# Heat and entropy flows in Carnot groups

### Luigi Ambrosio

Scuola Normale Superiore, Pisa, Italy### Giorgio Stefani

Scuola Normale Superiore, Pisa, Italy

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## 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.

## Cite this article

Luigi Ambrosio, Giorgio Stefani, Heat and entropy flows in Carnot groups. Rev. Mat. Iberoam. 36 (2020), no. 1, pp. 257–290

DOI 10.4171/RMI/1129