JournalsrmiVol. 30, No. 1pp. 133–163

Hamilton Jacobi equations on metric spaces and transport entropy inequalities

  • Nathael Gozlan

    Université de Marne-la-Vallée, Marne la Vallée, France
  • Cyril Roberto

    Université Paris Ouest Nanterre la Défense, France
  • Paul-Marie Samson

    Université de Marne-la-Vallée, Marne la Vallée, France
Hamilton Jacobi equations on metric spaces and transport entropy inequalities cover
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Abstract

We prove a 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 generality that the log-Sobolev inequality is equivalent to a 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 cc-convex functions.

Cite this article

Nathael Gozlan, Cyril Roberto, Paul-Marie Samson, Hamilton Jacobi equations on metric spaces and transport entropy inequalities. Rev. Mat. Iberoam. 30 (2014), no. 1, pp. 133–163

DOI 10.4171/RMI/772