{
  "abstract": "We consider the stochastic heat equation $\\partial_t u(t,x) = Δu(t,x) + \\dot{W}_α(t,x)$ on a bounded Lipschitz domain with zero Dirichlet boundary condition and zero initial condition, where $\\dot{W}_α$ is a Gaussian noise that is white in time and whose spatial covariance is the kernel of $(-Δ)^{-α}$ with $α>0$. We prove that a unique pointwise defined mild solution exists if and only if $α>d/2-1$. In this case, if in addition the domain is $C^2$, we also establish spatial and temporal Holder regularity of the solution. When $d/2-1<α<d/2$, we show that the Holder exponents are optimal and obtain exact local and uniform moduli of continuity, a Chung-type law of the iterated logarithm, and sharp small ball probability estimates for the solution.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08718",
  "author_total": 2,
  "authors": [
    {
      "name": "Jingwu Hu"
    },
    {
      "name": "Cheuk Yin Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fwtzt0vztfxp1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08718",
  "fulltext_url": null,
  "id": "fwtzt0vztfxp1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08718",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7212051798",
  "pdf_url": "https://arxiv.org/pdf/2609.08718",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fwtzt0vztfxp1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08718",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08718"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Sharp regularity and small ball probabilities for the stochastic heat equation on bounded domains",
  "title_text": "Sharp regularity and small ball probabilities for the stochastic heat equation on bounded domains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/fwtzt0vztfxp1",
  "venue": "arXiv",
  "versions": []
}