{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T17:08:57Z","timestamp":1778692137063,"version":"3.51.4"},"reference-count":24,"publisher":"AIP Publishing","issue":"5","funder":[{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-26-1-0011"],"award-info":[{"award-number":["FA9550-26-1-0011"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Texas Advanced Computing Center","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2026,5,1]]},"abstract":"<jats:p>We develop a practical framework for distinguishing diffusive stochastic processes from deterministic signals using only a single discrete time series. Our approach is based on classical excursion and crossing theorems for continuous semimartingales, which correlates number N\u03b5 of excursions of magnitude at least \u03b5 with the quadratic variation [X]T of the process. The scaling law holds universally for all continuous semimartingales with finite quadratic variation, including general Ito diffusions with nonlinear or state-dependent volatility, but fails sharply for deterministic systems\u2014thereby providing a theoretically certified method of distinguishing between these dynamics, as opposed to the subjective entropy or recurrence based state of the art methods. We construct a robust data-driven diffusion test. The method compares the empirical excursion counts against the theoretical expectation. The resulting ratio K(\u03b5)=N\u03b5emp\/N\u03b5theory is then summarized by a log\u2013log slope deviation measuring the \u03b5\u22122 law that provides a classification into diffusion-like or not. We demonstrate the method on canonical stochastic systems, some periodic and chaotic maps and systems with additive white noise, as well as the stochastic Duffing system. The approach is nonparametric, model-free, and relies only on the universal small-scale structure of continuous semimartingales.<\/jats:p>","DOI":"10.1063\/5.0324348","type":"journal-article","created":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T16:10:16Z","timestamp":1778688616000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Detecting stochasticity in discrete signals via nonparametric excursion theorem"],"prefix":"10.1063","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7232-0973","authenticated-orcid":false,"given":"Sunia","family":"Tanweer","sequence":"first","affiliation":[{"name":"Department of Computational Mathematics, Sciences and Engineering, Michigan State University , Michigan 48824,","place":["USA"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7817-7445","authenticated-orcid":false,"given":"Firas A.","family":"Khasawneh","sequence":"additional","affiliation":[{"name":"Department of Computational Mathematics, Sciences and Engineering, Michigan State University , Michigan 48824,","place":["USA"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"317","published-online":{"date-parts":[[2026,5,13]]},"reference":[{"key":"2026051312101122800_c1","doi-asserted-by":"publisher","first-page":"20150349","DOI":"10.1098\/rspa.2015.0349","article-title":"Signatures of chaotic and stochastic dynamics uncovered with \u03b5-recurrence networks","volume":"471","year":"2015","journal-title":"Proc. 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