{
  "id": "1911.06668",
  "version": "v1",
  "published": "2019-11-15T14:40:42.000Z",
  "updated": "2019-11-15T14:40:42.000Z",
  "title": "Statistics of first-passage Brownian functionals",
  "authors": [
    "Satya N. Majumdar",
    "Baruch Meerson"
  ],
  "comment": "22 pages, 5 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We study the distribution of first-passage functionals ${\\cal A}= \\int_0^{t_f} x^n(t)\\, dt$, where $x(t)$ is a Brownian motion (with or without drift) with diffusion constant $D$, starting at $x_0>0$, and $t_f$ is the first-passage time to the origin. In the driftless case, we compute exactly, for all $n>-2$, the probability density $P_n(A|x_0)=\\text{Prob}.(\\mathcal{A}=A)$. This probability density has an essential singular tail as $A\\to 0$ and a power-law tail $\\sim A^{-(n+3)/(n+2)}$ as $A\\to \\infty$. The former is reproduced by the optimal fluctuation method (OFM), which also predicts the optimal paths of the conditioned process for small $A$. For the case with a drift toward the origin, where no exact solution is known for general $n>-1$, the OFM predicts the distribution tails. For $A\\to 0$ it predicts the same essential singular tail as in the driftless case. For $A\\to \\infty$ it predicts a stretched exponential tail $-\\ln P_n(A|x_0)\\sim A^{-1/(n+1)}$ for all $n>0$. In the limit of large P\\'eclet number $\\text{Pe}= \\mu x_0/(2D)\\gg 1$, where $\\mu$ is the drift velocity, the OFM predicts a large-deviation scaling for all $A$: $-\\ln P_n(A|x_0)\\simeq\\text{Pe}\\, \\Phi_n\\left(z= A/\\bar{A}\\right)$, where $\\bar{A}=x_0^{n+1}/{\\mu(n+1)}$ is the mean value of $\\mathcal{A}$. We compute the rate function $\\Phi_n(z)$ analytically for all $n>-1$. For $n>0$ $\\Phi_n(z)$ is analytic for all $z$, but for $-1<n<0$ it is non-analytic at $z=1$, implying a dynamical phase transition. The order of this transition is $2$ for $-1/2<n<0$, while for $-1<n<-1/2$ the order of transition changes continuously with $n$. Finally, we apply the OFM to the case of $\\mu<0$ (drift away from the origin). We show that, when the process is conditioned on reaching the origin, the distribution of $\\mathcal{A}$ coincides with the distribution of $\\mathcal{A}$ for $\\mu>0$ with the same $|\\mu|$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-15T14:40:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first-passage brownian functionals",
      "essential singular tail",
      "probability density",
      "statistics",
      "distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}