Hamilton-Jacobi Equations on Graph and Applications

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.

On a mixture of Brenier and Strassen Theorems

“…In particular, Kantorovich type duality formulas are obtained [33,Theorem 9.6] under the assumption that c is convex with respect to the p variable (and some additional mild regularity conditions). We refer to [22,31,32,66,68] for works directly connected to [33] and to [65] for an up-to-date survey of applications of weak transport costs to concentration of measure. Besides their many applications in the field of functional inequalities and concentration of measure, it turns out that weak transport costs are also interesting in themselves as a natural generalization of the transportation problem.…”

“…In particular, Kantorovich type duality formulas are obtained [, Theorem 9.6] under the assumption that is convex with respect to the variable (and some additional mild regularity conditions). We refer to for works directly connected to and to for an up‐to‐date survey of applications of weak transport costs to concentration of measure.…”

Curvature and transport inequalities for Markov chains in discrete spaces

“…Later, the second author remarked (see [39]) that the operator Q t satisfies a discrete version of the Hamilton-Jacobi equation: for all t > 0…”

Kantorovich duality for general transport costs and applications

“…One of the main motivations behind this work, and a few satellite papers by the same authors and Y. Shu [29,55,28,30], is to understand what can replace each term in the chain of implications:…”