  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.
