{
  "id": "1809.07547",
  "version": "v1",
  "published": "2018-09-20T09:33:58.000Z",
  "updated": "2018-09-20T09:33:58.000Z",
  "title": "Quasi-stationarity for one-dimensional renormalized Brownian motion",
  "authors": [
    "William Oçafrain"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We are interested in the quasi-stationarity of the time-inhomogeneous Markov process X t = B t (t + 1) $\\kappa$ where (B t) t$\\ge$0 is a one-dimensional Brownian motion and $\\kappa$ $\\in$ (0, $\\infty$). We first show that the law of X t conditioned not to go out from (--1, 1) until the time t converges weakly towards the Dirac measure $\\delta$ 0 when $\\kappa$ > 1 2 as t goes to infinity. Then we show that this conditioned probability converges weakly towards the quasi-stationary distribution of an Ornstein-Uhlenbeck process when $\\kappa$ = 1 2. Finally, when $\\kappa$ < 1 2 , it is shown that the conditioned probability converges towards the quasi-stationary distribution of a Brownian motion. We also prove the existence of a Q-process and a quasi-ergodic distribution for $\\kappa$ = 1 2 and $\\kappa$ < 1 2 .",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-20T09:33:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "one-dimensional renormalized brownian motion",
      "quasi-stationarity",
      "conditioned probability converges",
      "quasi-stationary distribution",
      "one-dimensional brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}