{ "schemaVersion": "1.6", "slug": "spectrally-negative-resetting-extremes", "title": "Fixed-horizon extremes of spectrally negative Lévy processes with Poisson resetting", "shortTitle": "After a high peak: near the top, or back home", "url": "https://evidencepress.org/releases/spectrally-negative-resetting-extremes/", "oneLine": "For a stationary spectrally negative Lévy process with Poisson resetting, a high maximum in a fixed window leaves the endpoint either just below the peak (no reset yet) or at an ordinary level (reset since), with explicit odds, an explicit total-variation error and the constant left open in Remark 3 of Dębicki, Hashorva and Michna (2026).", "abstract": "A process that wanders randomly but is sent back to a home position at random times is a basic model in search, foraging, queues with catastrophes and risk theory. This candidate takes the process to be spectrally negative Lévy (no upward jumps), resets it at the times of a Poisson clock and starts it in its long-run state, then asks where it is at the end of a fixed window given that its maximum over the window was very high. Up to an error smaller than any exponential order, there are only two answers: either no reset has happened and the process ends close to its peak, or a reset has happened and it ends at an ordinary level. The paper gives the odds of the two answers, the full endpoint law in each case, an explicit total-variation error bound uniform over all endpoint sets (including sets that move with the threshold), and a construction of the whole conditional path, in which the overshoot of the peak is exponential and independent of everything else. The leading constants come from a standard first-cycle computation; the uniform error bound, the conditional path law and its quantitative corollaries are the new content. For Brownian motion the result supplies the constant left unspecified in Remark 3 of Dębicki, Hashorva and Michna (2026) and proves that their two bounds are strict. The theorem is exact for bounded-variation inputs above an explicit threshold and extends to random reset positions. Analytic values and an exact-sampling Monte Carlo of the original process agree with the formulas. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-10-27", "dateModified": "2026-10-27", "version": "1.0.0-candidate", "doi": "10.5281/zenodo.23689354", "doiUrl": "https://doi.org/10.5281/zenodo.23689354", "conceptDoi": "10.5281/zenodo.23689353", "pdfUrl": "https://github.com/ipitchford/spectrally-negative-resetting-extremes/releases/download/v1.0.0-candidate/spectrally-negative-resetting-extremes-1.0.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23689354", "repoUrl": "https://github.com/ipitchford/spectrally-negative-resetting-extremes", "releaseUrl": "https://github.com/ipitchford/spectrally-negative-resetting-extremes/releases/tag/v1.0.0-candidate", "markdownUrl": "https://evidencepress.org/releases/spectrally-negative-resetting-extremes/index.md", "bibtexUrl": "https://evidencepress.org/releases/spectrally-negative-resetting-extremes/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/spectrally-negative-resetting-extremes.mp3", "imageUrl": "https://evidencepress.org/assets/og/spectrally-negative-resetting-extremes.png?v=e69ce1ea5d", "coverArtUrl": "https://evidencepress.org/assets/art/spectrally-negative-resetting-extremes.svg?v=4c9eb28f91", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/spectrally-negative-resetting-extremes.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/spectrally-negative-resetting-extremes.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematical candidate. Every theorem has a written proof; numerical checks are floating-point or Monte Carlo, not enclosures. Internal AI-assisted review, two AI-assisted proof audits and a verification re-review only. External specialist review, independent reproduction and formal verification have not been established." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23689354", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files by full download; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "code/reproduce.py reran every check from a clean copy of the final package (reduced Monte Carlo size); deterministic outputs agree with the committed copies to 1e-9 and the frozen archive's scripts and outputs are byte-identical to that copy." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "partial", "note": "The Brownian constant is checked by a closed form and independent quadrature, and the probability statements by an exact-sampling simulation of the original process adapted from a reviewer-written sampler; this is implementation diversity inside the producer workflow, not external reproduction." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal review and proof audits are not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Checks need Python 3.12 with pinned NumPy, SciPy and Matplotlib (requirements.txt); not containerised. negative_controls.py needs only the standard library." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical research", "AI-assisted internal review, proof audits (including one by a different model family) and verification re-review", "Claude Code: final editorial fixes, publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted review and proof audits, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the final manuscript (27 October 2026, 19 pages), its check scripts and stored outputs, and the frozen release archive. The versioned GitHub and Zenodo research archives were checked against the frozen release assets on 27 October 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "stochastic resetting", "spectrally negative Lévy processes", "extremes", "running maximum", "total variation", "Wiener–Hopf factorisation", "Esscher transform", "Brownian motion" ], "keyResults": [ "Two-profile endpoint theorem (Theorem 3.1, proof): ‖P(H_T>u, Y_T∈·) − e^{−β(u−r)}{Q_T(·−u)+K_T}‖_var ≤ (aβ/(γ−β)+λT)e^{Tψ(γ)−γ(u−r)} for every γ>β, uniformly over all Borel endpoint sets, including sets that move with u; the error is smaller than any exponential order.", "Conditional path (Theorem 5.1, Propositions 5.2–5.3, proof): given a high maximum, the first reset is absent with probability A(T)/D_T (0.531 in the standard Brownian case); the overshoot is Exp(β), independent of the marks, and its mean tends to 1/β for every input.", "The constant in Remark 3 of Dębicki–Hashorva–Michna (Corollary 6.1, proof): C_T = e^{βr}[Φ(√(2λT)) + λ∫₀ᵀΦ(√(2λs))p_{T−s}(0)ds]; both of their bounds are strict; C_1 = 1.32858 for λ = T = 1, r = 0.", "Exactness and extensions (Corollary 7.2, Theorem 7.3, proof): the decomposition is exact for bounded-variation inputs once u − r ≥ dT; for random reset positions the leading term gains the factor E e^{βR}, the initial reset position is exponentially tilted, and the first-reset odds are unchanged." ], "reviews": [], "evidencePackage": "Written proofs of every theorem in a 19-page manuscript, line-checked by two AI-assisted proof audits whose gaps were repaired; closed form against independent quadrature for the Brownian constant; floating-point reflection-principle brackets; an exact-sampling Monte Carlo of the original reset process (4×10⁷ Brownian paths per experiment, no time discretisation, plus a bounded-variation replicate); semantic negative controls; a clean replay of every script.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Dębicki, Hashorva and Michna (2026), Distributional and extremal behaviour of Brownian motion with exponential resetting. Methodology and Computing in Applied Probability 28(3), article 79 (source of the question: Remark 3).", "url": "https://doi.org/10.1007/s11009-026-10316-4", "doi": "10.1007/s11009-026-10316-4" }, { "citation": "Evans and Majumdar (2011), Diffusion with stochastic resetting. Physical Review Letters 106, 160601.", "url": "https://doi.org/10.1103/PhysRevLett.106.160601", "doi": "10.1103/PhysRevLett.106.160601" }, { "citation": "Evans, Majumdar and Schehr (2020), Stochastic resetting and applications. Journal of Physics A 53, 193001 (review).", "url": "https://doi.org/10.1088/1751-8121/ab7cfe", "doi": "10.1088/1751-8121/ab7cfe" }, { "citation": "Singh and Pal (2021), Extremal statistics for stochastic resetting systems. Physical Review E 103, 052119.", "url": "https://doi.org/10.1103/PhysRevE.103.052119", "doi": "10.1103/PhysRevE.103.052119" }, { "citation": "Majumdar, Mounaix, Sabhapandit and Schehr (2022), Record statistics for random walks and Lévy flights with resetting. Journal of Physics A 55, 034002.", "url": "https://doi.org/10.1088/1751-8121/ac3fc1", "doi": "10.1088/1751-8121/ac3fc1" }, { "citation": "Mori, Majumdar and Schehr (2021), Distribution of the time of the maximum for stationary processes. Europhysics Letters 135, 30003.", "url": "https://doi.org/10.1209/0295-5075/ac19ee", "doi": "10.1209/0295-5075/ac19ee" }, { "citation": "Majumdar, Pal and Schehr (2020), Extreme value statistics of correlated random variables: a pedagogical review. Physics Reports 840, 1–32.", "url": "https://doi.org/10.1016/j.physrep.2019.10.005", "doi": "10.1016/j.physrep.2019.10.005" }, { "citation": "Kyprianou (2014), Fluctuations of Lévy Processes with Applications. Universitext, Springer.", "url": "https://doi.org/10.1007/978-3-642-37632-0", "doi": "10.1007/978-3-642-37632-0" }, { "citation": "Kuznetsov, Kyprianou and Rivero (2012), The theory of scale functions for spectrally negative Lévy processes. Lévy Matters II, Lecture Notes in Mathematics, 97–186.", "url": "https://doi.org/10.1007/978-3-642-31407-0_2", "doi": "10.1007/978-3-642-31407-0_2" }, { "citation": "Pal, Kundu and Evans (2016), Diffusion under time-dependent resetting. Journal of Physics A 49, 225001.", "url": "https://doi.org/10.1088/1751-8113/49/22/225001", "doi": "10.1088/1751-8113/49/22/225001" }, { "citation": "Asmussen and Albrecher (2010), Ruin Probabilities. World Scientific.", "url": "https://doi.org/10.1142/7431", "doi": "10.1142/7431" }, { "citation": "Anonymous (2026), Exact exponential moments of interval quasi-stationary laws for Lévy processes, and the saturation of idealised QSD restart. Evidence Press release (companion: restart of a simulation from a quasi-stationary law, sharing the Esscher and Wiener–Hopf toolkit).", "url": "https://evidencepress.org/releases/levy-qsd-exponential-moments-restart/", "doi": null } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:spectrally-negative-resetting-extremes", "attemptIds": [ "ep-attempt:spectrally-negative-resetting-extremes-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "other", "description": "An explicit two-profile description, with uniform total-variation error, of where a stationary spectrally negative Lévy process with Poisson resetting ends given a high fixed-horizon maximum; the constant left open in Remark 3 of Dębicki, Hashorva and Michna (2026), with strict bounds.", "scope": "Spectrally negative Lévy inputs, Poisson reset clock, fixed or i.i.d. random reset positions with an exponential moment, stationary start, fixed horizon, threshold tending to infinity. 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