{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:44:11Z","timestamp":1787341451902,"version":"build-2736575974"},"reference-count":35,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-10-LABX-0098"],"award-info":[{"award-number":["ANR-10-LABX-0098"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Math."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>In this paper, we investigate a stochastic model describing the time evolution of a polymerization process. A polymer is a macromolecule resulting from the aggregation of several elementary subunits called monomers. Polymers can grow by the addition of monomers or can be split into several polymers. The initial state of the system consists mainly of monomers. We study the time evolution of the mass of polymers, in particular the asymptotic distribution of the first instant when the fraction of monomers used in polymers is above some positive threshold $\\delta$. A scaling approach is used by taking the mass $N$ as a scaling parameter. The mathematical model used in this paper includes a nucleation property: If $n_c$ is defined as the size of the nucleus, polymers with a size less than $n_c$ are quickly fragmented into smaller polymers at a rate proportional to $\\Phi(N)$ for some nondecreasing and unbounded function $\\Phi$. For polymers of size greater than $n_c$, fragmentation still occurs but at bounded rates. If $T^N$ is the instant of creation of the first polymer whose size is $n_c$, it is shown that, under appropriate conditions, the variable $T^N{\/}[\\Phi(N)^{n_c{-}2}{\/}N]$ converges in distribution, and that the first instant $L_\\delta^N$ when a fraction $\\delta$ of monomers is polymerized has the same order of magnitude. A feature originally proved for this model is the significant variability of the variable $T^N$. This is a well-known phenomenon observed in biological experiments but few mathematical models of the literature have this property. The results are proved via a series of technical estimates for occupation measures of some functionals of the corresponding Markov processes on fast time scales and by using coupling techniques.<\/jats:p>","DOI":"10.1137\/19m1237508","type":"journal-article","created":{"date-parts":[[2019,10,29]],"date-time":"2019-10-29T10:51:22Z","timestamp":1572346282000},"page":"2080-2106","source":"Crossref","is-referenced-by-count":1,"title":["On the Asymptotic Distribution of Nucleation Times of Polymerization Processes"],"prefix":"10.1137","volume":"79","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8122-8880","authenticated-orcid":true,"given":"Philippe","family":"Robert","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wen","family":"Sun","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,10,29]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"D. Aldous,\n                      Probability Approximations via the Poisson Clumping Heuristic\n                      , Appl. Math. Sci. 77, Springer-Verlag, New York, 1989.","DOI":"10.1007\/978-1-4757-6283-9"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.2307\/3318611"},{"key":"atypb3","first-page":"3","author":"Anderson D. F.","year":"2011","journal-title":"New York"},{"key":"atypb4","unstructured":"K. B. Athreya and P. E. Ney,\n                      Branching Processes\n                      , Die Grundlehren der mathematischen Wissenschaften, Band 196, Springer-Verlag, New York, Heidelberg, 1972."},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/BF01211070"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1002\/andp.19354160806"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"J. Bertoin,\n                      Random Fragmentation and Coagulation Processes\n                      , Cambridge Stud. Adv. Math. 102, Cambridge University Press, Cambridge, 2006,https:\/\/doi.org\/10.1017\/CBO9780511617768.","DOI":"10.1017\/CBO9780511617768"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176995481"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/16M1066920"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v19-3244"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1063\/1.4947472"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1993-1094553-1"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/0022-2836(86)90324-4"},{"key":"atypb14","doi-asserted-by":"crossref","unstructured":"J. Jacod and A. N. Shiryaev,\n                      Limit Theorems for Stochastic Processes\n                      , Grundlehren Math. Wiss. 288, Springer-Verlag, Berlin, 1987.","DOI":"10.1007\/978-3-662-02514-7_6"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1007\/s002200050368"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"N. V. Kampen,\n                      Stochastic Processes in Physics and Chemistry\n                      , North-Holland, Amsterdam, New York, 2007.","DOI":"10.1016\/B978-044452965-7\/50006-4"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/03830543"},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"D. Kashchiev,\n                      Nucleation\n                      , Butterworth-Heinemann, Oxford, 2000.","DOI":"10.1016\/B978-075064682-6\/50006-8"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1016\/j.bpj.2015.10.010"},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"J. Keilson,\n                      Markov Chains Models-Rarity and Exponentiality\n                      , Appl. Math. Sci. 28, Springer Verlag, New York, Berlin, 1979.","DOI":"10.1007\/978-1-4612-6200-8"},{"key":"atypb21","unstructured":"J. F. C. Kingman,\n                      Poisson Processes\n                      , Oxford Stud. Probab. 3, The Clarendon Press, Oxford University Press, New York, 1993."},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1214\/105051605000000494"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/j.bbapap.2008.10.016"},{"key":"atypb24","unstructured":"J. D. Murray,\n                      Mathematical biology. I. An introduction\n                      , Interdiscip. Appl. Math. 17, 3rd ed., Springer-Verlag, New York, 2002."},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/BF01213386"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.12732\/ijpam.v93i6.10"},{"key":"atypb27","unstructured":"L. Pujo-Menjouet,\n                      \u00c9tude de mod\u00e8les math\u00e9matiques issus de la biologie du cycle cellulaire et de la dynamique des prot\u00e9ines\n                      , habilitation \u00e0 diriger des recherches, Universit\u00e9 Claude Bernard Lyon 1 - Institut Camille Jordan, 2016,https:\/\/hal.inria.fr\/tel-01411371."},{"key":"atypb28","unstructured":"P. Robert,\n                      Stochastic Networks and Queues\n                      , Stochastic Modelling and Applied Probability Series, Springer-Verlag, Berlin, Heidelberg, 2003,http:\/\/www.springer.com\/math\/probability\/book\/978-3-540-00657-2."},{"key":"atypb29","unstructured":"W. Sun,\n                      A Functional Central Limit Theorem for the Becker-D\u00f6ring Model\n                      , preprint,https:\/\/arxiv.org\/abs\/1710.04059, 2017."},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1214\/15-AAP1167"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.113.098101"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1016\/0301-4622(75)80013-5"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0711664105"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1063\/1.4940033"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1063\/1.4772598"}],"container-title":["SIAM Journal on Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/19M1237508","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:20:59Z","timestamp":1787340059000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/19M1237508"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":35,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/19M1237508"],"URL":"https:\/\/doi.org\/10.1137\/19m1237508","relation":{},"ISSN":["0036-1399","1095-712X"],"issn-type":[{"value":"0036-1399","type":"print"},{"value":"1095-712X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}