A Strong Law of Large Numbers for Super-Critical Branching Brownian Motion with Absorption

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 the population survives forever with positive probability if and only if the branching rate is sufficiently large. 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 L¹. 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 (Stoch Process Appl 7(1):9–47, 1978) from 1978, for which no proof was available until now.