*Submitted Paper*

**Inserted:** 4 jan 2018

**Last Updated:** 29 oct 2019

**Year:** 2018

**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.

**Keywords:**
entropy, Gradient Flow, Carnot group, sub-elliptic heat equation

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