In this talk we consider a stochastic delay differential equation driven by a general Levy process. Both, the drift and the noise term may depend on the past, but only the drift term is assumed to be linear. It turns out that the segment process is eventually Feller, but in general not eventually strong Feller on the Skorokhod space.

Therefore, standard methods fail to prove the existence of an invariant measure. But the existence can be derived by proving tightness of the segments using semimartingale characteristics and the Krylov-Bogoliubov method. Some ideas of this approach will be introduced in the talk.

A counterexample shows that the stationary solution in completely general situations may not be unique, but in more specific cases uniqueness isestablished.