| Abstract: |
| A well-known conjecture in physical literature states that high frequency waves propagating over long distances through turbulence eventually become complex Gaussian distributed. The intensity of such wave fields then follows an exponential law, consistent with speckle formation observed in physical experiments. Though fairly well-accepted and intuitive, this conjecture is not entirely supported by any detailed mathematical derivation. In this talk, I will discuss some recent results demonstrating the Gaussian conjecture in a weak-coupling regime of the paraxial approximation.
The paraxial approximation is a high frequency approximation of the Helmholtz equation, where backscattering is ignored. This takes the form of a Schr\{o}dinger equation with a random potential and is often used to model laser propagation through turbulence. The proof relies on the asymptotic closeness of statistical moments of the wavefield under the paraxial approximation, its white noise limit and the complex Gaussian distribution itself. I will describe two scaling regimes, one is a kinetic scaling where the second moment is given by a transport equation and a second diffusive scaling, where the second moment follows an anomalous diffusion. In both cases, the limiting complex Gaussian distribution is fully characterized by its first and second moments. An additional stochastic continuity/tightness criterion allows to show the convergence of these distributions over spaces of H\{o}lder-continuous functions.
This is joint work with Guillaume Bal. |
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