# Couplings in $L^p$ distance of two Brownian motions and their Lévy 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 Lévy 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 ≥ 0. Actually, we prove the stronger result that they can not stay bounded in L p for p ≥ 2. We also study the coupling by reflection, and show that it stays bounded in L p for 0 ≤ p < 1. Finally, we explain how the results generalise to the Heisenberg groups of higher dimension
Type de document :
Pré-publication, Document de travail
2019
Domaine :

https://hal.archives-ouvertes.fr/hal-01671676
Contributeur : Michel Bonnefont <>
Soumis le : mardi 8 janvier 2019 - 13:51:32
Dernière modification le : jeudi 10 janvier 2019 - 01:24:50

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### Identifiants

• HAL Id : hal-01671676, version 3
• ARXIV : 1801.04109

### Citation

Michel Bonnefont, Nicolas Juillet. Couplings in $L^p$ distance of two Brownian motions and their Lévy area. 2019. 〈hal-01671676v3〉

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