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

TL;DR

This paper determines the wave speed of a Fisher-KPP equation influenced by jumps, using probabilistic duality with branching Brownian motions, providing an explicit formula based on the noise's impact distribution.

Contribution

It introduces a novel explicit formula for the wave speed of a Fisher-KPP equation with jump-driven noise, using a duality approach with coordinated branching Brownian motions.

Findings

Explicit wave speed formula involving the impact distribution.

Unique wave speed exists and is positive.

Method based on upper and lower bounds via quenched duality.

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.