On Harnack inequalities and optimal transportation

Dominique Bakry, Ivan Gentil, Michel Ledoux

Published 2012-10-17, updated 2013-07-08Version 4

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.