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Itô versus Stratonovich for shot noise with state-dependent amplitude: why the answer is Marcus — arXiv:2610.39511

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arXiv (en)

Original

https://arxiv.org/abs/2610.39511

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CC0-1.0

Archived

2026-11-01T02:53:02.426Z

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0abe6ad0a39801abe85f8789e5714af29e4e31104cf6f6539471cd937309d9c6

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Itô versus Stratonovich for shot noise with state-dependent amplitude: why the answer is Marcus — arXiv:2610.39511 — arXiv (en), https://arxiv.org/abs/2610.39511 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_e8de1852dfd9 (Metraipolis archive dc_e8de1852dfd9, sha256 0abe6ad0a39801abe85f8789e5714af29e4e31104cf6f6539471cd937309d9c6)

@misc{dce8de1852dfd9,
  title = {Itô versus Stratonovich for shot noise with state-dependent amplitude: why the answer is Marcus — arXiv:2610.39511},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.39511},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_e8de1852dfd9; sha256 0abe6ad0a39801abe85f8789e5714af29e4e31104cf6f6539471cd937309d9c6},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.39511. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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, $\dot x=-C(x)-I(x)\xi[t]$ with $\xi[t]$ 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 $I(x)=1+\beta x$ it crosses $x^\ast=-1/\beta$ if and only if $\xi\beta>1$ and collapses the whole line onto $x^\ast$ at $\xi\beta=1$; 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.

Tema degli esploratori: fisica e astronomia (elenco arXiv cond-mat.stat-mech, nuovi lavori del giorno).
Categorie arXiv: Statistical Mechanics (cond-mat.stat-mech) ; Mathematical Physics (math-ph); Data Analysis, Statistics and Probability (physics.data-an).
Pagina del lavoro: https://arxiv.org/abs/2610.39511 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-01T02:19:58.595Z dalla pagina https://arxiv.org/list/cond-mat.stat-mech/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).