Heat and entropy flows in Carnot groups

Abstract

We prove the correspondence between the solutions of the sub-elliptic heat equation in a Carnot group $\mathbb G$ and the gradient flows of the relative entropy functional in the Wasserstein space of probability measures on $\mathbb G$. Our result completely answers a question left open in a previous paper by N. Juillet, where the same correspondence was proved for $\mathbb G = \mathbb H^n$, the $n$-dimensional Heisenberg group.