  We study Markov interval maps with random holes. The holes are not
necessarily elements of the Markov partition. Under a suitable, and physically
relevant, assumption on the noise, we show that the transfer operator
associated with the random open system can be reduced to a transfer operator
associated with the closed deterministic system. Exploiting this fact, we show
that the random open system admits a unique (meaningful) absolutely continuous
conditionally stationary measure. Moreover, we prove the existence of a unique
probability equilibrium measure supported on the survival set, and we study its
Hausdorff dimension.
