{
  "id": "2306.10922",
  "version": "v1",
  "published": "2023-06-19T13:33:27.000Z",
  "updated": "2023-06-19T13:33:27.000Z",
  "title": "Fractional Brownian motion with deterministic drift: How critical is drift regularity in hitting probabilities",
  "authors": [
    "Mohamed Erraoui",
    "Youssef Hakiki"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $B^{H}$ be a $d$-dimensional fractional Brownian motion with Hurst index $H\\in(0,1)$, $f:[0,1]\\longrightarrow\\mathbb{R}^{d}$ a Borel function, and $E\\subset[0,1]$, $F\\subset\\mathbb{R}^{d}$ are given Borel sets. The focus of this paper is on hitting probabilities of the non-centered Gaussian process $B^{H}+f$. It aims to highlight how each component $f$, $E$ and $F$ is involved in determining the upper and lower bounds of $\\mathbb{P}\\{(B^H+f)(E)\\cap F\\neq \\emptyset \\}$. When $F$ is a singleton and $f$ is a general measurable drift, some new estimates are obtained for the last probability by means of suitables Hausdorff measure and capacity of the graph $Gr_E(f)$. As application we deal with the issue of polarity of points for $(B^H+f)\\vert_E$ (the restriction of $B^H+f$ to the subset $E\\subset (0,\\infty)$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-19T13:33:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J45",
      "60G17",
      "28A78"
    ],
    "keywords": [
      "probability",
      "hitting probabilities",
      "deterministic drift",
      "drift regularity",
      "dimensional fractional brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}