Couplings in $L^p$ distance of two Brownian motions and their Lévy area

Michel Bonnefont (1) , Nicolas Juillet (2)

1 IMB - Institut de Mathématiques de Bordeaux
2 IRMA - Institut de Recherche Mathématique Avancée

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

Domaines: Probabilités [math.PR]

Michel Bonnefont, Nicolas Juillet. Couplings in $L^p$ distance of two Brownian motions and their Lévy area. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2020, ⟨10.1214/19-AIHP972⟩. ⟨hal-01671676v4⟩