Spread rate of branching Brownian motions

TL;DR

This paper determines the exponential growth rate of particles outside a growing ball in branching Brownian motion, linking it to the principal eigenvalue of an associated operator, with applications to various models.

Contribution

It introduces a method to calculate the growth rate of branching Brownian particles outside a time-dependent radius, considering singular and small branching rate measures.

Findings

Growth rate linked to principal eigenvalue of Schrödinger-type operator

Upper bound of particle range depends on branching mechanism

Applicable to models with singular and measure-dependent branching rates

Abstract

We find the exponential growth rate of the population outside a ball with time dependent radius for a branching Brownian motion in Euclidean space. We then see that the upper bound of the particle range is determined by the principal eigenvalue of the Schr\"odinger type operator associated with the branching rate measure and branching mechanism. We assume that the branching rate measure is small enough at infinity, and can be singular with respect to the Lebesgue measure. We finally apply our results to several concrete models.