Guillaume BalDepartment of Applied Physics & Applied Mathematics, Columbia University, New York, NY 10027Josselin GarnierLaboratoire de Probabilit´es et Mod`eles Al´eatoires & Laboratoire Jacques-Louis Lions, Universit´e Paris VII, 2 Place Jussieu, 75251 Paris Cedex 5, FranceYu GuDepartment of Applied Physics & Applied Mathematics, Columbia University, New York, NY 10027Wenjia JingDepartment of Applied Physics & Applied Mathematics, Columbia University, New York, NY 10027
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.
Habib AmmariD ́epartement de Math ́ematiques et Applications, ́Ecole Normale Sup ́erieure, 75230 Paris Cedex05, FranceJosselin GarnierLaboratoire de Probabilit ́es et Mod`eles Al ́eatoires & Laboratoire Jacques-Louis Lions, Universit ́eParis VII, 75205 Paris Cedex 13, FranceWenjia JingD ́epartement de Math ́ematiques et Applications, ́Ecole Normale Sup ́erieure, 75230 Paris Cedex05, France
We consider reflector imaging in a weakly random waveguide. We address the situation in which the source is farther from the reflector to be imaged than the energy equipartition distance, but the receiver array is closer to the reflector to be imaged than the energy equipartition distance. As a consequence, the reflector is illuminated by a partially coherent field and the signals recorded by the receiver array are noisy. This paper shows that migration of the recorded signals cannot give a good image, but an appropriate migration of the cross correlations of the recorded signals can give a very good image. The resolution and stability analysis of this original functional shows that the reflector can be localized with an accuracy of the order of the wavelength even when the receiver array has small aperture, and that broadband sources are necessary to ensure statistical stability, whatever the aperture of the array.