Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media

This work is devoted to the asymptotic analysis of high frequency wave propagation in random media with long-range dependence. We are interested in two asymptotic regimes, that we investigate simultaneously: the paraxial approximation, where the wave is collimated and propagates along a privileged direction of propagation, and the white-noise limit, where random uctuations in the background are well approximated in a statistical sense by a fractional white noise. The fractional nature of the uctuations is reminiscent of the long-range correlations in the underlying random medium. A typical physical setting is laser beam propagation in turbulent atmosphere. Starting from the high frequency wave equation with fast non-Gaussian random oscillations in the velocity eld, we derive the fractional Itô-Schrödinger equation, that is a Schrödinger equation with potential equal to a fractional white noise. The proof involves a ne analysis of the backscattering and of the coupling between the propagating and evanescent modes. Because of the long-range dependence, classical di usion-approximation theorems for equations with random coe cients do not apply, and we therefore use moment techniques to study the convergence.