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.