The unscaled paths of branching Brownian motion

TL;DR

This paper investigates the growth and fluctuations of particle paths in dyadic branching Brownian motion without rescaling, providing new probabilistic results on large deviations, oscillations, and frontier behavior.

Contribution

It introduces a novel probabilistic approach to analyze unscaled paths in branching Brownian motion, extending understanding of particle growth and fluctuations.

Findings

Particles can oscillate dramatically in number.

New large deviations probabilities are established.

Results include behavior near the frontier of the model.

Abstract

For a set $A\subset C[0,\infty)$, we give new results on the growth of the number of particles in a dyadic branching Brownian motion whose paths fall within A. We show that it is possible to work without rescaling the paths. We give large deviations probabilities as well as a more sophisticated proof of a result on growth in the number of particles along certain sets of paths. Our results reveal that the number of particles can oscillate dramatically. As a byproduct of our methods we also obtain new results on the number of particles near the frontier of the model. The methods used are entirely probabilistic.