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

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