Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential

TL;DR

This paper analyzes a branching Brownian motion with inhomogeneous breeding rates based on distance from the origin, providing large deviations, growth rates, and optimal paths for particles.

Contribution

It introduces a detailed analysis of large deviations and growth dynamics in a branching Brownian motion with distance-dependent breeding rates.

Findings

Derived large deviations probabilities for particle paths

Calculated growth rates along optimal paths

Identified the total population growth rate

Abstract

We consider a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power $p$, for $p\in[0,2)$. The asymptotic behaviour of the right-most particle for this system is already known; in this article we give large deviations probabilities for particles following "difficult" paths, growth rates along "easy" paths, the total population growth rate, and we derive the optimal paths which particles must follow to achieve this growth rate.