Diffusion by Optimal Transport in the Heisenberg Group

In this talk, we will consider the hypoelliptic diffusion, the “heat diffusion” of the subRiemannian Heisenberg group $\mathbb{H}$. We will show that in the Wasserstein space $\mathcal{P}_2(\mathbb{H})$, the space of probability measures with finite second moment, it is a curve driven by the gradient flow of the Boltzmann entropy, $\mathrm{Ent}: \mathcal{P}_2\to \mathbb{R}\cup\{\infty\}$. Conversely any gradient flow curve of $\mathrm{Ent}$ satisfies the hypoelliptic heat equation.
This illustrates and completes the theory of Ambrosio, Gigli ans Savaré about the gradient flows of $\mathrm{Ent}$ on the Wasserstein spaces of some very general metric spaces.