White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation for Wave Beams in Turbulent Media

TL;DR

This paper derives the white-noise and geometrical optics limits of the Wigner-Moyal equation for wave beams propagating in turbulent media, showing convergence to Gaussian white-noise models under certain scaling regimes.

Contribution

It establishes the rigorous convergence of the Wigner-Moyal equation to Gaussian white-noise models in wave and geometrical optics regimes, including the characterization via martingale problems.

Findings

Wave beam regime leads to white-noise scaling limit.

Convergence to Gaussian white-noise model characterized by stochastic differential equations.

Closed-form equations govern the n-point correlation functions in the white-noise model.

Abstract

Starting with the Wigner distribution formulation for beam wave propagation in H\"{o}lder continuous non-Gaussian random refractive index fields we show that the wave beam regime naturally leads to the white-noise scaling limit and converges to a Gaussian white-noise model which is characterized by the martingale problem associated to a stochastic differential-integral equation of the It\^o type. In the simultaneous geometrical optics the convergence to the Gaussian white-noise model for the Liouville equation is also established if the ultraviolet cutoff or the Fresnel number vanishes sufficiently slowly. The advantage of the Gaussian white-noise model is that its $n$-point correlation functions are governed by closed form equations.