{
  "id": "1910.07626",
  "version": "v1",
  "published": "2019-10-16T21:37:44.000Z",
  "updated": "2019-10-16T21:37:44.000Z",
  "title": "Diffusions on a space of interval partitions: Poisson-Dirichlet stationary distributions",
  "authors": [
    "Noah Forman",
    "Soumik Pal",
    "Douglas Rizzolo",
    "Matthias Winkel"
  ],
  "comment": "50 pages, 8 figures. Following arXiv:1909.02584 [math.PR], this is the second in a sequence of three papers that will collectively supersede arXiv:1609.06706 [math.PR]",
  "categories": [
    "math.PR"
  ],
  "abstract": "We introduce diffusions on a space of interval partitions of the unit interval that are stationary with the Poisson-Dirichlet laws with parameters $(\\alpha,0)$ and $(\\alpha,\\alpha)$. The construction has two steps. The first is a general construction of interval partition processes obtained previously, by decorating the jumps of a L\\'evy process with independent excursions. Here, we focus on the second step, which requires explicit transition kernels and what we call pseudo-stationarity. This allows us to study processes obtained from the original construction via scaling and time-change. In a sequel paper, we establish connections to diffusions on decreasing sequences introduced by Ethier and Kurtz (1981) and Petrov (2009). The latter diffusions are continuum limits of up-down Markov chains on Chinese restaurant processes. Our construction is also a step towards resolving longstanding conjectures by Feng and Sun on measure-valued Poisson-Dirichlet diffusions, and by Aldous on a continuum-tree-valued diffusion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-16T21:37:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J25",
      "60J60",
      "60J80",
      "60G18",
      "60G52",
      "60G55"
    ],
    "keywords": [
      "poisson-dirichlet stationary distributions",
      "explicit transition kernels",
      "interval partition processes",
      "up-down markov chains",
      "chinese restaurant processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}