{
  "abstract": "We study the Bayesian recovery of the initial state in a semilinear time-fractional subdiffusion equation from noisy random space-time point observations. A rescaled Gaussian prior based on a Whittle--Matérn process is assigned to the unknown initial condition. We prove that the solution has \\(\\dot H^3\\)-regularity when the nonlinearity satisfies a Lipschitz condition in the Sobolev norm. We then establish posterior contraction rates for the prediction error in the \\(L^2\\)-norm and for the parameter in Sobolev norms. The rates are polynomial in the sample size, with exponent depending on the prior smoothness and the spatial dimension. Moreover, we prove a minimax lower bound by constructing a wavelet-packing set and controlling the Kullback--Leibler divergences.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.26536",
  "author_total": 3,
  "authors": [
    {
      "name": "Haoyu Lu"
    },
    {
      "name": "Shaokang Zu"
    },
    {
      "name": "Junxiong Jia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e6sbmhkyv4kn6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.26536",
  "fulltext_url": null,
  "id": "e6sbmhkyv4kn6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.26536",
  "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"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2608.26536",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e6sbmhkyv4kn6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2608.26536",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.26536"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Posterior Consistency for Recovering Initial States in Nonlinear Subdiffusion Equations",
  "title_text": "Posterior Consistency for Recovering Initial States in Nonlinear Subdiffusion Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:46Z",
  "url": "https://research.draconitic.com/paper/e6sbmhkyv4kn6",
  "venue": "arXiv",
  "versions": []
}