Revista Matemática Iberoamericana


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Volume 36, Issue 1, 2020, pp. 257–290
DOI: 10.4171/rmi/1129

Published online: 2019-10-21

Heat and entropy flows in Carnot groups

Luigi Ambrosio[1] and Giorgio Stefani[2]

(1) Scuola Normale Superiore, Pisa, Italy
(2) Scuola Normale Superiore, Pisa, Italy

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: Carnot group, entropy, gradient flow, sub-elliptic heat equation

Ambrosio Luigi, Stefani Giorgio: Heat and entropy flows in Carnot groups. Rev. Mat. Iberoam. 36 (2020), 257-290. doi: 10.4171/rmi/1129