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

TL;DR

This paper investigates a non-local Fisher-KPP equation linked to a branching Brownian motion model with mass decay, establishing solution behavior and connecting the particle system's hydrodynamics to the PDE.

Contribution

It proves convergence properties of the PDE solutions and demonstrates that the particle system's hydrodynamic limit aligns with the non-local Fisher-KPP equation, extending prior results.

Findings

Solutions converge to 1 behind the front

Solutions are globally bounded under certain conditions

Hydrodynamic limit of the particle system matches the PDE

Abstract

In this work we study a non-local version of the Fisher-KPP equation, \begin{equation*} \begin{cases} \frac{\partial u}{\partial t}=\tfrac{1}{2}\Delta u +u (1- \phi \ast u), \quad t>0, \quad x\in \mathbb{R}, u(0,x)=u_0(x), \quad x\in \mathbb{R} \end{cases} \end{equation*} and its relation to $\textit{branching Brownian motion with decay of mass}$ as introduced by Addario-Berry and Penington (2017), i.e. a particle system consisting of a standard branching Brownian motion (BBM) with a competitive interaction between nearby particles. Particles in the BBM with decay of mass have a position in $\mathbb{R}$ and a mass, branch at rate 1 into two daughter particles of the same mass and position, and move independently as Brownian motions. Particles lose mass at a rate proportional to the mass in a neighbourhood around them (as measured by the function $\phi$). We obtain two types of results. First, we study the behaviour of solutions to the partial differential equation above. We show that, under suitable conditions on $\phi$ and $u_0$, the solutions converge to 1 behind the front and are globally bounded, improving recent results of Hamel and Ryzhik (arxiv:arXiv:1307.3001). Second, we show that the hydrodynamic limit of the BBM with decay of mass is the solution of the non-local Fisher-KPP equation. We then harness this to obtain several new results concerning the behaviour of the particle system.