{
  "id": "2205.02063",
  "version": "v1",
  "published": "2022-05-04T13:53:29.000Z",
  "updated": "2022-05-04T13:53:29.000Z",
  "title": "Diffusive search with resetting via the Brownian bridge",
  "authors": [
    "Ross G. Pinsky"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Fix $D>0$. For $r>0$, let $X^{(r)}(\\cdot)$ be a Brownian motion with diffusion coefficient $D$, equipped with an exponential clock with rate $r$, so that when the clock rings the process jumps to the origin, where it resets and begins anew. Denote expectations with respect to this process by $E_0^{(r)}$. This process, by now rather well-studied, is called Brownian motion with resetting. For $T>0$, consider also a process $X^{\\text{bb};T}(\\cdot)$ that performs a Brownian bridge with diffusion coefficient $D$ and bridge interval $T$, and then at time $T$ resets and starts anew from the origin. Denote expectations with respect to this process by $E_0^{\\text{bb};T}$. The two resetting processes, one with jumps and the other continuous, search for a random target $a\\in\\mathbb{R}$ that has a known distribution $\\mu$. Letting $\\tau_a$ denote the hitting time of $a$, the expected time to locate the target is $\\int_{\\mathbb{R}}\\big(E_0^{(r)}\\tau_a\\big)\\mu(da)$ for the first process and $\\int_{\\mathbb{R}}\\big(E_0^{\\text{bb};T}\\tau_a\\big)\\mu(da)$ for the second process. It is known that $E_0^{(r)}\\tau_a=\\frac{e^{\\sqrt{\\frac{2r}D}\\thinspace |a|}-1}r$. We calculate $E_0^{\\text{bb};T}\\tau_a$. Then we calculate $\\int_{\\mathbb{R}}\\big(E_0^{(r)}\\tau_a\\big)\\mu(da)$ and $\\int_{\\mathbb{R}}\\big(E_0^{\\text{bb};T}\\tau_a\\big)\\mu(da)$ in the case that $\\mu$ is a centered Gaussian distribution with variance $\\sigma^2$, which is the mean-squared distance of the target from the origin. Finally, we compare $\\inf_{r>0}\\int_{\\mathbb{R}}\\big(E_0^{(r)}\\tau_a\\big)\\mu(da)$ to $\\inf_{T>0}\\int_{\\mathbb{R}}\\big(E_0^{\\text{bb};T}\\tau_a\\big)\\mu(da)$, obtaining $\\inf_{r>0}\\int_{\\mathbb{R}}\\big(E_0^{(r)}\\tau_a\\big)\\mu(da)\\approx3.548\\frac{\\sigma^2}D$ and $\\inf_{T>0}\\int_{\\mathbb{R}}\\big(E_0^{\\text{bb};T}\\tau_a\\big)\\mu(da)\\approx4.847\\frac{\\sigma^2}D$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-04T13:53:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J60"
    ],
    "keywords": [
      "brownian bridge",
      "diffusive search",
      "diffusion coefficient",
      "denote expectations",
      "brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}