Diffusion by Optimal Transport in the Heisenberg Group

The Heisenberg group Hn is apparently one of the meeting point of various mathematical and scientifical domains. This can be partially explained by the fact that the first Heisenberg group appears as the less elaborated Lie group after the Euclidean spaces. It includes only a few non-commutativity and its Lie algebra presents a unique non-trivial relation (that can be interpreted in quantic physic as the uncertainty principle). That is a reason why it is occuring so much in different domains. The subRiemannian distance, called for Hn Carnot-Carathéodory distance dc provides an exotic structure that still allows elementary computations. It makes of Hn one of the spaces aimed by some theory on metric spaces (see e.g. [12] for geometric measure theory, [19] for conformal geometry, [6] for embedding problems). This paper is devoted to a metric approach of the hypoelliptic diffusion in the Heisenberg group. We will consider in the main theorems (Theorem 0.1 and Theorem 0.2) some curves of P2(Hn)– a space of probability measures called Wasserstein space –that are defined as the gradient flow of the Boltzmann entropy with respect to the Wasserstein metric structure on P(Hn). We prove that these curves correspond to the solutions of the hypoelliptic heat equation provided by ∆H, the subRiemmannian “sum of square” operator, the so-called Kohn operator. The hypoellipticity of this operator is the consequence of a famous theorem by Hörmander [14] about “sum of square” operators. Roughly speaking we give a metric way to characterize the classical diffusion of the Heisenberg group by using a gradient flow in the Wasserstein space P2(Hn). We say now a little more about the origin of this approach. The breakthrough on this topic are the seminal papers of Otto ([15, 24] the first one with Jordan and Kinderlehrer) where the Wasserstein space P2(R) is considered for the first time formally as an infinite dimensional Riemannian manifold. This approach is sometimes called “Otto calculus” as in [28]. Otto and his coauthors realized that the solutions of the heat equation are densities of measures describing a special curve on P2(R) and justify it at the formal level. The Boltzmann entropy (with