Annals of Applied Probability

Exponential growth rates in a typed branching diffusion

Y. Git, J. W. Harris, and S. C. Harris

Full-text: Open access

Abstract

We study the high temperature phase of a family of typed branching diffusions initially studied in [Astérisque 236 (1996) 133–154] and [Lecture Notes in Math. 1729 (2000) 239–256 Springer, Berlin]. The primary aim is to establish some almost-sure limit results for the long-term behavior of this particle system, namely the speed at which the population of particles colonizes both space and type dimensions, as well as the rate at which the population grows within this asymptotic shape. Our approach will include identification of an explicit two-phase mechanism by which particles can build up in sufficient numbers with spatial positions near −γt and type positions near $\kappa \sqrt{t}$ at large times t. The proofs involve the application of a variety of martingale techniques—most importantly a “spine” construction involving a change of measure with an additive martingale. In addition to the model’s intrinsic interest, the methodologies presented contain ideas that will adapt to other branching settings. We also briefly discuss applications to traveling wave solutions of an associated reaction–diffusion equation.

Article information

Source
Ann. Appl. Probab., Volume 17, Number 2 (2007), 609-653.

Dates
First available in Project Euclid: 19 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1174323258

Digital Object Identifier
doi:10.1214/105051606000000853

Mathematical Reviews number (MathSciNet)
MR2308337

Zentralblatt MATH identifier
1131.60077

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Spatial branching process branching diffusion multi-type branching process additive martingales spine decomposition

Citation

Git, Y.; Harris, J. W.; Harris, S. C. Exponential growth rates in a typed branching diffusion. Ann. Appl. Probab. 17 (2007), no. 2, 609--653. doi:10.1214/105051606000000853. https://projecteuclid.org/euclid.aoap/1174323258


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