  We study a one-dimensional elliptic problem with highly oscillatory random
diffusion coefficient. We derive a homogenized solution and a so-called
Gaussian corrector. We also prove a "pointwise" large deviation principle (LDP)
for the full solution and approximate this LDP with a more tractable form.
These results allow one to access the limits of Gaussian correctors. In
general, the corrector does not capture the large deviation behavior.
Applications to uncertainty quantification are considered.
