  We consider multiple time scales systems of stochastic differential equations
with small noise in random environments. We prove a quenched large deviations
principle with explicit characterization of the action functional. The random
medium is assumed to be stationary and ergodic. In the course of the proof we
also prove related quenched ergodic theorems for controlled diffusion processes
in random environments that are of independent interest. The proof relies
entirely on probabilistic arguments, allowing to obtain detailed information on
how the rare event occurs. We derive a control, equivalently a change of
measure, that leads to the large deviations lower bound. This information on
the change of measure can motivate the design of asymptotically efficient Monte
Carlo importance sampling schemes for multiscale systems in random
environments.
