{
  "id": "1009.2732",
  "version": "v1",
  "published": "2010-09-14T18:53:30.000Z",
  "updated": "2010-09-14T18:53:30.000Z",
  "title": "Current fluctuations for independent random walks in multiple dimensions",
  "authors": [
    "Rohini Kumar"
  ],
  "comment": "31 pages; accepted for publication in Journal of Theoretical Probability",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider a system of particles evolving as independent and identically distributed (i.i.d.) random walks. Initial fluctuations in the particle density get translated over time with velocity $\\vec{v}$, the common mean velocity of the random walks. Consider a box centered around an observer who starts at the origin and moves with constant velocity $\\vec{v}$. To observe interesting fluctuations beyond the translation of initial density fluctuations, we measure the net flux of particles over time into this moving box. We call this the ``box-current\" process. We generalize this current process to a distribution valued process. Scaling time by $n$ and space by $\\sqrt{n}$ gives current fluctuations of order $n^{d/4}$ where $d$ is the space dimension. The scaling limit of the normalized current process is a distribution valued Gaussian process with given covariance. The limiting current process is equal in distribution to the solution of a given stochastic partial differential equation which is related to the generalized Ornstein-Uhlenbeck process.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-09-14T18:53:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60F10"
    ],
    "keywords": [
      "independent random walks",
      "current fluctuations",
      "multiple dimensions",
      "current process",
      "stochastic partial differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.2732K"
    }
  }
}