Branching Brownian Motion with Decay of Mass and the Nonlocal Fisher-KPP Equation

In this work we study a nonlocal version of the Fisher-KPP equation,{ partial differential u partial differential t=12 Delta u+u(1-phi*u),t>0, x is an element of Double-struck capital R,u(0,x)=u0(x),x is an element of Double-struck capital Rand its relation to a branching Brownian motion with decay of mass as introduced in Addario-Berry and Penington (2015) , 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 Double-struck capital 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 neighborhood around them (as measured by the function phi). We obtain two types of results. First, we study the behavior 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. Second, we show that the hydrodynamic limit of the BBM with decay of mass is the solution of the nonlocal Fisher-KPP equation. We then harness this to obtain several new results concerning the behavior of the particle system. (c) 2019 Wiley Periodicals, Inc.