Heat and entropy flows in Carnot groups

TL;DR

This paper establishes a fundamental link between heat flow solutions and entropy gradient flows in Carnot groups, extending previous results from the Heisenberg group to more general structures.

Contribution

It generalizes the known correspondence between heat equations and entropy flows from the Heisenberg group to all Carnot groups, solving an open problem.

Findings

Proves the equivalence between sub-elliptic heat solutions and entropy gradient flows in Carnot groups.

Extends previous results from the Heisenberg group to general Carnot groups.

Provides a complete answer to an open question in the mathematical analysis of sub-elliptic operators.

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.