  We study the large deviations principle for locally periodic stochastic
differential equations with small noise and fast oscillating coefficients.
There are three possible regimes depending on how fast the intensity of the
noise goes to zero relative to the homogenization parameter. We use weak
convergence methods which provide convenient representations for the action
functional for all three regimes. Along the way we study weak limits of related
controlled SDEs with fast oscillating coefficients and derive, in some cases, a
control that nearly achieves the large deviations lower bound at the prelimit
level. This control is useful for designing efficient importance sampling
schemes for multiscale diffusions driven by small noise.
