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# Communications in Mathematical Sciences

## Volume 14 (2016)

### Number 4

### Population stabilization in branching Brownian motion with absorption and drift

Pages: 973 – 985

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n4.a5

#### Author

#### Abstract

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

Published 6 April 2016