The number of absorbed individuals in branching Brownian motion with a barrier
Résumé
We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift $c$. At the point $x > 0$, we add an absorbing barrier, i.e.\ individuals touching the barrier are instantly killed without producing offspring. It is known that there is a critical drift $c_0$, such that this process becomes extinct almost surely if and only if $c \ge c_0$. In this case, if $Z_x$ denotes the number of individuals absorbed at the barrier, we give an equivalent for $P(Z_x=n)$ as $n$ goes to infinity. In the case of a $b$-ary offspring distribution, this answers a conjecture by David Aldous about the total progeny of the process, originally stated for the branching random walk.
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