BONNEFONT, Michel; JUILLET, Nicolas

doi:10.1214/19-AIHP972

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BONNEFONT, Michel
Institut de Mathématiques de Bordeaux [IMB]

JUILLET, Nicolas
Institut de Recherche Mathématique Avancée [IRMA]

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Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques. 2020

Institut Henri Poincaré (IHP)

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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< Leer menos

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

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