  We investigate the effects of random perturbations on fully chaotic open
systems. Perturbations can be applied to each trajectory independently (white
noise) or simultaneously to all trajectories (random map). We compare these two
scenarios by generalizing the theory of open chaotic systems and introducing a
time-dependent conditionally-map-invariant measure. For the same perturbation
strength we show that the escape rate of the random map is always larger than
that of the noisy map. In random maps we show that the escape rate $\kappa$ and
dimensions $D$ of the relevant fractal sets often depend nonmonotonically on
the intensity of the random perturbation. We discuss the accuracy (bias) and
precision (variance) of finite-size estimators of $\kappa$ and $D$, and show
that the improvement of the precision of the estimations with the number of
trajectories $N$ is extremely slow ($\propto 1/\ln N$). We also argue that the
finite-size $D$ estimators are typically biased. General theoretical results
are combined with analytical calculations and numerical simulations in
area-preserving baker maps.
