{
  "abstract": "We study the stochastic delay equation \\begin{equation*} dX(t)=[-γX(t)+r f(X(t-τ))]\\,dt+σX(t)\\,dW(t), \\end{equation*} where $γ,r,τ\\in(0,\\infty)$, $σ\\in[0,\\infty)$, and with $f$ either globally Lipschitz or globally bounded and locally Lipschitz, with $f(0)=0$ and $f(x)\\ge0$ for all $x>0$. Under the condition \\begin{equation*} r\\liminf_{x\\to 0^{+}}\\frac{f(x)}{x}>γ+\\frac{1}{2}σ^2, \\end{equation*} independent of the time delay $τ,$ we obtain that every solution $X$ with nonnegative initial history and $X(0)>0$ is bounded away from zero in probability. This result then confirms a recent conjecture of van den Bosch, van Gaans, and Verduyn Lunel, and yields a non-trivial invariant measure for, e.g., the Mackey--Glass or Nicholson's blowflies equation with multiplicative noise. To get to this result, we apply a transformation to the solution process, essentially rewriting the equation using a stochastic integrating factor. A comparison with a linear equation gives contraction estimates for negative conditional moments and hence boundedness away from zero in probability. The same argument yields persistence whenever the linear comparison equation has a positive first Lyapunov exponent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21599",
  "author_total": 2,
  "authors": [
    {
      "name": "Mark van den Bosch"
    },
    {
      "name": "John Ioannis Stavroulakis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d38syz12bp7ye.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21599",
  "fulltext_url": null,
  "id": "d38syz12bp7ye",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21599",
  "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.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.21599",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d38syz12bp7ye.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21599",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21599"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Persistence in probability for stochastic delay equations with multiplicative noise",
  "title_text": "Persistence in probability for stochastic delay equations with multiplicative noise",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/d38syz12bp7ye",
  "venue": "arXiv",
  "versions": []
}