The wave speed of an FKPP equation with jumps via coordinated branching

Abstract

We consider a Fisher–KPP equation with nonlinear selection driven by a Poisson random measure. We prove that the equation admits a unique wave speed $\mathfrak{s}>0$ given by

$$\frac{{\mathfrak{s}}^2}{2}={\int }_{[0,1]}\frac{log(1+y)}{y}\mathfrak{R}(\mathrm{d}y),$$

where $\mathfrak{R}$ is the intensity of the impacts of the driving noise. Our arguments are based on upper and lower bounds via a quenched duality with a coordinated system of branching Brownian motions.