Couplings in $L^p$ distance of two Brownian motions and their L{\'e}vy area

Abstract

We study co-adapted couplings of (canonical hypoelliptic) diffu-sions on the (subRiemannian) Heisenberg group, that we call (Heisenberg) Brow-nian motions and are the joint laws of a planar Brownian motion with its L{\'e}vy area. We show that contrary to the situation observed on Riemannian manifolds of non-negative Ricci curvature, for any co-adapted coupling, two Heisenberg Brownian motions starting at two given points can not stay at bounded distance for all time t $\ge$ 0. Actually, we prove the stronger result that they can not stay bounded in L p for p $\ge$ 2. We also study the coupling by reflection, and show that it stays bounded in L p for 0 $\le$ p < 1. Finally, we explain how the results generalise to the Heisenberg groups of higher dimension