We introduce two-frequency Wigner distribution in the setting of parabolic approximation to study the scaling limits of the wave propagation in a turbulent medium at two different frequencies. We show that the two-frequency Wigner distribution satisfies a closed-form equation (the two-frequency Wigner–Moyal equation). In the white-noise limit we show the convergence of weak solutions of the two-frequency Wigner–Moyal equation to a Markovian model and thus prove rigorously the Markovian approximation with power-spectral densities widely used in the physics literature. We also prove the convergence of the simultaneous geometrical optics limit whose mean field equation has a simple, universal form and is exactly solvable
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Fannjiang, A.C. White-Noise and Geometrical OpticsLimits of Wigner–Moyal Equation for Beam Waves in Turbulent Media II: Two-Frequency Formulation. J Stat Phys 120, 543–586 (2005). https://doi.org/10.1007/s10955-005-5961-1
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DOI: https://doi.org/10.1007/s10955-005-5961-1