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Branching Brownian motion with decay of mass and the non-...
Louigi Addario-Berry, Julien Berestycki, Sarah Penington · 2017-12-22 · via math.PR updates on arXiv.org

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}Δu +u (1- φ\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 $φ$). 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 $φ$ 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.