One-dimensional space-discrete transport subject to Levy perturbations

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.