For a continuous-time Bienaymé–Galton–Watson process, X, with immigration and culling, 0 as an absorbing state, call $X^q$ the process that results from killing X at rate $q\in (0,\mathrm{\infty })$, followed by stopping it on extinction or explosion. Then an explicit identification of the relevant harmonic functions of $X^q$ allows to determine the Laplace transforms (at argument q) of the first passage times downwards and of the explosion time for X. Strictly speaking, this is accomplished only when the killing rate q is sufficiently large (but always when the branching mechanism is not supercritical or if there is no culling). In particular, taking the limit $q\downarrow 0$ (whenever possible) yields the passage downwards and explosion probabilities for X. A number of other consequences of these results are presented.

对于连续时间的 Bienaymé-Galton-Watson 过程，X，具有迁移和剔除，0 作为吸收状态，调用$X^q$以速率杀死X所产生的过程$q\in (0,\mathrm{\infty })$，然后在灭绝或爆炸时停止它。然后明确识别相关的调和函数$X^q$允许确定向下的第一次通过时间和X的爆炸时间的拉普拉斯变换（在参数q处） 。严格来说，这只有在杀死率q足够大时才能实现（但总是在分支机制不是超临界或没有剔除的情况下）。特别是，采取限制$q\downarrow 0$（尽可能）产生X的向下通道和爆炸概率。提出了这些结果的许多其他后果。