Affiliations: Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, USA | Laboratoire de Probabilités et Modèles Aléatoires and Laboratoire Jacques-Louis Lions, Université Paris VII, Paris, France
Note: [] Corresponding author: Wenjia Jing, Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA. E-mail: [email protected].
Abstract: We consider an elliptic pseudo-differential equation with a highly oscillating linear potential modeled as a stationary ergodic random field. The random field is a function composed with a centered long-range correlated Gaussian process. In the limiting of vanishing correlation length, the heterogeneous solution converges to a deterministic solution obtained by averaging the random potential. We characterize the deterministic and stochastic correctors. With proper rescaling, the mean-zero stochastic corrector converges to a Gaussian random process in probability and weakly in the spatial variables. In addition, for two prototype equations involving the Laplacian and the fractional Laplacian operators, we prove that the limit holds in distribution in some Hilbert spaces. We also determine the size of the deterministic corrector when it is larger than the stochastic corrector. Depending on the correlation structure of the random field and on the singularities of the Green's function, we show that either the deterministic or the random part of the corrector dominates.
Keywords: corrector theory, random homogenization, long-range correlations, Gaussian random field, weak convergence of probability measures
