2017/08/28 by Louidor, Oren, Saglietti, Santiago
#60F99 #60J80 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1708.08440
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 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 from 1978, for which no proof was available until now.