Positive curvature property for some hypoelliptic heat kernels

TL;DR

This paper investigates hypoelliptic heat kernels from nilpotent Lie algebras, showing they satisfy a non-negative Ricci curvature condition on radial functions, leading to sharp gradient bounds.

Contribution

It demonstrates that certain hypoelliptic operators satisfy a $CD(0, abla)$ inequality on radial functions, extending known results from the Heisenberg group.

Findings

The sub-Laplace operator on radial functions satisfies a $CD(0, abla)$ inequality.

The operator does not satisfy any $CD(r, abla)$ inequality for $r>0$.

Sharp gradient bounds for the heat kernel are derived.

Abstract

In this note, we look at some hypoelliptic operators arising from nilpotent rank 2 Lie algebras. In particular, we concentrate on the diffusion generated by three Brownian motions and their three L\'evy areas, which is the simplest extension of the Laplacian on the Heisenberg group $\mathbb{H}$. In order to study contraction properties of the heat kernel, we show that, as in the case of the Heisenberg group, the restriction of the sub-Laplace operator acting on radial functions (which are defined in some precise way in the core of the paper) satisfies a non negative Ricci curvature condition (more precisely a $CD(0, \infty)$ inequality), whereas the operator itself does not satisfy any $CD(r,\infty)$ inequality. From this we may deduce some useful, sharp gradient bounds for the associated heat kernel.