  We consider systems of stochastic differential equations with multiple scales
and small noise and assume that the coefficients of the equations are ergodic
and stationary random fields. Our goal is to construct provably-efficient
importance sampling Monte Carlo methods that allow efficient computation of
rare event probabilities or expectations of functionals that can be associated
with rare events. Standard Monte Carlo algorithms perform poorly in the small
noise limit and hence fast simulations algorithms become relevant. The presence
of multiple scales complicates the design and the analysis of efficient
importance sampling schemes. An additional complication is the randomness of
the environment. We construct explicit changes of measures that are proven to
be logarithmic asymptotically efficient with probability one with respect to
the random environment (i.e., in the quenched sense). Numerical simulations
support the theoretical results.
