  We initiate the study of random iteration of automorphisms of real and
complex projective surfaces, or more generally compact K{\"a}hler surfaces,
focusing on the fundamental problem of classification of stationary measures.
We show that, in a number of cases, such stationary measures are invariant, and
provide criteria for uniqueness, smoothness and rigidity of invariant
probability measures. This involves a variety of tools from complex and
algebraic geometry, random products of matrices, non-uniform hyperbolicity, as
well as recent results of Brown and Rodriguez Hertz on random iteration of
surface diffeomorphisms.
