Branching Brownian motion: Almost sure growth along unscaled paths

TL;DR

This paper investigates the almost sure growth of particles in branching Brownian motion along fixed paths without rescaling, revealing oscillations and employing probabilistic spine techniques.

Contribution

It introduces new probabilistic results on particle growth along unscaled paths, utilizing spine methods to analyze oscillations in branching Brownian motion.

Findings

Number of particles can oscillate dramatically along certain paths

Growth can be counted without path rescaling

Probabilistic methods effectively analyze pathwise growth

Abstract

We give new results on the growth of the number of particles in a dyadic branching Brownian motion which follow within a fixed distance of a path $f:[0,\infty)\to \mathbb{R}$. We show that it is possible to count the number of particles without rescaling the paths. Our results reveal that the number of particles along certain paths can oscillate dramatically. The methods used are entirely probabilistic, taking advantage of the spine technique developed by, amongst others, Lyons et al, Kyprianou, and Hardy & Harris.