{
  "id": "1301.6354",
  "version": "v1",
  "published": "2013-01-27T14:09:35.000Z",
  "updated": "2013-01-27T14:09:35.000Z",
  "title": "Absolute Continuity under Time Shift of Trajectories and Related Stochastic Calculus",
  "authors": [
    "Jörg-Uwe Löbus"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "The paper is concerned with a class of two-sided stochastic processes of the form $X=W+A$. Here $W$ is a two-sided Brownian motion with random initial data at time zero and $A\\equiv A(W)$ is a function of $W$. Elements of the related stochastic calculus are introduced. In particular, the calculus is adjusted to the case when $A$ is a jump process. Absolute continuity of $(X,P_{\\sbnu})$ under time shift of trajectories is investigated. For example under various conditions on the initial density with respect to the Lebesgue measure, $m=d\\bnu/dx$, and on $A$ with $A_0=0$ we verify % {eqnarray*} \\frac{P_{\\sbnu}(dX_{\\cdot -t})}{P_{\\sbnu}(dX_\\cdot)}=\\frac{m(X_{-t})} {m(X_0)}\\cdot\\prod_i|\\nabla_{W_0}X_{-t}|_i {eqnarray*} % a.e. where the product is taken over all coordinates. Here $\\sum_i(\\nabla_{W_0}X_{-t})_i$ is the divergence of $X_{-t}$ with respect to the initial position. Crucial for this is the {\\it temporal homogeneity} in the sense that $X(W_{\\cdot +v}+A_v\\1)=X_{\\cdot+v}(W)$, $v\\in {\\Bbb R}$, where $A_v\\1$ is the trajectory taking the constant value $A_v (W)$. By means of such a density, partial integration relative to the generator of the process $X$ is established. Relative compactness of sequences of such processes is established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-27T14:09:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G44",
      "60H07",
      "60J65",
      "60J75"
    ],
    "keywords": [
      "related stochastic calculus",
      "time shift",
      "absolute continuity",
      "trajectory",
      "random initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.6354L"
    }
  }
}