{
  "abstract": "Consider the one-dimensional nonlinear stochastic heat equation $$ \\frac{\\partial u(t,x)}{\\partial t} = \\frac{\\partial^2 u(t,x)}{\\partial x^2} + σ(u(t,x))\\dot W(t,x), \\qquad t>0,\\ x\\in\\mathbb R, $$ where $\\dot W$ is a Gaussian noise that is white in time and fractional in space with Hurst parameter $H\\in(1/4,1/2)$. Hu et al. ({\\it Ann. Probab.} {\\bf 45} (2017), 4561--4616) established existence and uniqueness of the solution under the assumptions that $σ$ is differentiable, both $σ$ and $σ'$ are Lipschitz continuous, and $σ(0)=0$. Hu and Wang ({\\it Ann. Inst. Henri Poincaré Probab. Stat.} {\\bf 58} (2022), 379--423) subsequently removed the technical condition $σ(0)=0$ by developing the well-posedness theory in a suitable weighted space associated with the weight $λ(x)= c_H (1 + |x|^2)^{H-1}$. In this weighted framework, which allows $σ(0)\\ne0$, we establish temporal and spatial local linearizations: small increments of the nonlinear solution are approximated by the corresponding increments of the solution to the additive equation, multiplied by the local coefficient $σ(u(t,x))$. The main difficulty lies in handling the nonlocal fractional-difference structure induced by the rough spatial noise. We address this difficulty using weighted heat-kernel increment estimates. These approximations yield exact Lévy-type uniform moduli of continuity in both time and space, together with Khinchin and Chung laws of the iterated logarithm.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19535",
  "author_total": 4,
  "authors": [
    {
      "name": "Bin Qian"
    },
    {
      "name": "Mengxin Tong"
    },
    {
      "name": "Ran Wang"
    },
    {
      "name": "Yimin Xiao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0k28qx263yb33.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19535",
  "fulltext_url": null,
  "id": "0k28qx263yb33",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19535",
  "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",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7222039219",
  "pdf_url": "https://arxiv.org/pdf/2610.19535",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0k28qx263yb33.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19535",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19535"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Local Linearization and Exact Oscillations for the Nonlinear Stochastic Heat Equation with Spatially Rough Noise",
  "title_text": "Local Linearization and Exact Oscillations for the Nonlinear Stochastic Heat Equation with Spatially Rough Noise",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/0k28qx263yb33",
  "venue": "arXiv",
  "versions": []
}