{
  "abstract": "Let $P$ denote normalized nearest-neighbor averaging on $\\mathbb{Z}^d$ and set $A=I-P$. We consider $\\partial_tu+A^su=f(u),t>0, x\\in\\mathbb{Z}^d, 0<s<1.$ For a Fisher-KPP reaction with $a=f'(0)>0$, compactly supported data have logarithmic propagation rate $a/(d+2s)$. In the logistic case, each fixed level set lies between two constant multiples of $\\exp(at/(d+2s))$; this localization remains valid for data with the critical algebraic tail. Nondecreasing front-like data in one dimension instead propagate at rate $a/(2s)$, and the resulting acceleration rules out planar traveling fronts of finite speed in every dimension. We also analyze the singular limit $s\\uparrow1$. If $τ_s=-\\log(1-s)$, then throughout $at<τ_s$ compactly supported solutions follow the Wulff shape of the local nearest-neighbor equation. After a constant multiple of $τ_s$, their propagation is exponential. Half-space data exhibit the corresponding transition from the directional local speed to the exponent $a/(2s)$. The analysis rests on sharp bounds for the subordinated kernel, algebraic barriers, a compactness--Liouville argument, and a decomposition of the fractional walk into its nearest-neighbor and rare long-jump parts.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05124",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuanyang Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fgfmwpjfwk4f0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05124",
  "fulltext_url": null,
  "id": "fgfmwpjfwk4f0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05124",
  "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.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7211915682",
  "pdf_url": "https://arxiv.org/pdf/2609.05124",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fgfmwpjfwk4f0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.05124",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05124"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Propagation and the Local-to-Nonlocal Transition for Fractional Fisher-KPP Equations on Integer Lattices",
  "title_text": "Sharp Propagation and the Local-to-Nonlocal Transition for Fractional Fisher-KPP Equations on Integer Lattices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/fgfmwpjfwk4f0",
  "venue": "arXiv",
  "versions": []
}