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Hamilton Jacobi equations on metric spaces and transport-entropy inequalities

Abstract : We prove an Hopf-Lax-Oleinik formula for the solutions of some Hamilton- Jacobi equations on a general metric space. As a rst consequence, we show in full generality 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 transportentropy inequalities in metric space are characterized in terms of log-Sobolev inequalities restricted to the class of c-convex functions.
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https://hal-upec-upem.archives-ouvertes.fr/hal-00795829
Contributor : Nathael Gozlan <>
Submitted on : Friday, March 1, 2013 - 2:46:06 AM
Last modification on : Thursday, March 19, 2020 - 12:26:03 PM
Long-term archiving on: : Thursday, May 30, 2013 - 4:00:26 AM

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Nathael Gozlan, Cyril Roberto, Paul-Marie Samson. Hamilton Jacobi equations on metric spaces and transport-entropy inequalities. Revista Matemática Iberoamericana, European Mathematical Society, 2014, 30 (1), pp.133-163. ⟨10.4171/rmi/772⟩. ⟨hal-00795829⟩

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