A strong law of large numbers for super-critical branching Brownian motion with absorption
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Publication:2658882
DOI10.1007/S10955-020-02620-1zbMATH Open1458.60096arXiv1708.08440OpenAlexW3048117288MaRDI QIDQ2658882
Author name not available (Why is that?)
Publication date: 25 March 2021
Published in: (Search for Journal in Brave)
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 . 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.
Full work available at URL: https://arxiv.org/abs/1708.08440
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