On the Support of a hypoelliptic diffusion on the Heisenberg group

TL;DR

This paper offers an elementary proof characterizing the support of hypoelliptic Brownian motion on the Heisenberg group using control norms and horizontal curves.

Contribution

It provides a new, simplified proof of the support theorem for hypoelliptic diffusions on the Heisenberg group, linking support to finite energy horizontal curves.

Findings

Support characterized by finite energy horizontal curves

Elementary proof simplifies previous approaches

Clarifies the structure of hypoelliptic Brownian motion

Abstract

We provide an elementary proof of the support of the law of a hypoelliptic Brownian motion on the Heisenberg group $\mathbb{H}$. We consider a control norm associated to left-invariant vector fields on $\mathbb{H}$, and describe the support in terms of the space of finite energy horizontal curves.