Central Limits and Homogenization in Random Media

TLDR: In this paper, the rescaled difference between the perturbed and unperturbed solutions may be written asymptotically as explicit Gaussian processes, and the results are derived for more general elliptic equations with random coefficients in one dimension of space.

Abstract: We consider the perturbation of elliptic pseudodifferential operators $P(\mathbf{x},\mathbf{D})$ with more than square integrable Green's functions by random, rapidly varying, sufficiently mixing, potentials of the form $q(\frac{\mathbf{x}}{\varepsilon},\omega)$. We analyze the source and spectral problems associated with such operators and show that the rescaled difference between the perturbed and unperturbed solutions may be written asymptotically as $\varepsilon \to 0$ as explicit Gaussian processes. Such results may be seen as central limit corrections to homogenization (law of large numbers). Similar results are derived for more general elliptic equations with random coefficients in one dimension of space. The results are based on the availability of a rapidly converging integral formulation for the perturbed solutions and on the use of classical central limit results for random processes with appropriate mixing conditions.