Population stabilization in branching Brownian motion with absorption and drift

We consider, through PDE methods, branching Brownian motion with drift and absorption. It is well known that there exists a critical drift which separates those processes which die out almost surely from those which survive with positive probability. In this work, we consider lower-order corrections to the critical drift which ensures a nonnegative, bounded expected number of particles and convergence of this expectation to a limiting nonnegative number which is positive for some initial data. In particular, we show that the average number of particles stabilizes at the convergence rate $O(\log (t)/t)$ if and only if the multiplicative factor of the $O(t^{- 1/2})$ correction term is ${3\sqrt{\pi} t}^{- 1/2}$. Otherwise, the convergence rate is $O(1/ \sqrt{t})$. We point out some connections between this work and recent work investigating the expansion of the front location for the initial value problem in Fisher–KPP.

Keywords

branching Brownian motion, absorption, population dynamics, selection models, Fisher–KPP, delay, Bramson correction

2010 Mathematics Subject Classification

35K57, 92D25

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Published 5 June 2023