TY - JOUR
T1 - A lack of Ricci bounds for the entropic measure on Wasserstein space over the interval
AU - Chodosh, Otis
N1 - Funding Information:
This paper is a consolidated version of my essay written for Part III of the Mathematical Tripos at Cambridge University for the 2010–2011 academic year. I would like to thank Clément Mouhot for agreeing to set and mark the essay, assisting me in learning the material contained within,hisextensiveeditinghelp,suggestingtometheproblemofRicciboundson(P(X),dW), as well as his continued support after the conclusion of the program. I would additionally like to thank Cedric Villani, as well as the anonymous referee for their helpful comments and suggestions. I am grateful to the Cambridge Gates Trust for their financial support during my year at Cambridge. Preparation of this paper was partially completed while supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1147470.
PY - 2012/5/15
Y1 - 2012/5/15
N2 - We examine the entropic measure, recently constructed by von Renesse and Sturm, a measure over the metric space of probability measures on the unit interval equipped with the 2-Wasserstein distance. We show that equipped with this measure, Wasserstein space over the interval does not admit generalized Ricci lower bounds in the entropic displacement convexity sense of Lott-Villani-Sturm. We discuss why this is contrary to what one might expect from heuristic considerations.
AB - We examine the entropic measure, recently constructed by von Renesse and Sturm, a measure over the metric space of probability measures on the unit interval equipped with the 2-Wasserstein distance. We show that equipped with this measure, Wasserstein space over the interval does not admit generalized Ricci lower bounds in the entropic displacement convexity sense of Lott-Villani-Sturm. We discuss why this is contrary to what one might expect from heuristic considerations.
KW - Geometry of spaces of measures
KW - Lower bounds on curvature
KW - Optimal transport
KW - Wasserstein space
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U2 - 10.1016/j.jfa.2012.03.007
DO - 10.1016/j.jfa.2012.03.007
M3 - Article
AN - SCOPUS:84858796837
SN - 0022-1236
VL - 262
SP - 4570
EP - 4581
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 10
ER -