{
  "id": "2101.11895",
  "version": "v1",
  "published": "2021-01-28T09:56:40.000Z",
  "updated": "2021-01-28T09:56:40.000Z",
  "title": "Survival probability of a run-and-tumble particle in the presence of a drift",
  "authors": [
    "Benjamin De Bruyne",
    "Satya N. Majumdar",
    "Gregory Schehr"
  ],
  "comment": "58 pages, 14 figures",
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.soft",
    "math.PR"
  ],
  "abstract": "We consider a one-dimensional run-and-tumble particle, or persistent random walk, in the presence of an absorbing boundary located at the origin. After each tumbling event, which occurs at a constant rate $\\gamma$, the (new) velocity of the particle is drawn randomly from a distribution $W(v)$. We study the survival probability $S(x,t)$ of a particle starting from $x \\geq 0$ up to time $t$ and obtain an explicit expression for its double Laplace transform (with respect to both $x$ and $t$) for an arbitrary velocity distribution $W(v)$, not necessarily symmetric. This result is obtained as a consequence of Spitzer's formula, which is well known in the theory of random walks and can be viewed as a generalization of the Sparre Andersen theorem. We then apply this general result to the specific case of a two-state particle with velocity $\\pm v_0$, the so-called persistent random walk (PRW), and in the presence of a constant drift $\\mu$ and obtain an explicit expression for $S(x,t)$, for which we present more detailed results. Depending on the drift $\\mu$, we find a rich variety of behaviours for $S(x,t)$, leading to three distinct cases: (i) subcritical drift $-v_0\\!<\\!\\mu\\!<\\! v_0$, (ii) supercritical drift $\\mu < -v_0$ and (iii) critical drift $\\mu=-v_0$. In these three cases, we obtain exact analytical expressions for the survival probability $S(x,t)$ and establish connections with existing formulae in the mathematics literature. Finally, we discuss some applications of these results to record statistics and to the statistics of last-passage times.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-28T09:56:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "survival probability",
      "run-and-tumble particle",
      "persistent random walk",
      "explicit expression",
      "sparre andersen theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}