We consider the genealogical tree of a stationary continuous state branching process with immigration. For a sub-critical stable branching mechanism, we consider the genealogical tree of the extant population at some fixed time and prove that, up to a deterministic time-change, it is distributed as a continuous-time Galton–Watson process with immigration. We obtain similar results for a critical stable branching mechanism when only looking at immigrants arriving in some fixed time-interval. For a general sub-critical branching mechanism, we consider the number of individuals that give descendants in the extant population. The associated processes (forward or backward in time) are pure-death or pure-birth Markov processes, for which we compute the transition rates.

我们考虑具有移民的平稳连续状态分支过程的系谱树。对于亚临界稳定分支机制，我们考虑在某个固定时间现存种群的谱系树，并证明，直到确定性时间变化，它作为具有移民的连续时间 Galton-Watson 过程分布。当仅查看在某个固定时间间隔内到达的移民时，我们获得了关键稳定分支机制的类似结果。对于一般的亚临界分支机制，我们考虑在现存种群中产生后代的个体数量。相关的过程（在时间上向前或向后）是纯死亡或纯出生马尔可夫过程，我们计算转换率。