Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds

By Giuseppe Savar'e. 2007

89 citations recorded by the source corpus.

We present some new results concerning well-posedness of gradient flows generated by λ-convex functionals in a wide class of metric spaces, including Alexandrov spaces satisfying a lower curvature bound and the corresponding L2-Wasserstein spaces. Applications to the gradient flow of Entropy functionals in metric-measure spaces with Ricci curvature bounded from below and to the corresponding diffusion semigroup are also considered. These results have been announced during the workshop on “Optimal Transport: theory and applications” held in Pisa, November 2006. To cite this article: G. Savare, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5339fd16a3ba588902d00abddb72ad438a035b78

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Nonlinear mobility continuity equations and generalized displacement convexity (2009, background): “- The convergence of the so called “Minimizing movement” or JKO-scheme could be exploited in this and other situations: in the case of geodesically convex energies, further information on the Alexandrov curvature of the distance Wm, would be crucial, see [AGS05, Sa07 ].”
Duality on gradient estimates and Wasserstein controls (2009, methodology): “With the aid of Theorem 8 in [32], we can apply our duality to show an L(2)-gradient estimate for the heat semigroup.”
Heat Flow on Alexandrov Spaces (2010, methodology): “On Alexandrov spaces, the structure of the heat flow as gradient flow has already been studied in [20] for the Dirichlet energy approach and in [27] by the third author (see also [14, 38]) for the relative entropy approach.”
A User’s Guide to Optimal Transport (2013, background): “6, whose proof has been communicated to us b y Savaré (see also [72], [73]).”
Gradient flows and Evolution Variational Inequalities in metric spaces. I: Structural properties (2018, unknown): “It is a remarkable fact that in this case the gradient flow of E is a semigroup of operators in P2(X) satisfying St (αµ + (1 − α)σ) αSt (µ) + (1 − α)St (σ) for every α ∈ (0, 1), a property encoded by the logarithmic structure of E and the variational formulation of the EVIK-flow [82].”
On the heat flow on metric measure spaces: existence, uniqueness and stability (2010, background): “On the other side, Savaré showed that if a normalized metric metric space satisfy a certain Local Angle Condition, then the flow of the Entropy satisfies: the Evolution Variational Inequality, the λ-exponential contractivity and the linearity (see [11]).”
Uniform convexity and smoothness, and their applications in Finsler geometry (2009, background): “For instance, the 2-uniform smoothness finds its usefulness in Wasserstein geometry, as the L2-Wasserstein space over a compact geodesic space X is 2-uniformly smooth if and only if so is X [21,25].”
Local Poincaré inequalities from stable curvature conditions on metric spaces (2011, background): “condition is also stable under the measured Gromov-Hausdorff convergence (see for instance [19] for this type of statement).”
Gradient flows of the entropy for finite Markov chains (2011, background): “By now a similar interpretation of the heat flow has been established in a wide variety of settings, including Riemannian manifolds [10], Hilbert spaces [2], Wiener spaces [11], Finsler spaces [19], Alexandrov spaces [13] and metric measure spaces [12, 25].”
Geometry and Analysis of Dirichlet forms (2012, background): “Notice that, by [41, 33], compact Aleksandrov spaces with curvature bounded from below satisfy the assumptions of Corollary 6.”
On the differential structure of metric measure spaces and applications (2012, unknown): “Chapter 4 of [3] and [42]).”