Hamilton Jacobi equations on metric spaces and transport-entropy inequalities

By Nathael Gozlan, Cyril Roberto, Paul-Marie Samson. 2012

59 citations recorded by the source corpus.

We prove an Hopf-Lax-Oleinik formula for the solutions of some Hamilton- Jacobi equations on a general metric space. As a first consequence, we show in full gener- ality that the log-Sobolev inequality is equivalent to an hypercontractivity property of the Hamilton-Jacobi semi-group. As a second consequence, we prove that Talagrand's transport- entropy inequalities in metric space are characterized in terms of log-Sobolev inequalities restricted to the class of c-convex functions.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/927d746e55e70f7e0a4e5e317153bca561f50852

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In Applied Mathematics

Citations in the corpus

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below (2011, background): “6) (in connection to this, under less general assumptions on the metric structure, closely related results have been independently obtained in [19]).”
On a class of first order Hamilton–Jacobi equations in metric spaces (2014, background): “N. Gozlan, C. Roberto, and P. Samson, Hamilton-Jacobi equations on metric spaces and transport entropy inequalities.”
Kantorovich duality for general transport costs and applications (2014, background): “For instance, it is proven in full generality in [28], that if μ is a probability measure on a metric space (X, d) which satisfies T2, then its support is connected.”
Curvature and transport inequalities for Markov chains in discrete spaces (2015, background): “When X is a graph, π never satisfies T2, unless it is a Dirac measure (see for example, [16], or Section 2).”
Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. (2011, result): “ locally compact spaces and ﬁnite distances, but the proofs can be modiﬁed to deal with more general cases, see also Section 8. The results of this section overlap with those of the forthcoming paper [9] by Gozlan, Roberto and Samson, where the HL semigroup is used in connection with the proof of tr”
Characterization of a class of weak transport-entropy inequalities on the line (2015, methodology): “Then the weak transport-entropy inequalities of Item ( iii ) are obtained in their dual forms, involving inﬁmum convolution operators (Lemma 4.1), by means of the Hamilton-Jacobi semi-group approach of Bobkov, Gentil and Ledoux [4], an approach also generalized in [26, 21, 22].”