LIMIT THEOREMS FOR SMOLUCHOWSKI DYNAMICS ASSOCIATED WITH CRITICAL CONTINUOUS-STATE BRANCHING PROCESSES
We investigate the well-posedness and asymptotic self-similarity of solutions to a generalized Smoluchowski coagulation equation recently introduced by Bertoin and Le Gall in the context of continuous-state branching theory. In particular, this equation governs the evolution of the Lévy measure of a...
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| Veröffentlicht in: | The Annals of applied probability 2015-04, Vol.25 (2), p.675-713 |
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| container_end_page | 713 |
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| container_title | The Annals of applied probability |
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| creator | Iyer, Gautam Leger, Nicholas Pego, Robert L. |
| description | We investigate the well-posedness and asymptotic self-similarity of solutions to a generalized Smoluchowski coagulation equation recently introduced by Bertoin and Le Gall in the context of continuous-state branching theory. In particular, this equation governs the evolution of the Lévy measure of a critical continuous-state branching process which becomes extinct (i.e., is absorbed at zero) almost surely. We show that a nondegenerate scaling limit of the Lévy measure (and the process) exists if and only if the branching mechanism is regularly varying at 0. When the branching mechanism is regularly varying, we characterize nondegenerate scaling limits of arbitrary finite-measure solutions in terms of generalized Mittag–Leffler series. |
| doi_str_mv | 10.1214/14-AAP1008 |
| format | Article |
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In particular, this equation governs the evolution of the Lévy measure of a critical continuous-state branching process which becomes extinct (i.e., is absorbed at zero) almost surely. We show that a nondegenerate scaling limit of the Lévy measure (and the process) exists if and only if the branching mechanism is regularly varying at 0. When the branching mechanism is regularly varying, we characterize nondegenerate scaling limits of arbitrary finite-measure solutions in terms of generalized Mittag–Leffler series.</description><identifier>ISSN: 1050-5164</identifier><identifier>EISSN: 2168-8737</identifier><identifier>DOI: 10.1214/14-AAP1008</identifier><language>eng</language><publisher>Hayward: Institute of Mathematical Statistics</publisher><subject>35Q70 ; 60G18 ; 60J80 ; 82C28 ; Asymptotic methods ; Bernstein function ; coagulation ; Continuous-state branching process ; critical branching ; limit theorem ; Mittag–Leffler series ; regular variation ; scaling limit ; self-similar solution ; Smoluchowski equation ; Theorems</subject><ispartof>The Annals of applied probability, 2015-04, Vol.25 (2), p.675-713</ispartof><rights>Copyright © 2015 Institute of Mathematical Statistics</rights><rights>Copyright Institute of Mathematical Statistics Apr 2015</rights><rights>Copyright 2015 Institute of Mathematical Statistics</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c375t-3fb343f2397c50e532acd43b76f0f7ff8e75a199d46c333c5650e42d06a680ff3</citedby><cites>FETCH-LOGICAL-c375t-3fb343f2397c50e532acd43b76f0f7ff8e75a199d46c333c5650e42d06a680ff3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/24519932$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/24519932$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>230,314,776,780,799,828,881,921,27901,27902,57992,57996,58225,58229</link.rule.ids></links><search><creatorcontrib>Iyer, Gautam</creatorcontrib><creatorcontrib>Leger, Nicholas</creatorcontrib><creatorcontrib>Pego, Robert L.</creatorcontrib><title>LIMIT THEOREMS FOR SMOLUCHOWSKI DYNAMICS ASSOCIATED WITH CRITICAL CONTINUOUS-STATE BRANCHING PROCESSES</title><title>The Annals of applied probability</title><description>We investigate the well-posedness and asymptotic self-similarity of solutions to a generalized Smoluchowski coagulation equation recently introduced by Bertoin and Le Gall in the context of continuous-state branching theory. 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When the branching mechanism is regularly varying, we characterize nondegenerate scaling limits of arbitrary finite-measure solutions in terms of generalized Mittag–Leffler series.</description><subject>35Q70</subject><subject>60G18</subject><subject>60J80</subject><subject>82C28</subject><subject>Asymptotic methods</subject><subject>Bernstein function</subject><subject>coagulation</subject><subject>Continuous-state branching process</subject><subject>critical branching</subject><subject>limit theorem</subject><subject>Mittag–Leffler series</subject><subject>regular variation</subject><subject>scaling limit</subject><subject>self-similar solution</subject><subject>Smoluchowski equation</subject><subject>Theorems</subject><issn>1050-5164</issn><issn>2168-8737</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2015</creationdate><recordtype>article</recordtype><recordid>eNo9kU1rg0AQhpfSQtO0l94LC70VbPdbPdqNiUvVDa4SehJjXEhoa6rm0H9fQ0JOAzMPz8zwAvCI0SsmmL1h5gTBEiPkXYEJwcJzPJe612CCEUcOx4Ldgru-3yGEfOa7E2Bjlagc5lGoszAxcK4zaBIdFzLSK_Oh4OwzDRIlDQyM0VIFeTiDK5VHUGYqVzKIodRprtJCF8Yx-TiH71mQykilC7jMtAyNCc09uLHVV988nOsUFPMwl5ET68VR4tTU5YND7Zoyagn13ZqjhlNS1RtG166wyLrWeo3LK-z7GyZqSmnNxUgxskGiEh6ylk5BcPLuu3bX1ENzqL-2m3Lfbb-r7q9sq20pi_jcPZeqrfYlZoRRzjHxRsfzxfF7aPqh3LWH7mc8u8SCewwTgsRIvZyoumv7vmvsZQlG5TGLUVmesxjhpxO864e2u5CE8fEZSug_HeR8zQ</recordid><startdate>20150401</startdate><enddate>20150401</enddate><creator>Iyer, Gautam</creator><creator>Leger, Nicholas</creator><creator>Pego, Robert L.</creator><general>Institute of Mathematical Statistics</general><general>The Institute of Mathematical Statistics</general><scope>AAYXX</scope><scope>CITATION</scope><scope>JQ2</scope></search><sort><creationdate>20150401</creationdate><title>LIMIT THEOREMS FOR SMOLUCHOWSKI DYNAMICS ASSOCIATED WITH CRITICAL CONTINUOUS-STATE BRANCHING PROCESSES</title><author>Iyer, Gautam ; Leger, Nicholas ; Pego, Robert L.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c375t-3fb343f2397c50e532acd43b76f0f7ff8e75a199d46c333c5650e42d06a680ff3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2015</creationdate><topic>35Q70</topic><topic>60G18</topic><topic>60J80</topic><topic>82C28</topic><topic>Asymptotic methods</topic><topic>Bernstein function</topic><topic>coagulation</topic><topic>Continuous-state branching process</topic><topic>critical branching</topic><topic>limit theorem</topic><topic>Mittag–Leffler series</topic><topic>regular variation</topic><topic>scaling limit</topic><topic>self-similar solution</topic><topic>Smoluchowski equation</topic><topic>Theorems</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Iyer, Gautam</creatorcontrib><creatorcontrib>Leger, Nicholas</creatorcontrib><creatorcontrib>Pego, Robert L.</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Computer Science Collection</collection><jtitle>The Annals of applied probability</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Iyer, Gautam</au><au>Leger, Nicholas</au><au>Pego, Robert L.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>LIMIT THEOREMS FOR SMOLUCHOWSKI DYNAMICS ASSOCIATED WITH CRITICAL CONTINUOUS-STATE BRANCHING PROCESSES</atitle><jtitle>The Annals of applied probability</jtitle><date>2015-04-01</date><risdate>2015</risdate><volume>25</volume><issue>2</issue><spage>675</spage><epage>713</epage><pages>675-713</pages><issn>1050-5164</issn><eissn>2168-8737</eissn><abstract>We investigate the well-posedness and asymptotic self-similarity of solutions to a generalized Smoluchowski coagulation equation recently introduced by Bertoin and Le Gall in the context of continuous-state branching theory. In particular, this equation governs the evolution of the Lévy measure of a critical continuous-state branching process which becomes extinct (i.e., is absorbed at zero) almost surely. We show that a nondegenerate scaling limit of the Lévy measure (and the process) exists if and only if the branching mechanism is regularly varying at 0. When the branching mechanism is regularly varying, we characterize nondegenerate scaling limits of arbitrary finite-measure solutions in terms of generalized Mittag–Leffler series.</abstract><cop>Hayward</cop><pub>Institute of Mathematical Statistics</pub><doi>10.1214/14-AAP1008</doi><tpages>39</tpages><oa>free_for_read</oa></addata></record> |
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| source | Jstor Complete Legacy; EZB-FREE-00999 freely available EZB journals; Project Euclid Complete; JSTOR Mathematics & Statistics |
| subjects | 35Q70 60G18 60J80 82C28 Asymptotic methods Bernstein function coagulation Continuous-state branching process critical branching limit theorem Mittag–Leffler series regular variation scaling limit self-similar solution Smoluchowski equation Theorems |
| title | LIMIT THEOREMS FOR SMOLUCHOWSKI DYNAMICS ASSOCIATED WITH CRITICAL CONTINUOUS-STATE BRANCHING PROCESSES |
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