{
  "id": "math/0509422",
  "version": "v2",
  "published": "2005-09-19T13:10:14.000Z",
  "updated": "2006-08-03T12:15:17.000Z",
  "title": "Two-parameter $p, q$-variation Paths and Integrations of Local Times",
  "authors": [
    "Chunrong Feng",
    "Huaizhong Zhao"
  ],
  "journal": "Potential Analysis, Vol. 25 (2006), 165-204",
  "doi": "10.1007/s11118-006-9024-2",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we prove two main results. The first one is to give a new condition for the existence of two-parameter $p, q$-variation path integrals. Our condition of locally bounded $p,q$-variation is more natural and easy to verify than those of Young. This result can be easily generalized to multi-parameter case. The second result is to define the integral of local time $\\int_{-\\infty}^\\infty\\int_0^t g(s,x)d_{s,x}L_s(x)$ pathwise and then give generalized It$\\hat {\\rm o}$'s formula when $\\nabla^-f(s,x)$ is only of bounded $p,q$-variation in $(s,x)$. In the case that $g(s,x)=\\nabla^-f(s,x)$ is of locally bounded variation in $(s,x)$, the integral $\\int_{-\\infty}^\\infty\\int_0^t \\nabla^-f(s,x)d_{s,x}L_s(x)$ is the Lebesgue-Stieltjes integral and was used in Elworthy, Truman and Zhao \\cite{Zhao}. When $g(s,x)=\\nabla^-f(s,x)$ is of only locally $p, q$-variation, where $p\\geq 1$,$q\\geq 1$, and $2q+1>2pq$, the integral is a two-parameter Young integral of $p,q$-variation rather than a Lebesgue-Stieltjes integral. In the special case that $f(s,x)=f(x)$ is independent of $s$, we give a new condition for Meyer's formula and $\\int_{-\\infty}^\\infty L_t(x)d_x\\nabla^-f(x)$ is defined pathwise as a Young integral. For this we prove the local time $L_t(x)$ is of $p$-variation in $x$ for each $t\\geq 0$, for each $p>2$ almost surely ($p$-variation in the sense of Lyons and Young, i.e. $\\sup\\limits_{E: \\ a finite partition of [-N,N]} \\sum\\limits_{i=1}^m|L_t(x_i)-L_t(x_{i-1})|^p<\\infty$).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-08-03T12:15:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local time",
      "integrations",
      "lebesgue-stieltjes integral",
      "two-parameter young integral",
      "variation path integrals"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......9422F"
    }
  }
}