Hamilton–Jacobi semigroup on length spaces and applications

By J. Lott, Cédric Villani. 2006

66 citations recorded by the source corpus.

Abstract We define a Hamilton–Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to concentration of measure. Our main results are that (1) a Talagrand inequality on a measured length space implies a global Poincare inequality and (2) if the space satisfies a doubling condition, a local Poincare inequality and a log-Sobolev inequality then it also satisfies a Talagrand inequality.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d134e1cc040e26d31061546468f65a5941dc56f7

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

A characterization of dimension free concentration in terms of transportation inequalities (2008, background): “Following [29], one says that a probability measure μ on X verifies the inequality LSI+(C) if Entμ(f 2) ≤ C ∫ |∇−f |2 dμ”
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below (2011, background): “We remark that under this assumption it can be proved that the Hopf-Lax formula produces a semigroup (see for instance the proof in [28]), while in general only the inequality Qs+tf ≤ Qs(Qtf) holds.”
Metric viscosity solutions of Hamilton–Jacobi equations depending on local slopes (2015, background): “As regards equations in metric spaces, Hamilton–Jacobi semigroups, and pointwise differential inequalities involving local slopes satisfied by them have been investigated in [3, 4, 5, 6, 42, 43, 53], mostly in connection with applications to various functional inequalities.”
Geometry and Analysis of Dirichlet forms (2012, methodology): “To prove this, we recall the following result about the Hamilton-Jacobi semigroup established in [29, 5].”
Transport Inequalities. A Survey (2010, background): “This approach is at the origin of many subsequent developments (see for instance [48], [47], [25], [73] or [54]).”
Yau's gradient estimates on Alexandrov spaces (2010, background): “re exists t¯= t¯(Ω′,kukL∞) such that (5.1) Qtu(x) = min y∈Ω n u(y) + |xy|2 2t o for all x ∈ Ω′ and 0 <t <t¯. For convenience, we always set ut := Qtu in this section. The following was shown in [30] in framework of length spaces. YAU’S GRADIENT ESTIMATES 31 Lemma 5.1. (Lott–Villani [30]) (i) F”
Characterization of a class of weak transport-entropy inequalities on the line (2015, unknown): “1), by means of the Hamilton-Jacobi semi-group approach of Bobkov, Gentil and Ledoux [7], an approach also generalized in [32, 24, 25].”
Duality on gradient estimates and Wasserstein controls (2009, methodology): “In the case p > 1, we employ a general theory of Hamilton-Jacobi semigroup developed in [7, 23] to analyze the variational formula.”
Hamilton Jacobi equations on metric spaces and transport-entropy inequalities (2012, methodology): “In [20, Theorem 22.46], Villani proves (1.8) for all t and x on a Riemannian manifold.”