  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.
