  This paper presents a systematic numerical study of the effects of noise on
the invariant probability densities of dynamical systems with varying degrees
of hyperbolicity. It is found that the rate of convergence of invariant
densities in the small-noise limit is frequently governed by power laws. In
addition, a simple heuristic is proposed and found to correctly predict the
power law exponent in exponentially mixing systems. In systems which are not
exponentially mixing, the heuristic provides only an upper bound on the power
law exponent. As this numerical study requires the computation of invariant
densities across more than 2 decades of noise amplitudes, it also provides an
opportunity to discuss and compare standard numerical methods for computing
invariant probability densities.
