Hamilton Jacobi equations on metric spaces and transport-entropy inequalities

Nathael Gozlan; Cyril Roberto; Paul-Marie Samson

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arXiv · 1203.2783

Hamilton Jacobi equations on metric spaces and transport-entropy inequalities

Nathael Gozlan·Cyril Roberto·Paul-Marie Samson

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Abstract

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.

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Nathael Gozlan, Cyril Roberto, Paul-Marie Samson. 2012-03-13. Hamilton Jacobi equations on metric spaces and transport-entropy inequalities. https://arxiv.org/abs/1203.2783

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