Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure

```@article{Ambrosio2015RiemannianRC,
title={Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure},
author={Luigi Ambrosio and Nicola Gigli and Andrea Mondino and Tapio Rajala},
journal={Transactions of the American Mathematical Society},
year={2015},
volume={367},
pages={4661-4701}
}```
• L. Ambrosio, +1 author T. Rajala
• Published 2015
• Mathematics
• Transactions of the American Mathematical Society
In prior work (4) of the first two authors with Savare, 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) the RCD(K,∞) property is defined in three equivalent ways and several properties of RCD(K,∞) spaces, including the regularization properties… Expand
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References

SHOWING 1-10 OF 40 REFERENCES
Metric measure spaces with Riemannian Ricci curvature bounded from below
• Mathematics
• 2014
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 FinslerExpand
Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds
• Mathematics
• 2015
The aim of the present paper is to bridge the gap between the Bakry–Emery and the Lott–Sturm–Villani approaches to provide synthetic and abstract notions of lower Ricci curvature bounds. We startExpand
Ricci curvature for metric-measure spaces via optimal transport
• Mathematics
• 2004
We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of theExpand
On the differential structure of metric measure spaces and applications
The main goals of this paper are: i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of SobolevExpand
Ricci curvature of Markov chains on metric spaces
Abstract We define the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. This definition naturallyExpand
On the structure of spaces with Ricci curvature bounded below. II
• Mathematics
• 2000
In this paper and in we study the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a deExpand
Improved geodesics for the reduced curvature-dimension condition in branching metric spaces
In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition CD*(K,N) we always have geodesics in the Wasserstein space of probability measures that satisfyExpand
Eulerian Calculus for the Displacement Convexity in the Wasserstein Distance
• Mathematics, Computer Science
• SIAM J. Math. Anal.
• 2008
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 is given. Expand
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
• Mathematics
• 2014
This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces \$(X,\mathsf {d},\mathfrak {m})\$. Our main results are: A generalExpand
Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm
We construct geodesics in the Wasserstein space of probability measure along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of theExpand