The front location in BBM with decay of mass

TL;DR

This paper studies a modified branching Brownian motion model where particles' masses decay due to local competition, and finds that the front location scales as  t^{1/3} behind the standard BBM front over large times.

Contribution

It introduces a competitive interaction with mass decay in BBM and characterizes the front displacement scaling as  t^{1/3} in this new model.

Findings

Front location is  t^{1/3} behind standard BBM.

Mass decay influences the spatial spread of particles.

The model exhibits a different scaling behavior than classical BBM.

Abstract

We augment standard branching Brownian motion by adding a competitive interaction between nearby particles. Informally, when particles are in competition, the local resources are insufficient to cover the energetic cost of motion, so the particles' masses decay. In standard BBM, we may define the front displacement at time $t$ as the greatest distance of a particle from the origin. For the model with masses, it makes sense to instead define the front displacement as the distance at which the local mass density drops from $\Theta(1)$ to $o(1)$. We show that one can find arbitrarily large times $t$ for which this occurs at a distance $\Theta(t^{1/3})$ behind the front displacement for standard BBM.