Diffusion by optimal transport in Heisenberg groups

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.