{
  "id": "2501.14255",
  "version": "v1",
  "published": "2025-01-24T05:39:26.000Z",
  "updated": "2025-01-24T05:39:26.000Z",
  "title": "Hitting probabilities, thermal capacity, and Hausdorff dimension results for the Brownian sheet",
  "authors": [
    "Cheuk Yin Lee",
    "Yimin Xiao"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $W= \\{W(t): t \\in \\mathbb{R}_+^N \\}$ be an $(N, d)$-Brownian sheet and let $E \\subset (0, \\infty)^N$ and $F \\subset \\mathbb{R}^d$ be compact sets. We prove a necessary and sufficient condition for $W(E)$ to intersect $F$ with positive probability and determine the essential supremum of the Hausdorff dimension of the intersection set $W(E)\\cap F$ in terms of the thermal capacity of $E \\times F$. This extends the previous results of Khoshnevisan and Xiao (2015) for the Brownian motion and Khoshnevisan and Shi (1999) for the Brownian sheet in the special case when $E \\subset (0, \\infty)^N$ is an interval.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-24T05:39:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "brownian sheet",
      "hausdorff dimension results",
      "thermal capacity",
      "hitting probabilities",
      "probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}