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This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimal mass transportation and of its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal… (More)

In prior work [4] of the first two authors with Savaré, a new Riemannian notion of lower bound for Ricci curvature in the class of metric measure spaces (X, d,m) was introduced, and the corresponding class of spaces denoted by RCD(K,∞). This notion relates the CD(K,N) theory of Sturm and Lott-Villani, in the case N = ∞, to the Bakry-Emery approach. In [4]… (More)

We compare several notion of weak (modulus of) gradient in metric measure spaces and prove their equivalence. Using tools from optimal transportation theory we prove density in energy of Lipschitz maps independently of doubling and Poincaré assumptions on the metric measure space.

In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X, d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity… (More)

The aim of the present paper is to bridge the gap between the Bakry-Émery and the Lott-Sturm-Villani approaches to provide synthetic and abstract notions of lower Ricci curvature bounds. We start from a strongly local Dirichlet form E admitting a Carré du champ Γ in a Polish measure space (X,m) and a canonical distance dE that induces the original topology… (More)

Aim of this paper is to discuss convergence of pointed metric measure spaces in absence of any compactness condition. We propose various definitions, show that all of them are equivalent and that for doubling spaces these are also equivalent to the well known measured-Gromov-Hausdorff convergence. Then we show that the curvature conditions CD(K,∞) and… (More)

- Nicola Gigli, Jan Maas
- SIAM J. Math. Analysis
- 2013

This paper continues the investigation of ‘Wasserstein-like’ transportation distances for probability measures on discrete sets. We prove that the discrete transportation metrics on the d-dimensional discrete torus TN with mesh size 1 N converge, when N → ∞, to the standard 2-Wasserstein distance on the continuous torus in the sense of Gromov– Hausdorff.… (More)

- Nicola Gigli
- 2010

We do three things. First, we characterize the class of measures μ ∈P2(M) such that for any other ν ∈P2(M) there exists a unique optimal transport plan, and this plan is induced by a map. Second, we study the tangent space at any measure and we identify the class of measures for which the tangent space is an Hilbert space. Third, we prove that these two… (More)

We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiation and to the theory of Sobolev spaces… (More)

- Felix Otto, Nicola Gigli
- 2010

As noted by the second author in the context of unstable two-phase porous medium flow, entropy solutions of Burgers’ equation can be recovered from a minimizing movement scheme involving the Wasserstein metric in the limit of vanishing time step size [4]. In this paper, we give a simpler proof by verifying that the anti-derivative is a viscosity solution of… (More)