{
  "id": "math/0412410",
  "version": "v1",
  "published": "2004-12-20T19:37:20.000Z",
  "updated": "2004-12-20T19:37:20.000Z",
  "title": "On the invariant measure of a positive recurrent diffusion in R",
  "authors": [
    "Michele L. Baldini"
  ],
  "comment": "18 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Given an one-dimensional positive recurrent diffusion governed by the Stratonovich SDE \\[ X_t=x+\\int_0^t\\sigma(X_s)\\strat db(s)+\\int_0^t m(X_s) ds, \\] we show that the associated stochastic flow of diffeomorphisms focuses as fast as $ \\mathrm{exp}(-2t\\int_{R}\\frac{m^2}{\\sigma^2} d\\Pi)$, where $d\\Pi$ is the finite stationary measure. Moreover, if the drift is reversed and the diffeomorphism is inverted, then the path function so produced tends, independently of its starting point, to a single (random) point whose distribution is $d\\Pi$. Applications to stationary solutions of $X_t$, asymptotic behavior of solutions of SPDEs and random attractors are offered.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-12-20T19:37:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J60",
      "60H15"
    ],
    "keywords": [
      "invariant measure",
      "finite stationary measure",
      "stratonovich sde",
      "one-dimensional positive recurrent diffusion",
      "stochastic flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....12410B"
    }
  }
}