On Harnack inequalities and optimal transportation

TL;DR

This paper explores the relationship between Harnack inequalities for heat flows and optimal transportation, establishing new inequalities and properties that connect curvature bounds, heat kernel estimates, and Wasserstein distance contractions.

Contribution

It introduces novel links between Harnack inequalities and optimal transportation, including new isoperimetric-type inequalities and semigroup commutation properties.

Findings

Derived a new isoperimetric-type Harnack inequality.

Established commutation properties between heat and Hopf-Lax semigroups.

Connected curvature bounds with heat flow contraction in Wasserstein space.

Abstract

We develop connections between Harnack inequalities for the heat flow of diffusion operators with curvature bounded from below and optimal transportation. Through heat kernel inequalities, a new isoperimetric-type Harnack inequality is emphasized. Commutation properties between the heat and Hopf-Lax semigroups are developed consequently, providing direct access to the heat flow contraction property along Wasserstein distances.