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We study the limiting probability distribution of the homogenization errorfor second order elliptic equations in divergence form with highly oscillatoryperiodic conductivity coefficients and highly oscillatory stochastic potential.The effective conductivity coefficients are the same as those of the standardperiodic homogenization, and the effective potential is given by the mean. Weshow that in the short range correlation setting, the limiting distribution ofthe random part of the homogenization error, as random elements in properHilbert spaces, is Gaussian and can be characterized by the homogenized Green'sfunction, the homogenized solution and the statistics of the random potential.Similar results hold for random potentials that are functions of long rangecorrelated Gaussian random fields. These results generalize previous ones inthe setting with slowly varying diffusion coefficients, and the current settingwith fast oscillations in the differential operator requires new methods toprove compactness of the probability distributions of the random fluctuation.

Limiting distribution of elliptic homogenization error with periodic diffusion and random potential