part of [23].