On Harnack inequalities and optimal transportation

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.