Calculus of Variations and Geometric Measure Theory

L. Ambrosio - G. Stefani

Heat and entropy flows in Carnot groups

created by stefani on 04 Jan 2018
modified on 03 Oct 2021


Published Paper

Inserted: 4 jan 2018
Last Updated: 3 oct 2021

Journal: Rev. Mat. Iberoam.
Number: 1
Pages: 257--290
Year: 2020
Doi: 10.4171/rmi/1129

ArXiv: 1801.01300 PDF


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