{
  "id": "1106.5637",
  "version": "v2",
  "published": "2011-06-28T12:09:13.000Z",
  "updated": "2013-01-17T17:27:20.000Z",
  "title": "The Itô exponential on Lie Groups",
  "authors": [
    "Simão N. Stelmastchuk"
  ],
  "journal": "Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 7, 307 - 326",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $G$ be a Lie Group with a left invariant connection $\\nabla^{G}$. Denote by $\\g$ the Lie algebra of $G$, which is equipped with a connection $\\nabla^{\\g}$. Our main is to introduce the concept of the It\\^o exponential and the It\\^o logarithm, which take in account the geometry of the Lie group $G$ and the Lie algebra $\\g$. This definition characterize directly the martingales in $G$ with respect to the left invariant connection $\\nabla^{G}$. Further, if any $\\nabla^{\\g}$ geodesic in $\\g$ is send in a $\\nabla^{G}$ geodesic we can show that the It\\^o exponential and the It\\^o logarithm are the same that the stochastic exponential and the stochastic logarithm due to M. Hakim-Dowek and D. L\\'epingle in [10]. Consequently, we have a Campbell-Hausdorf formula. From this formula we show that the set of affine maps from $(M,\\nabla^{G})$ into $(G,\\nabla^{G})$ is a subgroup of the Loop group. As in general, the Lie algebra is considered as smooth manifold with a flat connection, we show a Campbell-Hausdorf formula for a flat connection on $\\g$ and a bi-invariant connection on $G$. To this main we introduce the definition of the null quadratic variation property. To end, we use the Campbell-Hausdorff formula to show that a product of harmonic maps with value in $G$ is a harmonic map.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-01-17T17:27:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E99",
      "53C43",
      "58E20",
      "58J65",
      "60H30"
    ],
    "keywords": [
      "lie group",
      "lie algebra",
      "left invariant connection",
      "null quadratic variation property",
      "flat connection"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.5637S"
    }
  }
}