We consider a (one-dimensional) branching Brownian motion process with a general offspring distribution having at least two moments, and in which all particles have a drift towards the origin where they are immediately absorbed. It is well-known that if and only if the branching rate is sufficiently large, then the population survives forever with positive probability. We show that throughout this super-critical regime, the number of particles inside any fixed set normalized by the mean population size converges to an explicit limit, almost surely and in $L^1$. As a consequence, we get that almost surely on the event of eternal survival, the empirical distribution of particles converges weakly to the (minimal) quasi-stationary distribution associated with the Markovian motion driving the particles. This proves a result of Kesten from 1978, for which no proof was available until now.

中文翻译：

---

具有吸收的超临界分支布朗运动的强大数定律

我们考虑一个（一维）分支布朗运动过程，其一般子代分布至少具有两个矩，其中所有粒子都向原点漂移，在那里它们立即被吸收。众所周知，当且仅当分支率足够大时，种群才能以正概率永远存活。我们表明，在整个超临界状态下，由平均种群大小归一化的任何固定集合内的粒子数量几乎可以肯定地收敛到一个明确的限制，并且在 $L^1$。因此，我们几乎可以肯定，在永恒存在的事件中，粒子的经验分布弱收敛到与驱动粒子的马尔可夫运动相关的（最小）准平稳分布。这证明了 Kesten 从 1978 年的结果，