  Conditioning Markov processes to avoid a set is a classical problem that has
been studied in many settings. In the present article we study the question if
a Levy process can be conditioned to avoid an interval and, if so, the path
behavior of the conditioned process. For Levy processes with finite second
moments we show that conditioning is possible and identify the conditioned
process as an h-transform of the original killed process. The h-transform is
explicit in terms of successive overshoot distributions and is used to prove
that the conditioned process diverges to plus infinity and minus infinity with
positive probabilities.
