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 f_0(x) = 1 ¿ (-x ¿ 0) from [17] 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. Keywords: stochastic PDE; KPP equation; white noise; travelling wave; initial conditions with compact support; recurrence