空间相关分支过程的渐近性和临界性

Pub Date : 2023-09-13 DOI:10.1080/17442508.2023.2256922
Ilie Grigorescu, Min Kang
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引用次数: 1

摘要

摘要研究了一个非保守半群(St)t≥0,该半群是由在Rd域中运动的粒子的分支过程确定的。当一个粒子在边界处被杀死时,在该区域的随机点上产生了平均数为K¯的新一代粒子。在分支之间,粒子由具有狄利克雷边界条件的扩散过程驱动。根据K¯−1的符号,我们区分了超/次临界状态,并确定了粒子总数n(t) ~ exp (α∗t)的确切指数率,其中α∗显式依赖于K¯。证明了Yaglom极限St/n(t)→ν,其中拟平稳分布ν由Dirichlet核在点α∗处的解决定。主要应用于粒子系统,其中半群的总质量归一化给出了贝克-斯奈彭分支扩散(BSBD)的流体动力学极限。由于ν是平衡状态下的渐近剖面,而拟平稳分布族ν由K¯索引,因此该模型提供了一个明确的自组织临界性的例子。关键词:SemigroupYaglom限制分支过程supercriticalqsdbak - sneppenflefleming - viotdirichlet核关键词:一级:60j3560j80二级:47D0760K35披露声明作者未报告潜在利益冲突。
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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).
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