Open Access
January, 1989 Travelling Waves in Inhomogeneous Branching Brownian Motions. II
S. Lalley, T. Sellke
Ann. Probab. 17(1): 116-127 (January, 1989). DOI: 10.1214/aop/1176991498


We study an inhomogeneous branching Brownian motion in which individual particles execute standard Brownian movements and reproduce at rates depending on their locations. The rate of reproduction for a particle located at $x$ is $\beta(x) = b + \beta_0(x)$, where $\beta_0(x)$ is a nonnegative, continuous, integrable function. Let $M(t)$ be the position of the rightmost particle at time $t$; then as $t \rightarrow \infty, M(t) - \operatorname{med}(M(t))$ converges in law to a location mixture of extreme value distributions. We determine $\operatorname{med}(M(t))$ to within a constant $+ o(1)$. The rate at which $\operatorname{med}(M(t)) \rightarrow \infty$ depends on the largest eigenvalue $\lambda$ of a differential operator involving $\beta(x)$; the cases $\lambda < 2, \lambda = 2$ and $\lambda > 2$ are qualitatively different.


Download Citation

S. Lalley. T. Sellke. "Travelling Waves in Inhomogeneous Branching Brownian Motions. II." Ann. Probab. 17 (1) 116 - 127, January, 1989.


Published: January, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0692.60064
MathSciNet: MR972775
Digital Object Identifier: 10.1214/aop/1176991498

Primary: 60J80
Secondary: 60F05 , 60G55

Keywords: extreme value distribution , Feynman-Kac formula , Inhomogeneous branching Brownian motion , Travelling wave

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 1 • January, 1989
Back to Top