{
  "id": "2001.04523",
  "version": "v1",
  "published": "2020-01-13T20:15:37.000Z",
  "updated": "2020-01-13T20:15:37.000Z",
  "title": "Quasi-Limiting Behavior of Drifted Brownian Motion",
  "authors": [
    "SangJoon Lee",
    "Iddo Ben-Ari"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "A Quasi-Stationary Distribution (QSD)for a Markov process with an almost surely hit absorbing state is a time-invariant initial distribution for the process conditioned on not being absorbed by any given time. An initial distribution for the process is in the domain of attraction of some QSD $\\nu$ if the distribution of the process a time $t$, conditioned not to be absorbed by time $t$ converges to $\\nu$. In this work study mostly Brownian motion with constant drift on the half line $[0,\\infty)$ absorbed at $0$. Previous work by Martinez et al. identifies all QSDs and provides a nearly complete characterization for their domain of attraction. Specifically, it was shown that if the distribution a well-defined exponential tail (including the case of lighter than any exponential tail), then it is in the domain of attraction of a QSD determined by the exponent. In this work we 1. Obtain a new approach to existing results, explaining the direct relation between a QSD and an initial distribution in its domain of attraction. 2. Study the behavior under a wide class of initial distributions whose tail is heavier than exponential, and obtain no-trivial limits under appropriate scaling.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-13T20:15:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "drifted brownian motion",
      "quasi-limiting behavior",
      "exponential tail",
      "attraction",
      "time-invariant initial distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}