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If the speed parameters of the randomisation are small, then this dynamics has one metastable uncertainty interval and two invariant attractors. The probabilities of leaving the metastable interval through the upper boundary are determined, as well as characteristics of the first crossing times. Invariant measures are also found. When and if the direction of the telegraph process velocity coincides with the direction of the periodic change in potential, the system can go into a metastable state, having received a time window for the interwell transition. The obtained results can be used as an alternative to stochastic resonance models.<\/jats:p>","DOI":"10.1017\/jpr.2025.10027","type":"journal-article","created":{"date-parts":[[2025,9,19]],"date-time":"2025-09-19T07:28:42Z","timestamp":1758266922000},"page":"204-224","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Metastability for telegraph processes in a double-well potential"],"prefix":"10.1017","volume":"63","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0242-0549","authenticated-orcid":false,"given":"Nikita","family":"Ratanov","sequence":"first","affiliation":[{"name":"Chelyabinsk State University"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2025,9,19]]},"reference":[{"key":"S0021900225100272_ref41","doi-asserted-by":"publisher","DOI":"10.1016\/j.cjph.2021.10.004"},{"key":"S0021900225100272_ref30","doi-asserted-by":"publisher","DOI":"10.64837\/GSA.10.3.5"},{"key":"S0021900225100272_ref37","first-page":"549","article-title":"Stochastic bistable systems: Competing hysteresis and phase coexistence","volume":"127","author":"Verma","year":"2018","journal-title":"Statist. Nonlin. Soft Matter Phys."},{"key":"S0021900225100272_ref21","unstructured":"[21] Milankovitch, M. (1930). Mathematische klimalehre und astronomische theorie der klimaschwankungen. In Handbuch der Klimatologie, Vol. I, eds W. K\u00f6ppen and R. Geiger. Borntr\u00e4ger, Berlin."},{"key":"S0021900225100272_ref18","doi-asserted-by":"crossref","unstructured":"[18] Herrmann, S. , Imkeller, P. , Pavlyukevich, I. and Peithmann, D. (2014). Stochastic Resonance: A Mathematical Approach in the Small Noise Limit (Math. Surv. Monographs 194). 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I. and Koralov, L. (2024). Metastability in parabolic equations and diffusion processes with a small parameter. Preprint, arXiv:2403.12333v1."},{"key":"S0021900225100272_ref10","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/2009\/05\/P05004"},{"key":"S0021900225100272_ref22","doi-asserted-by":"publisher","DOI":"10.1063\/1.1704269"},{"key":"S0021900225100272_ref19","doi-asserted-by":"publisher","DOI":"10.1216\/RMJ-1974-4-3-497"},{"key":"S0021900225100272_ref23","doi-asserted-by":"publisher","DOI":"10.5772\/intechopen.70178.10.5772\/intechopen.70178"},{"key":"S0021900225100272_ref28","doi-asserted-by":"publisher","DOI":"10.1007\/s11009-022-09956-z"},{"key":"S0021900225100272_ref17","doi-asserted-by":"publisher","DOI":"10.1016\/j.aop.2024.169857"},{"key":"S0021900225100272_ref38","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-010-9944-5"},{"key":"S0021900225100272_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-5193(05)80396-0"},{"key":"S0021900225100272_ref39","doi-asserted-by":"publisher","DOI":"10.1016\/0040-5809(89)90036-1"},{"key":"S0021900225100272_ref27","doi-asserted-by":"publisher","DOI":"10.1007\/s11009-020-09794-x"},{"key":"S0021900225100272_ref13","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.70.223"},{"key":"S0021900225100272_ref35","doi-asserted-by":"publisher","DOI":"10.1214\/23-AAP1930"},{"key":"S0021900225100272_ref16","unstructured":"[16] Gutkin, B. S. , Jost, J. and Tuckwell, H. C. (2008). The silencing of neuronal activity by noise and the phenomenon of inverse stochastic resonance. Preprint, arXiv:0805.0592 [q-bio.NC]."},{"key":"S0021900225100272_ref2","doi-asserted-by":"publisher","DOI":"10.1214\/19-AOP1356"},{"key":"S0021900225100272_ref26","doi-asserted-by":"publisher","DOI":"10.1016\/j.biosystems.2020.104190"},{"key":"S0021900225100272_ref34","volume-title":"Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry, and Engineering","author":"Strogatz","year":"2018"},{"key":"S0021900225100272_ref6","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202525400044"},{"key":"S0021900225100272_ref1","doi-asserted-by":"publisher","DOI":"10.1137\/0143037"},{"key":"S0021900225100272_ref20","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511535239"},{"key":"S0021900225100272_ref24","doi-asserted-by":"publisher","DOI":"10.1007\/BF01053950"},{"key":"S0021900225100272_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/JHEP08(2022)047"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900225100272","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T01:25:58Z","timestamp":1775525158000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900225100272\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,19]]},"references-count":41,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S0021900225100272"],"URL":"https:\/\/doi.org\/10.1017\/jpr.2025.10027","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,19]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. 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