Sarah Penington

Branching Brownian motion with decay of mass and the non-local Fisher-KPP equation

We add a competitive interaction between nearby particles in a branching Brownian motion (BBM). Each particle has a mass, which decays at rate proportional to the local mass density at its location. The total mass increases through branching events.
In standard BBM, we may define the front location at time t as the greatest distance of a particle from the origin. For BBM with decay of mass, it makes sense to instead define the front displacement as the distance at which the local mass density drops from Θ(1) to o(1). We can show that in a weak sense this front is ~ c t^{1/3} behind the front for standard BBM.
At large times, over a bounded time interval, the local mass density for BBM with decay of mass is well approximated by a solution of the non-local Fisher-KPP equation. This allows us to control the behaviour of the local mass density behind the front.
This is joint work with Louigi Addario-Berry and Julien Berestycki.