  Exponential functionals of L\'evy processes appear as stationary
distributions of generalized Ornstein-Uhlenbeck (GOU) processes. In this paper
we obtain the infinitesimal generator of the GOU process and show that it is a
Feller process. Further we use these results to investigate properties of the
mapping \Phi, which maps two independent L\'evy processes to their
corresponding exponential functional, where one of the processes is assumed to
be fixed. We show that in many cases this mapping is injective and give the
inverse mapping in terms of (L\'evy) characteristics. Also, continuity of \Phi
is treated and some results on its range are obtained.
