A New Class of First Order Displacement Convex Functionals

Abstract

New displacement convex functionals on $S^1$ are presented, and a novel Eulerian extension of the Otto calculus is developed. The proof of the displacement convexity of these functionals relies on this Eulerian calculus, which extends the Otto calculus from a purely Riemannian setting to general Lagrangians and provides a systematic way to compute displacement Hessians of functionals involving derivatives of densities. This resolves, in the special cases of $\mathbb{R}$ and $S^1$, a problem posed by Villani in Open Problem 15.11 of [Optimal Transport: Old and New, Grundlehren Math. Wiss. 338, Springer, Berlin, 2009].