Optimal Transport and Ricci Curvature: Wasserstein Space Over the Interval

Abstract

In this essay, we discuss the notion of optimal transport on geodesic measure spaces and the associated (2-)Wasserstein distance. We then examine displacement convexity of the entropy functional on the space of probability measures. In particular, we give a detailed proof that the Lott-Villani-Sturm notion of generalized Ricci bounds agree with the classical notion on smooth manifolds. We also give the proof that generalized Ricci bounds are preserved under Gromov-Hausdorff convergence. In particular, we examine in detail the space of probability measures over the interval, $P(X)$ equipped with the Wasserstein metric $d^W$. We show that this metric space is isometric to a totally convex subset of a Hilbert space, $L^2[0,1]$, which allows for concrete calculations, contrary to the usual state of affairs in the theory of optimal transport. We prove explicitly that $(P(X),d^W)$ has vanishing Alexandrov curvature, and give an easy to work with expression for the entropy functional on this space. In addition, we examine finite dimensional Gromov-Hausdorff approximations to this space, and use these to construct a measure on the limit space, the entropic measure first considered by Von Renesse and Sturm. We examine properties of the measure, in particular explaining why one would expect it to have generalized Ricci lower bounds. We then show that this is in fact not true. We also discuss the possibility and consequences of finding a different measure which does admit generalized Ricci lower bounds.Comment: 47 pages, 9 figure