Heat Flow on Alexandrov Spaces

By Nicola Gigli, Kazumasa Kuwada, Shin-ichi Ohta. 2010

153 citations recorded by the source corpus.

We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}L^2\end{align*} \end{document}‐space produces the same evolution as the gradient flow of the relative entropy in the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}L^2\end{align*} \end{document}‐Wasserstein space. This means that the heat flow is well‐defined by either one of the two gradient flows. Combining properties of these flows, we are able to deduce the Lipschitz continuity of the heat kernel as well as Bakry‐Émery gradient estimates and the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\Gamma_2\end{align*} \end{document}‐condition. Our identification is established by purely metric means, unlike preceding results relying on PDE techniques. Our approach generalizes to the case of heat flow with drift. © 2012 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c74f1ee62e5d5fded1ac266ea02511d37ac466ad

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Citations in the corpus

Non-collapsed spaces with Ricci curvature bounded from below (2017, unknown): “( This definition is the result of a research program devoted to the understanding of the heat flow [27, 32, 3] on CD ( K, N ) spaces, and in particular of the introduction of the class of RCD ( K, ∞ ) spaces -”
A User’s Guide to Optimal Transport (2013, background): “Progresses in this direction have been made in [8], where the notion of spaces with Riemannian Ricci bounded below is introduced: shortly said, these spaces are the subclass of CD(K,N) spaces where the heat flow (studied in [45], [53], [7]) is linear.”
Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. (2011, methodology): “This lemma has been introduced in [17] and then used in [9, 2] to study the heat flow on metric measure spaces.”
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below (2011, methodology): “In connection with (ii), notice that the equivalence so far has been proved in Euclidean spaces by Jordan-Kinderleher-Otto, in the seminal paper [24], in Riemannian manifolds by Erbar and Villani [13, 41], in Hilbert spaces by [6], in Finsler spaces by Ohta-Sturm [32] and eventually in Alexandrov sp”
Ricci Curvature of Finite Markov Chains via Convexity of the Entropy (2011, background): “In this sense, W takes over the role of the Wasserstein metric, since it is known since the seminal work by Jordan, Kinderlehrer, and Otto that the heat flow on Rn is the gradient flow of the entropy [23] (see [2, 18, 19, 20, 32, 36] for variations and generalisations).”
Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below (2013, background): “In [4], [5] (see also [24] for the first progresses in this direction) it has been shown that the RCD(K,∞) condition is equivalent to the (properly written/understood) Bochner inequality”
On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces (2013, methodology): “We sketch here an argument following basically the ideas developed in [21] in the setting of Alexandrov spaces.”
Equivalent definitions of BV space and of total variation on metric measure spaces (2014, methodology): “This lemma, in the Wp case, has been introduced in [25] and then used in [17, 5] to study the heat flow on metric measure spaces.”