{
  "abstract": "For Gaussian white noise the Itô and Stratonovich prescriptions differ by $O(dt)$ per time step, which accumulates into the familiar spurious drift $\\tfrac12 D\\,II'$. We show that for renewal shot noise with state-dependent gain, $\\dot x=-C(x)-I(x)ξ[t]$ with $ξ[t]$ a train of Dirac spikes of finite random amplitude, the two prescriptions differ by a finite amount at every event, and that where $I$ vanishes the difference can become qualitative. The Marcus kick, the jump-process counterpart of the Stratonovich prescription, is the flow map of $dx/ds=I(x)$: no kick can cross a zero $x^\\ast$ of $I$, for any jump and waiting-time statistics. The Itô kick is only the tangent map of that flow. For $I(x)=1+βx$ it crosses $x^\\ast=-1/β$ if and only if $ξβ>1$ and collapses the whole line onto $x^\\ast$ at $ξβ=1$; below threshold it is itself a Marcus kick with a remapped jump law. Bounded jump laws thus have a threshold, while unbounded ones leak across the barrier for \\emph{any} parameters. The discrepancy cannot be absorbed into a renormalised drift beyond lowest order, and only the Marcus kick is generated by a single operator, which makes the master equation of Bianucci, Bologna and Mannella (2026) intrinsically Marcus. Numerically, the standard Heun kick is a second-order truncation of the Marcus map that preserves the barrier only because its order is even; it shortens every jump, biasing second moments in proportion to the squared jump amplitude and the jump kurtosis, and depleting the region near the barrier. We give the exact-flow integrator that replaces it.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39511",
  "author_total": 2,
  "authors": [
    {
      "name": "Marco Bianucci"
    },
    {
      "name": "Riccardo Mannella"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/17k1nwecwng0w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39511",
  "fulltext_url": null,
  "id": "17k1nwecwng0w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39511",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "physics.data-an"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39511",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/17k1nwecwng0w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:23Z",
      "native_id": "2610.39511",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39511"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.data-an",
      "subfield": "computational"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Itô versus Stratonovich for shot noise with state-dependent amplitude: why the answer is Marcus",
  "title_text": "Itô versus Stratonovich for shot noise with state-dependent amplitude: why the answer is Marcus",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/17k1nwecwng0w",
  "venue": "arXiv",
  "versions": []
}