Exponential growth rates in a typed branching diffusion

TL;DR

This paper analyzes the long-term behavior of a typed branching diffusion process, establishing almost-sure limits for population growth and spatial-type colonization speeds using martingale techniques.

Contribution

It introduces explicit two-phase mechanisms and martingale-based methods to determine asymptotic growth and spread rates in typed branching diffusions.

Findings

Particles build up near -γt in space and κ√t in type at large times

Almost-sure limits for population growth rates are established

Methodologies can be adapted to other branching processes

Abstract

We study the high temperature phase of a family of typed branching diffusions initially studied in [Ast\'{e}risque 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 $-\gamma t$ 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.