On the equivalence of the entropic curvature-dimension condition and bochner's inequality on metric measure spaces
TLDR
In this paper, the equivalence of the curvature dimension bounds of Lott-Sturm-Villani and Bakry-Emery in complete generality for infinitesimally Hilbertian metric measure spaces was established.Abstract:
We prove the equivalence of the curvature-dimension bounds of Lott–Sturm–Villani (via entropy and optimal transport) and of Bakry–Emery (via energy and $$\Gamma _2$$
-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric measure spaces. Moreover, we deduce new contraction bounds for the heat flow on Riemannian manifolds and on mms in terms of the $$L^2$$
-Wasserstein distance.read more
Citations
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Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures
TL;DR: In this paper, the authors developed a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces, and introduced the new Hellinger-Kantorovich distance between measures in metric spaces.
Journal ArticleDOI
Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds
Fabio Cavalletti,Andrea Mondino +1 more
TL;DR: In this paper, it was shown that the classic Levy-Gromov isoperimetric inequality holds for non-smooth metric spaces with Ricci curvature lower bounds.
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The Globalization Theorem for the Curvature Dimension Condition
Fabio Cavalletti,Emanuel Milman +1 more
TL;DR: The Lott-Sturm-Villani Curvature-Dimension condition provides a synthetic notion for a metric-measure space to have Ricci-curvature bounded from below and dimension bounded from above as discussed by the authors.
Book
Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces
TL;DR: In this article, a new characterisation of the curvature dimension condition in the context of metric measure spaces (X, d,m) is provided, taking into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N -dimensional entropies.
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Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds
Andrea Mondino,Aaron Naber +1 more
TL;DR: In this article, it was shown that a metric measure space with finite dimensional lower Ricci curvature bounds and whose Sobolev space is Hilbert is rectifiable, i.e. the point at which the tangent cone is unique and euclidean of dimension at most n.
References
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Book
Optimal Transport: Old and New
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
Book
Gradient Flows: In Metric Spaces and in the Space of Probability Measures
TL;DR: In this article, Gradient flows and curves of Maximal slopes of the Wasserstein distance along geodesics are used to measure the optimal transportation problem in the space of probability measures.
Book
Heat kernels and spectral theory
TL;DR: In this paper, the authors introduce the concept of Logarithmic Sobolev inequalities and Gaussian bounds on heat kernels, as well as Riemannian manifolds.
Book
A Course in Metric Geometry
TL;DR: In this article, a large-scale Geometry Spaces of Curvature Bounded Above Spaces of Bounded Curvatures Bounded Below Bibliography Index is presented. But it is based on the Riemannian metric space.
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On the parabolic kernel of the Schrödinger operator
Peter Li,Shing-Tung Yau +1 more
TL;DR: Etude des equations paraboliques du type (Δ−q/x,t)−∂/∂t)u(x, t)=0 sur une variete riemannienne generale as discussed by the authors.