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

TL;DR

This paper investigates the behavior of co-adapted couplings of Heisenberg Brownian motions, revealing they cannot remain bounded in L^p for p ≥ 2, contrasting with Riemannian cases, and explores coupling by reflection.

Contribution

It demonstrates the non-existence of bounded co-adapted couplings in L^p for p ≥ 2 on the Heisenberg group, extending the analysis to higher dimensions.

Findings

Couplings cannot stay bounded in L^p for p ≥ 2.

Reflection coupling remains bounded in L^p for 0 ≤ p < 1.

Results extend to higher-dimensional Heisenberg groups.

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