Transport inequalities, gradient estimates, entropy and Ricci curvature

By Max-K. von Renesse, Karl‐Theodor Sturm. 2005

547 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/09298e1d2778dbc25e1d918dc832eca94e1dbd69

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature (2016, background): “ hfor a (not necessarily symmetric) 2-tensor T and a function hprovided T (X,X) ≥ h(x)|X|2, X∈ TxM,x∈ M. There exist many inequalities on Pt which are equivalent to the curvature condition (1.1), see [5, 19, 22, 39] for details. In particular, for any constant K∈ R, the Wasserstein distance inequali”
A User’s Guide to Optimal Transport (2013, methodology): “M/;W2/ and the bound from below on the Ricci curvature is has been given by Sturm and von Renesse in [76].”
Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates (2018, background): “Examples where the quasi Bakry-Émery estimate is satisfied include: Riemannian manifolds with non negative Ricci curvature (in that case C = 1, see [95]), some metric graphs like the Walsh spider (see [18, Example 5.”
Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension (2014, background): “The CD( ρ, N ) condition has been an object of extensive study over the last two decades (see e.g. also [4,18,19,31,33,46,51,58,61,62] and the references therein), especially since Perelman’s work on the Poincar´e Conjecture [57] and the extension of the Curvature-Dimension condition to the…”
Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds (2009, background): “One of them is the use of optimal transportation for an alternative definition of Ricci curvature lower bound developed in a series of papers [23, 9, 26 ].”
Ricci Curvature on Alexandrov spaces and Rigidity Theorems (2009, background): “1] and von Renesse, Sturm in [RS, S4] proved that (M,d,vol) satisﬁes CD(∞,k) if and only if Ric(M) > k. Indeed, they proved a stronger weighted version (see Theorem 7.3 in [LV1] and Theorem 1.1 in [RS], Theorem 1.3 in [S4]). Let φbe a smooth function on Mwith R M e −φdvol= 1. Lott and Villani in ”
Bochner-Weitzenb\"ock formula and Li-Yau estimates on Finsler manifolds (2011, background): “he bound RicN(v) ≥KF(v)2 is equivalent to Lott, Villani and the second author’s curvature-dimension condition CD(K,N). This extends the corresponding result on (weighted) Riemannian manifolds (due to [vRS], [St1], [St2], [St3], [LV2], [LV3]), and has many analytic and geometric applications (see [Oh”
Duality on gradient estimates and Wasserstein controls (2009, background): “Sturm [37] unified several formulations of this kind for linear heat equation on a complete Riemannian manifold.”
Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries (2011, background): “Let us note explicitly that the function f1 constructed in [47] satisfies (i)-(iii) above.”
Semi Log-Concave Markov Diffusions (2013, background): “Actually as shown first by Sturm and Von Renesse [44], this contraction property is equivalent to positive curvature K.”