{
  "abstract": "We investigate diffusion processes on disconnected fractal sets in one-sided Brownian environments. On the real line, it is established that the process exhibits either diffusion or trapping, each occurring with probability $1/2$. In this study, we demonstrate that for fractal sets, this behavior is governed by the geometric parameters $r$ (the reciprocal of the similitude ratio) and $N$ (the number of contraction mappings), which define the fractal structure. Two distinct regimes emerge: a diffusive regime on the environment-free side and a localization regime on the side influenced by the environment. The transition between these regimes is determined by whether the random environment first hits the threshold $\\log r$ or $-\\log N$. Consequently, the probability of diffusion versus trapping is explicitly characterized by the Hausdorff dimension $d_f = \\log N / \\log r$. This result demonstrates that fractal geometry scales the limiting distributions and dictates the stochastic regime.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08195",
  "author_total": 1,
  "authors": [
    {
      "name": "Hiroshi Takahashi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/93x0nd7z6rej5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08195",
  "fulltext_url": null,
  "id": "93x0nd7z6rej5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08195",
  "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": "W7212039287",
  "pdf_url": "https://arxiv.org/pdf/2609.08195",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/93x0nd7z6rej5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08195",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08195"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Geometric influence on the limiting behavior of diffusion processes in one-sided Brownian environments on disconnected fractal sets",
  "title_text": "Geometric influence on the limiting behavior of diffusion processes in one-sided Brownian environments on disconnected fractal sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/93x0nd7z6rej5",
  "venue": "arXiv",
  "versions": []
}