The Annals of Probability

Travelling Waves in Inhomogeneous Branching Brownian Motions. II

S. Lalley and T. Sellke

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Abstract

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.

Article information

Source
Ann. Probab., Volume 17, Number 1 (1989), 116-127.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991498

Digital Object Identifier
doi:10.1214/aop/1176991498

Mathematical Reviews number (MathSciNet)
MR972775

Zentralblatt MATH identifier
0692.60064

JSTOR
links.jstor.org

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G55: Point processes 60F05: Central limit and other weak theorems

Keywords
Inhomogeneous branching Brownian motion travelling wave extreme value distribution Feynman-Kac formula

Citation

Lalley, S.; Sellke, T. Travelling Waves in Inhomogeneous Branching Brownian Motions. II. Ann. Probab. 17 (1989), no. 1, 116--127. doi:10.1214/aop/1176991498. https://projecteuclid.org/euclid.aop/1176991498


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