Coupling in the Heisenberg group and its applications to gradient estimates

TL;DR

This paper develops a non-Markovian coupling for hypoelliptic diffusions in the Heisenberg group, leading to new gradient estimates for harmonic functions associated with the hypoelliptic Laplacian.

Contribution

It introduces a novel non-Markovian coupling method for hypoelliptic diffusions in the Heisenberg group and applies it to derive gradient estimates.

Findings

Established bounds on the coupling rate

Derived upper and lower bounds on total variation distance

Proved gradient estimates for harmonic functions

Abstract

We construct a non-Markovian coupling for hypoelliptic diffusions which are Brownian motions in the three-dimensional Heisenberg group. We then derive properties of this coupling such as estimates on the coupling rate, and upper and lower bounds on the total variation distance between the laws of the Brownian motions. Finally we use these properties to prove gradient estimates for harmonic functions for the hypoelliptic Laplacian which is the generator of Brownian motion in the Heisenberg group.