{"title":"空间相关分支过程的渐近性和临界性","authors":"Ilie Grigorescu, Min Kang","doi":"10.1080/17442508.2023.2256922","DOIUrl":null,"url":null,"abstract":"AbstractWe investigate a non-conservative semigroup (St)t≥0 determined by a branching process tracing the evolution of particles moving in a domain in Rd. When a particle is killed at the boundary, a new generation of particles with mean number K¯ is born at a random point in the domain. Between branching, the particles are driven by a diffusion process with Dirichlet boundary conditions. According to the sign of K¯−1, we distinguish super/sub-critical regimes and determine the exact exponential rate for the total number of particles n(t)∼exp(α∗t), with α∗ depending explicitly on K¯. We prove the Yaglom limit St/n(t)→ν, where the quasi-stationary distribution ν is determined by the resolvent of the Dirichlet kernel at the point α∗. The main application is in particle systems, where the normalization of the semigroup by its total mass gives the hydrodynamic limit of the Bak-Sneppen branching diffusions (BSBD). Since ν is the asymptotic profile under equilibrium, and the family of quasi-stationary distributions ν is indexed by K¯, the model provides an explicit example of self-organized criticality.Keywords: SemigroupYaglom limitbranching processessupercriticalqsdBak-SneppenFleming-ViotDirichlet kernelKey Words and Phrases: Primary: 60J3560J80Secondary: 47D0760K35 Disclosure statementNo potential conflict of interest was reported by the author(s).","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Asymptotics and criticality for a space-dependent branching process\",\"authors\":\"Ilie Grigorescu, Min Kang\",\"doi\":\"10.1080/17442508.2023.2256922\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"AbstractWe investigate a non-conservative semigroup (St)t≥0 determined by a branching process tracing the evolution of particles moving in a domain in Rd. When a particle is killed at the boundary, a new generation of particles with mean number K¯ is born at a random point in the domain. Between branching, the particles are driven by a diffusion process with Dirichlet boundary conditions. According to the sign of K¯−1, we distinguish super/sub-critical regimes and determine the exact exponential rate for the total number of particles n(t)∼exp(α∗t), with α∗ depending explicitly on K¯. We prove the Yaglom limit St/n(t)→ν, where the quasi-stationary distribution ν is determined by the resolvent of the Dirichlet kernel at the point α∗. The main application is in particle systems, where the normalization of the semigroup by its total mass gives the hydrodynamic limit of the Bak-Sneppen branching diffusions (BSBD). Since ν is the asymptotic profile under equilibrium, and the family of quasi-stationary distributions ν is indexed by K¯, the model provides an explicit example of self-organized criticality.Keywords: SemigroupYaglom limitbranching processessupercriticalqsdBak-SneppenFleming-ViotDirichlet kernelKey Words and Phrases: Primary: 60J3560J80Secondary: 47D0760K35 Disclosure statementNo potential conflict of interest was reported by the author(s).\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-09-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1080/17442508.2023.2256922\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1080/17442508.2023.2256922","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Asymptotics and criticality for a space-dependent branching process
AbstractWe investigate a non-conservative semigroup (St)t≥0 determined by a branching process tracing the evolution of particles moving in a domain in Rd. When a particle is killed at the boundary, a new generation of particles with mean number K¯ is born at a random point in the domain. Between branching, the particles are driven by a diffusion process with Dirichlet boundary conditions. According to the sign of K¯−1, we distinguish super/sub-critical regimes and determine the exact exponential rate for the total number of particles n(t)∼exp(α∗t), with α∗ depending explicitly on K¯. We prove the Yaglom limit St/n(t)→ν, where the quasi-stationary distribution ν is determined by the resolvent of the Dirichlet kernel at the point α∗. The main application is in particle systems, where the normalization of the semigroup by its total mass gives the hydrodynamic limit of the Bak-Sneppen branching diffusions (BSBD). Since ν is the asymptotic profile under equilibrium, and the family of quasi-stationary distributions ν is indexed by K¯, the model provides an explicit example of self-organized criticality.Keywords: SemigroupYaglom limitbranching processessupercriticalqsdBak-SneppenFleming-ViotDirichlet kernelKey Words and Phrases: Primary: 60J3560J80Secondary: 47D0760K35 Disclosure statementNo potential conflict of interest was reported by the author(s).