{
  "id": "1406.1742",
  "version": "v2",
  "published": "2014-06-06T17:08:46.000Z",
  "updated": "2014-12-25T10:26:46.000Z",
  "title": "Sharp asymptotics for the quasi-stationary distribution of birth-and-death processes",
  "authors": [
    "J. -R. Chazottes",
    "P. Collet",
    "S. Méléard"
  ],
  "comment": "48 pages, corrected typos, more details. To appear in Probab. Th. & Rel. Fields (2015)",
  "categories": [
    "math.PR",
    "q-bio.PE"
  ],
  "abstract": "We study a general class of birth-and-death processes with state space $\\mathbb{N}$ that describes the size of a population going to extinction with probability one. This class contains the logistic case. The scale of the population is measured in terms of a `carrying capacity' $K$. When $K$ is large, the process is expected to stay close to its deterministic equilibrium during a long time but ultimately goes extinct. Our aim is to quantify the behavior of the process and the mean time to extinction in the quasi-stationary distribution as a function of $K$, for large $K$. We also give a quantitative description of this quasi-stationary distribution. It turns out to be close to a Gaussian distribution centered about the deterministic long-time equilibrium, when $K$ is large. Our analysis relies on precise estimates of the maximal eigenvalue, of the corresponding eigenvector and of the spectral gap of a self-adjoint operator associated with the semigroup of the process.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-06T17:08:46.000Z",
      "comment": "44 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-25T10:26:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quasi-stationary distribution",
      "birth-and-death processes",
      "sharp asymptotics",
      "deterministic long-time equilibrium",
      "maximal eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.1742C"
    }
  }
}