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This paper deals with the asymptotic behavior of random oscillatory integralsin the presence of long-range dependence. As a byproduct, we solve thecorrector problem in random homogenization of one-dimensional ellipticequations with highly oscillatory random coefficients displaying long-rangedependence, by proving convergence to stochastic integrals with respect toHermite processes.

Convergence of random oscillatory integrals in the presence of long-range dependence and application to homogenization