{"title":"多类型连续状态分支过程的消灭时间","authors":"L. Chaumont, M. Marolleau","doi":"10.1214/22-aihp1279","DOIUrl":null,"url":null,"abstract":"A multitype continuous-state branching process (MCSBP) ${\\rm Z}=({\\rm Z}_{t})_{t\\geq 0}$, is a Markov process with values in $[0,\\infty)^{d}$ that satisfies the branching property. Its distribution is characterised by its branching mechanism, that is the data of $d$ Laplace exponents of $\\mathbb{R}^d$-valued spectrally positive L\\'evy processes, each one having $d-1$ increasing components. We give an expression of the probability for a MCSBP to tend to 0 at infinity in term of its branching mechanism. Then we prove that this extinction holds at a finite time if and only if some condition bearing on the branching mechanism holds. This condition extends Grey's condition that is well known for $d=1$. Our arguments bear on elements of fluctuation theory for spectrally positive additive L\\'evy fields recently obtained in \\cite{cma1} and an extension of the Lamperti representation in higher dimension proved in \\cite{cpgub}.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"8 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2021-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Extinction times of multitype continuous-state branching processes\",\"authors\":\"L. Chaumont, M. Marolleau\",\"doi\":\"10.1214/22-aihp1279\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A multitype continuous-state branching process (MCSBP) ${\\\\rm Z}=({\\\\rm Z}_{t})_{t\\\\geq 0}$, is a Markov process with values in $[0,\\\\infty)^{d}$ that satisfies the branching property. Its distribution is characterised by its branching mechanism, that is the data of $d$ Laplace exponents of $\\\\mathbb{R}^d$-valued spectrally positive L\\\\'evy processes, each one having $d-1$ increasing components. We give an expression of the probability for a MCSBP to tend to 0 at infinity in term of its branching mechanism. Then we prove that this extinction holds at a finite time if and only if some condition bearing on the branching mechanism holds. This condition extends Grey's condition that is well known for $d=1$. Our arguments bear on elements of fluctuation theory for spectrally positive additive L\\\\'evy fields recently obtained in \\\\cite{cma1} and an extension of the Lamperti representation in higher dimension proved in \\\\cite{cpgub}.\",\"PeriodicalId\":42884,\"journal\":{\"name\":\"Annales de l Institut Henri Poincare D\",\"volume\":\"8 1\",\"pages\":\"\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2021-09-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annales de l Institut Henri Poincare D\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1214/22-aihp1279\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales de l Institut Henri Poincare D","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/22-aihp1279","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
Extinction times of multitype continuous-state branching processes
A multitype continuous-state branching process (MCSBP) ${\rm Z}=({\rm Z}_{t})_{t\geq 0}$, is a Markov process with values in $[0,\infty)^{d}$ that satisfies the branching property. Its distribution is characterised by its branching mechanism, that is the data of $d$ Laplace exponents of $\mathbb{R}^d$-valued spectrally positive L\'evy processes, each one having $d-1$ increasing components. We give an expression of the probability for a MCSBP to tend to 0 at infinity in term of its branching mechanism. Then we prove that this extinction holds at a finite time if and only if some condition bearing on the branching mechanism holds. This condition extends Grey's condition that is well known for $d=1$. Our arguments bear on elements of fluctuation theory for spectrally positive additive L\'evy fields recently obtained in \cite{cma1} and an extension of the Lamperti representation in higher dimension proved in \cite{cpgub}.