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

We consider the one-dimensional KPP-equation driven by space–time white noise and extend the construction of travelling wave solutions arising from initial data f0(x)=1∧(−x∨0) from (Tribe, 1996) to f0 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 to the SPDE 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.