  For random compositions of independent and identically distributed measurable
maps on a Polish space, we study the existence and finitude of absolutely
continuous ergodic stationary probability measures (which are, in particular,
physical measures) whose basins of attraction cover the whole space almost
everywhere. We characterize and hierarchize such random maps in terms of their
associated Markov operators, as well as show the difference between classes in
the hierarchy by plenty of examples, including additive noise, multiplicative
noise, and iterated function systems. We also provide sufficient practical
conditions for a random map to belong to these classes. For instance, we
establish that any continuous random map on a compact Riemannian manifold with
absolutely continuous transition probability has finitely many physical
measures whose basins of attraction cover Lebesgue almost all the manifold.
