Brownian Bees with Drift: Finding the Criticality

This dissertation examines the impact of a drift {\mu} on Brownian Bees, which is a type of branching Brownian motion that retains only the N closest particles to the origin. The selection effect in the 0-drift system ensures that it remains recurrent and close to the origin. The study presents two novel findings that establish a threshold for {\mu}: below this value, the system remains recurrent, and above it, the system becomes transient. Moreover, the paper proves convergence to a unique invariant distribution for the small drift case. The research also explores N-BBM, a variant of branching Brownian motion where the N leftmost particles are retained, and presents one new result and further discussion on this topic.