{
  "id": "1801.04109",
  "version": "v1",
  "published": "2018-01-12T09:48:17.000Z",
  "updated": "2018-01-12T09:48:17.000Z",
  "title": "Couplings in L^p distance of two Brownian motions and their L{é}vy area",
  "authors": [
    "Michel Bonnefont",
    "Nicolas Juillet"
  ],
  "categories": [
    "math.PR"
  ],
  "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",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-12T09:48:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vy area",
      "heisenberg group",
      "planar brownian motion",
      "study co-adapted couplings",
      "brow-nian motions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}