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White-noise and geometrical optics limits of Wigner-Moyal equation for beam waves in turbulent media. II: Two-frequency formulation. (English) Zbl 1081.78004
The subject of the paper is the understanding of stochastic pulse propagation analysis which is usually based on spectral decomposition of the time dependent signal into time-harmonic wave fields. So the complete information about transient propagation requires a solution for the statistical moments of the wave field at different frequencies and locations. The first aim of this work is to establish a general two-frequency framework and the second to use this framework to get the concrete rigorous results. To study the scaling limits of the wave propagation in a turbulent medium at two different frequencies the author introduces a two-frequency Wigner distribution in the setting of the parabolic approximation. He shows that the two-frequency Wigner distribution satisfies a closed-form equation (the two-frequency Wigner-Moyal equation). It was shown, in the white-noise limit, the convergence of weak solutions of the two-frequency Wigner-Moyal equation to a Markovian model and thus proved rigorously the Markovian approximation with power-spectral densities. He also proved the convergence of the simultaneous geometrical optics limit whose mean field equation has a simple, universal form and is exactly solvable.
For Part I, cf. Commun. Math. Phys. 254, No. 2, 289–322 (2005).

78A05 Geometric optics
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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[3] Bailly F., Fouque J.-P. High frequency wave propagation in random media.unpublished · Zbl 0859.60061
[5] Fannjiang A. White-noise and geometrical optics limits of Wigner–Moyal equation for wave beams in turbulent media, Commun. Math. Phys., in press · Zbl 1077.60050
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