Archive for Rational Mechanics and Analysis

, Volume 226, Issue 3, pp 1061–1138 | Cite as

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

  • Christophe GomezEmail author
  • Olivier Pinaud


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 fluctuations in the background are well approximated in a statistical sense by a fractional white noise. The fractional nature of the fluctuations 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 field, we derive the fractional Itô–Schrödinger equation, that is, a Schrödinger equation with potential equal to a fractional white noise. The proof involves a fine analysis of the backscattering and of the coupling between the propagating and evanescent modes. Because of the long-range dependence, classical diffusion-approximation theorems for equations with random coefficients do not apply, and we therefore use moment techniques to study the convergence.


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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Aix Marseille Univ, CNRS, Centrale Marseille, I2MMarseilleFrance
  2. 2.Department of MathematicsColorado State UniversityFort CollinsUSA

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