标题： 临界多类型分支系统：灭绝结果 显示英文标题

标题： Critical Multitype Branching Systems: Extinction Results

摘要： 我们考虑在$\R^d$中的一个临界分支粒子系统，该系统由有限种类型的个体组成$i\in\{1,...,K\}$。每种类型为$i$的个体独立地根据对称的$\alpha_i$稳定运动移动。我们假设粒子寿命和子代分布是依赖于类型的。在分支系统中通常的独立性假设下，我们在以下情况下证明了灭绝定理：(1) 所有粒子寿命都有有限均值，或者 (2) 有一种类型的寿命分布具有重尾，其他寿命具有有限均值。通过在情况 (2) 中假设最活跃的粒子类型对应于有限均值的寿命，我们得到了更复杂的动态：在这种情况下，种群的局部灭绝由长寿命类型的参数（子代变异性、活跃度、寿命）与其他最活跃类型的参数之间的相互作用决定。证明基于对相关马尔可夫更新过程的占用时间的精确分析，这本身具有独立的兴趣。 显示英文摘要

摘要： We consider a critical branching particle system in $\R^d$, composed of individuals of a finite number of types $i\in\{1,...,K\}$. Each individual of type $i$ moves independently according to a symmetric $\alpha_i$-stable motion. We assume that the particle lifetimes and offspring distributions are type-dependent. Under the usual independence assumptions in branching systems, we prove extinction theorems in the following cases: (1) all the particle lifetimes have finite mean, or (2) there is a type whose lifetime distribution has heavy tail, and the other lifetimes have finite mean. We get a more complex dynamics by assuming in case (2) that the most mobile particle type corresponds to a finite-mean lifetime: in this case, local extinction of the population is determined by an interaction of the parameters (offspring variability, mobility, longevity) of the long-living type and those of the most mobile type. The proofs are based on a precise analysis of the occupation times of a related Markov renewal process, which is of independent interest.