The recurrence criterion for uniqueness of invariant measures in random dynamical systems

Let $\mathcal{D}$ be a random dynamical system, and let its associated fundamental Markov system be a directed Markov system whose vertices represent the states of the construction. The system is recurrent when every vertex is reached from any other vertex by a finite path. A Borel probability measure $\mu$ is invariant for $\mathcal{D}$ when it is preserved by the associated Markov operator.

Recurrence criterion. The random dynamical system has a unique invariant Borel probability measure if and only if its associated fundamental Markov system is recurrent.

This proposed criterion is intended to resolve a necessary-and-sufficient-condition question for the stability of random dynamical systems that the source describes as having been open for more than 70 years. The statement also observes that a recurrent Markov system is necessarily countable, since every vertex can be coded by a finite path; no resolution status is supplied in the source.