  We study the limiting probability distribution of the homogenization error
for second order elliptic equations in divergence form with highly oscillatory
periodic conductivity coefficients and highly oscillatory stochastic potential.
The effective conductivity coefficients are the same as those of the standard
periodic homogenization, and the effective potential is given by the mean. We
show that in the short range correlation setting, the limiting distribution of
the random part of the homogenization error, as random elements in proper
Hilbert spaces, is Gaussian and can be characterized by the homogenized Green's
function, the homogenized solution and the statistics of the random potential.
Similar results hold for random potentials that are functions of long range
correlated Gaussian random fields. These results generalize previous ones in
the setting with slowly varying diffusion coefficients, and the current setting
with fast oscillations in the differential operator requires new methods to
prove compactness of the probability distributions of the random fluctuation.
