Revista Matemática Iberoamericana


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Volume 30, Issue 1, 2014, pp. 133–163
DOI: 10.4171/RMI/772

Published online: 2014-03-23

Hamilton Jacobi equations on metric spaces and transport entropy inequalities

Nathael Gozlan[1], Cyril Roberto[2] and Paul-Marie Samson[3]

(1) Université de Marne-la-Vallée, Marne la Vallée, France
(2) Université Paris Ouest Nanterre la Défense, France
(3) Université de Marne-la-Vallée, Marne la Vallée, France

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 $c$-convex functions.

Keywords: Transport inequalities, Hamilton–Jacobi equations, logarithmic Sobolev inequalities, metric spaces

Gozlan Nathael, Roberto Cyril, Samson Paul-Marie: Hamilton Jacobi equations on metric spaces and transport entropy inequalities. Rev. Mat. Iberoamericana 30 (2014), 133-163. doi: 10.4171/RMI/772