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

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.