Maximal displacement and population growth for branching Brownian motions

TL;DR

This paper investigates the maximal displacement and population growth in branching Brownian motions, analyzing their growth rates, deviations, and extinction probabilities using spectral theory and eigenvalues.

Contribution

It introduces a spectral approach to quantify growth and deviations in branching Brownian motions, especially near critical phases and extinction scenarios.

Findings

Growth rates are characterized by the principal eigenvalue.

Upper deviation probabilities decay at a specific rate.

Population growth behavior is detailed at the critical phase.

Abstract

We study the maximal displacement and related population for a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of an associated Schr\"odinger type operator. We first determine their growth rates on the survival event. We then establish the upper deviation for the maximal displacement under the possibility of extinction. Under the non-extinction condition, we further discuss the decay rate of the upper deviation probability and the population growth at the critical phase.