00830nas a2200121 4500008004100000245005800041210005800099260003100157520043000188100001700618700002200635856005100657 2014 en d00aLecture notes on gradient flows and optimal transport0 aLecture notes on gradient flows and optimal transport bCambridge University Press3 aWe present a short overview on the strongest variational formulation for gradient flows of geodesically λ-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures. These notes are based on a series of lectures given by the second author for the Summer School "Optimal transportation: Theory and applications" in Grenoble during the week of June 22-26, 2009.1 aDaneri, Sara1 aSavarè, Giuseppe uhttp://urania.sissa.it/xmlui/handle/1963/3509300962nas a2200121 4500008004300000245008100043210006900124260000900193520056300202100001700765700002200782856003600804 2008 en_Ud 00aEulerian calculus for the displacement convexity in the Wasserstein distance0 aEulerian calculus for the displacement convexity in the Wasserst bSIAM3 aIn this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals defined 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 and Westdickenberg [SIAM J. Math. Anal., 37 (2005), pp. 1227-1255] and on the metric characterization of the gradient flows generated by the functionals in the Wasserstein space.1 aDaneri, Sara1 aSavarè, Giuseppe uhttp://hdl.handle.net/1963/3413