Hamilton Jacobi equations on metric spaces and transport-entropy inequalities

TL;DR

This paper establishes a connection between Hamilton-Jacobi equations, transport-entropy inequalities, and log-Sobolev inequalities on metric spaces, providing new characterizations and formulas for these fundamental concepts.

Contribution

It introduces a Hopf-Lax-Oleinik formula for Hamilton-Jacobi equations on metric spaces and links log-Sobolev inequalities with hypercontractivity and transport-entropy inequalities.

Findings

Log-Sobolev inequality is equivalent to hypercontractivity of the Hamilton-Jacobi semi-group.

Transport-entropy inequalities are characterized by log-Sobolev inequalities on c-convex functions.

A general Hopf-Lax-Oleinik formula for Hamilton-Jacobi equations on metric spaces.

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.