{
  "id": "2307.16886",
  "version": "v1",
  "published": "2023-07-31T17:52:15.000Z",
  "updated": "2023-07-31T17:52:15.000Z",
  "title": "Irregularity scales for Gaussian processes: Hausdorff dimensions and hitting probabilities",
  "authors": [
    "Youssef Hakiki",
    "Frederi Viens"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X$ be a $d$-dimensional Gaussian process in $[0,1]$, where the component are independent copies of a scalar Gaussian process $X_0$ on $[0,1]$ with a given general variance function $\\gamma^2(r)=\\operatorname{Var}\\left(X_0(r)\\right)$ and a canonical metric $\\delta(t,s):=(\\mathbb{E}\\left(X_0(t)-X_0(s)\\right)^2)^{1/2}$ which is commensurate with $\\gamma(t-s)$. Under a weak regularity condition on $\\gamma$, referred to below as $\\mathbf{(C_{0+})}$, which allows $\\gamma$ to be far from H\\\"older-continuous, we prove that for any Borel set $E\\subset [0,1]$, the Hausdorff dimension of the image $X(E)$ and of the graph $Gr_E(X)$ are constant almost surely. Furthermore, we show that these constants can be explicitly expressed in terms of $\\dim_{\\delta}(E)$ and $d$. However, when $\\mathbf{(C_{0+})}$ is not satisfied, the classical methods may yield different upper and lower bounds for the underlying Hausdorff dimensions. This case is illustrated via a class of highly irregular processes known as logBm. Even in such cases, we employ a new method to establish that the Hausdorff dimensions of $X(E)$ and $Gr_E(X)$ are almost surely constant. The method uses the Karhunen-Lo\\`eve expansion of $X$ to prove that these Hausdorff dimensions are measurable with respect to the expansion's tail sigma-field. Under similarly mild conditions on $\\gamma$, we derive upper and lower bounds on the probability that the process $X$ can reach the Borel set $F$ in $\\mathbb{R}^d$ from the Borel set $E$ in $[0,1]$. These bounds are obtained by considering the Hausdorff measure and the Bessel-Riesz capacity of $E\\times F$ in an appropriate metric $\\rho_{\\delta}$ on the product space, relative to appropriate orders. Moreover, we demonstrate that the dimension $d$ plays a critical role in determining whether $X\\lvert_E$ hits $F$ or not.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-31T17:52:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J45",
      "60G17",
      "28A78",
      "60G15"
    ],
    "keywords": [
      "hausdorff dimension",
      "probability",
      "irregularity scales",
      "hitting probabilities",
      "borel set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}