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

Finitude of physical measures for random maps