Hamilton Jacobi equations on metric spaces and transport-entropy inequalities

Nathael Gozlan (1) , Cyril Roberto (2) , Paul-Marie Samson (1)

1 Laboratoire d'Analyse et de Mathématiques Appliquées
2 MODAL'X - Modélisation aléatoire de Paris X

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.

Domaines: Probabilités [math.PR]

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