{
  "abstract": "We study Brownian motion killed upon exiting arbitrary open sets $Ω\\subset\\mathbb R^n$, $n\\in[3,\\infty)\\cap\\mathbb N$. Let $α\\in(0,1]$. The infimal radius $d_α(x)$ at which the absorber near $x$ carries an $α$-fraction of the Newtonian capacity of the ball of the same radius provides a local measure of electrostatic trapping. We prove that this capacitary loading forces an absorption probability of at least $c_nα$ within a dimension-dependent multiple of the diffusive time scale $d_α(x)^2$. Independently, if $p\\in (0,1)$ and $τ_p(x)$ is the time by which Brownian motion started at $x$ has been absorbed with probability at least $p$, then \\begin{align*} \\int_Ω\\frac{|u\\left(x\\right)|^2}{τ_p\\left(x\\right)}\\,\\mathrm d x \\le\\frac{2}{p^2}\\int_Ω|\\nabla u\\left(x\\right)|^2\\,\\mathrm d x \\end{align*} for every $u\\in C_{\\mathrm{c}}^\\infty(Ω)$. Together these statements give a capacitary Hardy inequality with constant $C_nα^{-2}$ on every open set, answering a problem of Maz'ya, where $C_n$ is a positive constant and, moreover, the exponent of $α^{-2}$ is sharp. As applications, we derive corresponding estimates for survival probabilities, a corresponding spectral lower bound (including a recovery of the Maz'ya--Shubin lower bound), heat dissipation, Schrödinger forms, and Fisher information.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.26663",
  "author_total": 5,
  "authors": [
    {
      "name": "Yiqun Chen"
    },
    {
      "name": "Jie Xiao"
    },
    {
      "name": "Dachun Yang"
    },
    {
      "name": "Wen Yuan"
    },
    {
      "name": "Yangyang Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6fs6yh52bad3h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.26663",
  "fulltext_url": null,
  "id": "6fs6yh52bad3h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.26663",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7204560280",
  "pdf_url": "https://arxiv.org/pdf/2608.26663",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6fs6yh52bad3h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2608.26663",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.26663"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Capacitary Trapping and Brownian Escape: A Sharp Hardy Principle for Absorbing Media",
  "title_text": "Capacitary Trapping and Brownian Escape: A Sharp Hardy Principle for Absorbing Media",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/6fs6yh52bad3h",
  "venue": "arXiv",
  "versions": []
}