On the heat flow on metric measure spaces: existence, uniqueness and stability

By Nicola Gigli. 2010

96 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ccf961474a7d0a147d5b0a25b729c813fd97ae5a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

An Overview of the Proof of the Splitting Theorem in Spaces with Non-Negative Ricci Curvature (2014, background): “This has been done by the author in [21], where it has been shown that such gradient ow is unique.”
On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces (2013, background): “Under the assumption that (X, d,m) satisfies CD(K,∞) it has been proven in [20] and more generally in [6, Thm.”
The Splitting Theorem in Non-Smooth Context (2013, unknown): “– In accordance with the smooth case ([42]), the gradient flow in L2(X,m) of the generically non-quadratic - natural Dirichlet energy on (X, d,m) and the gradient flow in (P2(X),W2) of the relative entropy functional coincide ([32], [37], [5]).”
Metric measure spaces with Riemannian Ricci curvature bounded from below (2011, methodology): “However, a consequence of the analysis of tangent spaces done in [10] is that a Finsler manifold arises as a limit of Riemannian manifolds with Ricci curvature uniformly bounded from below and dimension uniformly bounded from above if and only if it is Riemannian (the case of possibly unbounded dime”
On the differential structure of metric measure spaces and applications (2012, background): “v) If a sequence of K-geodesically convex functionals Γ-converges to a limit functional, then gradient flows of the approximating sequence converge to gradient flows of the limit functional (Theorem 21 of [19])”
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below (2011, background): “In Section 9 we show, still following to a large extent [16], the crucial lower semicontinuity of |DEntm| in LSV spaces; this shows that all existence and uniqueness results of Section 8.”
{Euclidean, metric, and Wasserstein} gradient flows: an overview (2016, background): “It has been recently studied by Ambrosio, Gigli, Savaré and new collaborators (Kuwada and Ohta) in a series of papers ([49, 51, 9]), and has been the starting point of recent researches on the differential structure of metric measure spaces.”
Geometry and Analysis of Dirichlet forms (2012, background): “Recently, under the compactness of X and weak λ-displacement convexity of U∞, Gigli [12] obtained the existence, uniqueness and stability of the gradient flow of U∞; for the basics of the theory of gradient flows see [1].”
Gradient flows of the entropy for jump processes (2012, background): “Recently, it has been shown that the gradient flow characterization provides a good framework to study stability properties of diffusion processes under changes of the driving potential or the underlying geometry [3], [16].”
A User’s Guide to Optimal Transport (2013, background): “K;N / spaces where the heat flow (studied in [8, 45, 53]) is linear.”