{
  "id": "math/0611292",
  "version": "v2",
  "published": "2006-11-09T20:31:53.000Z",
  "updated": "2009-08-26T08:32:25.000Z",
  "title": "Consistent families of Brownian motions and stochastic flows of kernels",
  "authors": [
    "Chris Howitt",
    "Jon Warren"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/08-AOP431 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2009, Vol. 37, No. 4, 1237-1272",
  "doi": "10.1214/08-AOP431",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the following mechanism for the random evolution of a distribution of mass on the integer lattice ${\\mathbf{Z}}$. At unit rate, independently for each site, the mass at the site is split into two parts by choosing a random proportion distributed according to some specified probability measure on $[0,1]$ and dividing the mass in that proportion. One part then moves to each of the two adjacent sites. This paper considers a continuous analogue of this evolution, which may be described by means of a stochastic flow of kernels, the theory of which was developed by Le Jan and Raimond. One of their results is that such a flow is characterized by specifying its $N$ point motions, which form a consistent family of Brownian motions. This means for each dimension $N$ we have a diffusion in ${\\mathbf{R}}^N$, whose $N$ coordinates are all Brownian motions. Any $M$ coordinates taken from the $N$-dimensional process are distributed as the $M$-dimensional process in the family. Moreover, in this setting, the only interactions between coordinates are local: when coordinates differ in value they evolve independently of each other. In this paper we explain how such multidimensional diffusions may be constructed and characterized via martingale problems.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-08-26T08:32:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J60",
      "60K35",
      "60K35"
    ],
    "keywords": [
      "brownian motions",
      "stochastic flow",
      "consistent family",
      "coordinates",
      "dimensional process"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....11292H"
    }
  }
}