{
  "abstract": "We establish the Varadhan small-time asymptotics for the quenched and annealed heat kernels of one-dimensional diffusions in a random environment with generator $\\mathcal L_W f(x)=e^{-ρ(x,W)}(e^{a(x,W)}f'(x))'$. The coefficients $a$ and $ρ$ are continuous in space and satisfy a local exponential moment condition. We assume that the law of the intrinsic coordinate map $Λ_W$ has compact support $\\mathscr L$ consisting of strictly increasing functions. Let $q^W(t,x,y)$ and $q(t,x,y)=\\mathbb E[q^W(t,x,y)]$ denote the quenched and annealed heat kernels with respect to Lebesgue measure, respectively. We prove that, for almost every environment $W$, $\\lim_{t\\downarrow0}t\\log q^W(t,x,y)=-\\frac12|Λ_W(y)-Λ_W(x)|^2$, and that $\\lim_{t\\downarrow0}t\\log q(t,x,y)=-\\frac12\\min_{Λ\\in\\mathscr L}|Λ(y)-Λ(x)|^2$. Both limits hold uniformly on compact subsets of $\\mathbb R^2$. The framework includes Brox diffusion, formally described by $dX_t=dB_t-\\frac12\\dot W(X_t)\\,dt$, where $B$ is a standard Brownian motion and $W$ is an independent two-sided Brownian motion.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21345",
  "author_total": 3,
  "authors": [
    {
      "name": "Yiduo Wang"
    },
    {
      "name": "Saisai Yang"
    },
    {
      "name": "Tusheng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/frr9edt8zs9dc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21345",
  "fulltext_url": null,
  "id": "frr9edt8zs9dc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21345",
  "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.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7213900364",
  "pdf_url": "https://arxiv.org/pdf/2609.21345",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/frr9edt8zs9dc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21345",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21345"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Small-time asymptotics of heat kernels of one-dimensional diffusions in a random environment",
  "title_text": "Small-time asymptotics of heat kernels of one-dimensional diffusions in a random environment",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/frr9edt8zs9dc",
  "venue": "arXiv",
  "versions": []
}