A Lack of Ricci Bounds for the Entropic Measure on Wasserstein space over the Interval

TL;DR

This paper demonstrates that the entropic measure on Wasserstein space over the interval lacks generalized Ricci lower bounds, challenging expectations based on heuristic reasoning in optimal transport geometry.

Contribution

It provides a rigorous proof that the entropic measure does not admit Ricci bounds, countering prior heuristic assumptions in the field.

Findings

Entropic measure on Wasserstein space over the interval has no Ricci lower bounds.

This result challenges previous heuristic expectations.

The paper discusses implications for optimal transport geometry.

Abstract

This is a condensed form of the author's essay, which can be found at [arXiv:1105.2883]. We prove that the entropic measure constructed by von Renesse-Sturm over Wasserstein space on the unit interval (probability measures on the unit interval equipped with the 2-Wasserstein metric) does not admit generalized Ricci lower bounds in the sense of Lott-Villani-Sturm. We discuss why this is surprising, considering various heuristic arguments.