Hamilton-Jacobi equations on graph and applications

TL;DR

This paper extends Hamilton-Jacobi equations to graphs by defining new gradient and convolution operators, linking these to discrete inequalities like log-Sobolev and Talagrand's transport, with potential applications in analysis on graphs.

Contribution

It introduces a novel framework for Hamilton-Jacobi equations on graphs, including gradient and convolution concepts, and connects these to important discrete functional inequalities.

Findings

Hypercontractivity of infimal-convolution operators on graphs

Connection to discrete log-Sobolev inequalities

Relation to Talagrand's transport inequality

Abstract

This paper introduces a notion of gradient and an infimal-convolution operator that extend properties of solutions of Hamilton Jacobi equations to more general spaces, in particular to graphs. As a main application, the hypercontractivity of this class of infimal-convolution operators is connected to some discrete version of the log-Sobolev inequality and to a discrete version of Talagrand's transport inequality.