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

TL;DR

This paper establishes a strong law of large numbers for super-critical branching Brownian motion with absorption, showing convergence of normalized particle counts and empirical distribution to explicit limits, confirming a longstanding conjecture.

Contribution

It proves a strong law of large numbers for the process, providing the first rigorous proof of Kesten's 1978 conjecture on the asymptotic behavior of particles.

Findings

Normalized particle counts converge almost surely and in L^1

Empirical distribution converges weakly to the quasi-stationary distribution

Results hold throughout the super-critical regime

Abstract

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.