  In this paper we study a one-dimensional space-discrete transport equation
subject to additive Levy forcing. The explicit form of the solutions allows
their analytic study. In particular we discuss the invariance of the covariance
structure of the stationary distribution for Levy perturbations with finite
second moment. The situation of more general Levy perturbations lacking the
second moment is considered as well. We moreover show that some of the
properties of the solutions are pertinent to a discrete system and are not
reproduced by its continuous analogue.
