Survival probabilities for branching Brownian motion with absorption. (English) Zbl 1132.60059

Authors’ abstract: We study a branching Brownian motion (BBM) with absorption, in which particles moving as Brownian motions with drift \(-\rho\), undergo dyadic branching at rate \(\beta>0\), and are killed on hitting the origin. In the case \(\rho > \sqrt{2 \beta}\) the extinction time for this process \(\zeta\) is known to be finite almost surely. The main result of this article is a large-time asymptotic formula for the survival probability \(P^x(\zeta > t)\) in the case \(\rho > \sqrt {2 \beta}\), where \(P^x\) is the law of the BBM with absorption started from a single particle at the position \(x>0\). We also introduce an additive martingale \(V\) for the BBM with absorption, and then ascertain the convergence properties of \(V\). Finally, we use \(V\) in a ‘spine’ change of measure and interpret this in terms of ‘conditioning the BBM to survive forever’ when \(\rho > \sqrt{2 \beta}\), in the sense that it is the large \(t\)-limit of the conditional probabilities \(P^x(A| \zeta>t+s)\), for \(A\in \mathcal{F}_s\).


60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G46 Martingales and classical analysis
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