{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T22:56:17Z","timestamp":1772060177204,"version":"3.50.1"},"reference-count":204,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,16]],"date-time":"2025-06-16T00:00:00Z","timestamp":1750032000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper introduces and explores a novel class of Brown and Levy steady-state motions. These motions generalize, respectively, the Ornstein-Uhlenbeck process (OUP) and the Levy-driven OUP. As the OUP and the Levy-driven OUP: the motions are Markov; their dynamics are Langevin; and their steady-state distributions are, respectively, Gauss and Levy. As the Levy-driven OUP: the motions can display the Noah effect (heavy-tailed amplitudal fluctuations); and their memory structure is tunable. And, as Gaussian-stationary processes: the motions can display the Joseph effect (long-ranged temporal dependencies); and their correlation structure is tunable. The motions have two parameters: a critical exponent which determines the Noah effect and the memory structure; and a clock function which determines the Joseph effect and the correlation structure. The novel class is a compelling stochastic model due to the following combination of facts: on the one hand the motions are tractable and amenable to analysis and use; on the other hand the model is versatile and the motions display a host of both regular and anomalous features.<\/jats:p>","DOI":"10.3390\/e27060643","type":"journal-article","created":{"date-parts":[[2025,6,16]],"date-time":"2025-06-16T10:47:22Z","timestamp":1750070842000},"page":"643","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Brown and Levy Steady-State Motions"],"prefix":"10.3390","volume":"27","author":[{"given":"Iddo","family":"Eliazar","sequence":"first","affiliation":[{"name":"School of Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1103\/PhysRev.36.823","article-title":"On the theory of the Brownian motion","volume":"36","author":"Uhlenbeck","year":"1930","journal-title":"Phys. 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