Diffusion by optimal transport in Heisenberg groups

  • Nicolas Juillet


We prove that the hypoelliptic diffusion of the Heisenberg group \({\mathbb{H }}_n\) describes, in the space of probability measures over \({\mathbb{H }}_n\), a curve driven by the gradient flow of the Boltzmann entropy \({{\mathrm{Ent}}}\), in the sense of optimal transport. We prove that conversely any gradient flow curve of \({{\mathrm{Ent}}}\) satisfy the hypoelliptic heat equation. This occurs in the subRiemannian \({\mathbb{H }}_n\), which is not a space with a lower Ricci curvature bound in the metric sense of Lott–Villani and Sturm.


Heisenberg group Optimal transport theory Gradient flow 

Mathematics Subject Classification (2000)

28A33 53C17 60J60 



I wish to thank Hervé Pajot and Karl-Theodor Sturm that were my advisers when I started this research. I thank Luigi Ambrosio, Nicola Gigli and Cédric Villani for helpful discussions as well as my ex-colleagues Matthias Erbar and Miguel Rodrigues. The referee of the first version gave me good advice (for instance on alternative proofs in Proposition 3.1 and Paragraph 4.1) and the motivation for avoiding a non-necessary extra hypothesis in Paragraph 4.2. I would like to thank her or him very much. I thank also the other referees for their careful reading.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institut de Recherche Mathématique Avancée, UMR 7501Université de Strasbourg et CNRSStrasbourgFrance

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