# Abstract:

This is a joint work with M. Volker Mayer It concerns random dynamics of transcendental functions $f:\C\to \cbar$. We will establish the existence of randmom conformal measures and their invariant versions. An appropriately defined spectral gap property will be shown. In classical situations there is a natural and powerful proof of this property Which stems from Birkhoff's Contraction Principle for operators preserving a positive cone. This method however fails in our non-compact situation. We will nevertheless define appropriate invariant cones of positive functions and will revive an old approach of Bowen to overcome this difficulty.