{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:32:39Z","timestamp":1760059959109,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,7,23]],"date-time":"2025-07-23T00:00:00Z","timestamp":1753228800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12271029","11901570","KM202411417001"],"award-info":[{"award-number":["12271029","11901570","KM202411417001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"R&amp;D Program of Beijing Municipal Education Commission","award":["12271029","11901570","KM202411417001"],"award-info":[{"award-number":["12271029","11901570","KM202411417001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We show that a class of downwardly skip-free processes can be regarded as the Siegmund dual of upwardly skip-free processes, which have been extensively studied in the literature. For such downwardly skip-free processes, using the duality method and existing results on upwardly skip-free processes, we provide explicit expressions for the extinction probability, the mean occupation time, the mean explosion time, and the mean extinction time. Additionally, we provide two examples to illustrate the main contributions of this work.<\/jats:p>","DOI":"10.3390\/axioms14080555","type":"journal-article","created":{"date-parts":[[2025,7,23]],"date-time":"2025-07-23T14:22:44Z","timestamp":1753280564000},"page":"555","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Dual Processes of a Class of Downwardly Skip-Free Processes"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3949-9529","authenticated-orcid":false,"given":"Pei-Sen","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100872, China"}]},{"given":"Pan","family":"Zhao","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Physics, Beijing Union University, Beijing 100101, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Chen, M.F. 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Appl."},{"key":"ref_9","first-page":"681","article-title":"Criteria for zero-exit (-entrance) of single-birth (-death) Q-matrices","volume":"57","author":"Wang","year":"2014","journal-title":"Acta. Math. Sin. (Chin. Ser.)"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1016\/j.jmaa.2005.05.076","article-title":"Dual and Feller-Reuter-Riley transition functions","volume":"313","author":"Li","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1112\/jlms\/s2-5.2.267","article-title":"The Feller property for Markov semigroups on a countable state space","volume":"5","author":"Reuter","year":"1972","journal-title":"J. London Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/S0167-7152(01)00130-4","article-title":"Strong ergodicity of monotone transition functions","volume":"55","author":"Zhang","year":"2001","journal-title":"Statist. Probab. Lett."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/555\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:14:39Z","timestamp":1760033679000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/555"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,23]]},"references-count":12,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["axioms14080555"],"URL":"https:\/\/doi.org\/10.3390\/axioms14080555","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,7,23]]}}}