Right marker speeds of solutions to the KPP equation with noise

TL;DR

This paper studies the stochastic KPP equation with noise, demonstrating the existence of wave solutions traveling at positive deterministic speeds for parameters above a critical threshold, using stopping times and averaging techniques.

Contribution

It establishes the existence of stochastic wave solutions with positive speeds in the noisy KPP equation and provides conditions for initial states to achieve these speeds.

Findings

Existence of stochastic wave solutions traveling at positive speeds

Conditions on initial conditions for attaining wave speed

Application of stopping times and averaging techniques to analyze solutions

Abstract

We consider the one-dimensional KPP-equation driven by space-time white noise. We show that for all parameters above the critical value for survival, there exist stochastic wavelike solutions which travel with a deterministic positive linear speed. We further give a sufficient condition on the initial condition of a solution to attain this speed. Our approach is in the spirit of corresponding results for the nearest-neighbor contact process respectively oriented percolation. Here, the main difficulty arises from the moderate size of the parameter and the long range interaction. Stopping times and averaging techniques are used to overcome this difficulty.