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

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.