Travelling wave solutions to the KPP equation with branching noise arising from initial conditions with compact support

TL;DR

This paper extends the construction of traveling wave solutions for the stochastic KPP equation with branching noise from step initial data to more general initial conditions with compact support, and explores their implications.

Contribution

It introduces a method to construct traveling wave solutions from general initial conditions and demonstrates their use in analyzing the support's recurrence and stochastic domination.

Findings

Traveling wave solutions exist for a broader class of initial conditions.

Support of solutions is shown to be recurrent.

New upper measures enable stochastic domination of solutions.

Abstract

We consider the one-dimensional KPP-equation driven by space-time white noise and extend the construction of travelling wave solutions arising from Heavyside initial data from [Tribe, 1996, MR1396765] to non-negative continuous functions with compact support. As an application the existence of travelling wave solutions is used to prove that the support of any solution is recurrent. As a by-product, several upper measures are introduced that allow for a stochastic domination of any solution to the SPDE at a fixed point in time.