On Harnack inequalities and optimal transportation

Dominique Bakry (1, 2) , Ivan Gentil (3) , Michel Ledoux (1, 2)

1 IUF - Institut universitaire de France
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 EDPA - Équations aux dérivées partielles, analyse

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.

Domaines: Probabilités [math.PR]

Dominique Bakry, Ivan Gentil, Michel Ledoux. On Harnack inequalities and optimal transportation. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2015, 14, pp.705-727. ⟨hal-00742654v5⟩