Eulerian Calculus for the Displacement Convexity in the Wasserstein Distance

Abstract

In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals dened on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and regularity results for optimal transport maps on Riemannian manifolds, but it is based on the Eulerian point of view recently introduced by and on the metric characterization of the gradient ows generated by the functionals in the Wasserstein space.

Optimal transport and Perelman’s reduced volume

“…We expect that one can prove Proposition 16 using the Eulerian approach and a density argument, along the lines of [5], but we do not pursue this here.…”

“…These proofs can then be considered as formal proofs of the analogous statements on the space of all probability measures P (M). The rigorous proofs of the statements on P (M) are usually done by the Lagrangian approach, but one can also use the density of P ∞ (M) in P (M) [5,20]. Most of the calculations in this section can be extracted from [19] and [20].…”

“…(See, however, the papers of Daneri-Savaré [5] and Otto-Westdickenberg [20], which prove results about optimal transport in P (M) using the Eulerian approach along with density arguments.) Much of the present paper consists of describing an Otto calculus which is adapted for the optimal transport of measures under a Ricci flow background.…”

“…Sections 4-7 contain calculations in the Eulerian framework under a Ricci flow background. We expect that one can extend these calculations to rigorous proofs on P (M), by adapting the density methods of [5] or [20] to the setting of time-dependent cost functions. We do not address this issue here.…”

Example of a displacement convex functional of first order

“…Displacement convexity on Riemannian manifolds for internal energy (under Ricci curvature bounds) was predicted by Otto and Villani [21] and proved by Cordero-Erausquin, McCann, and Schmuckenschläger in [8]. Recently Otto and Westdickenberg [22] have introduced techniques that were further developed by Daneri and Savare [11] to show geodesic convexity of functionals on manifolds using a purely local, Eulerian framework. The connection between bounds on displacement convexity and bounds on Ricci curvature has found important applications, see papers by Lott and Villani [16], Sturm and von Renesse [25], Sturm [24,23] and references therein.…”