Eulerian Calculus for the Displacement Convexity in the Wasserstein Distance

In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals de ned 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 Otto-Westdickenberg in [19] and on the metric characterization of the gradient ows generated by the functionals in the Wasserstein space.