Diffusion by optimal transport in Heisenberg groups

Nicolas Juillet (1, 2, 3)

1 IF - Institut Fourier
2 Institut für angewandte Mathematik
3 IRMA - Institut de Recherche Mathématique Avancée

We prove that the hypoelliptic diffusion of the Heisenberg group Hn describes, in the space of probability measures over Hn a gradient flow of the Boltzmann entropy Ent_\infty in the sense of optimal transport. We prove that conversely any gradient flow of Ent_\infty that satisfy a further derivability assumption satisfy the hypoelliptic heat equation.

Mots clés: groupe de Heisenberg; diffusion hypoelliptique; espace de Wasserstein

Domaines: Analyse fonctionnelle [math.FA]; Géométrie métrique [math.MG]; Probabilités [math.PR]

Nicolas Juillet. Diffusion by optimal transport in Heisenberg groups. Calculus of Variations and Partial Differential Equations, 2014, 50 (3-4), pp.693 - 721. ⟨10.1007/s00526-013-0652-2⟩. ⟨hal-00372216v2⟩