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 Schrödinger equation with a random potential and is often used to model laser propagation through turbulence. In particular, I will describe a diffusive scaling where the limiting probability distribution of the wavefield is completely described by a second moment which follows an anomalous diffusion. 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. An additional stochastic continuity/tightness criterion allows to show the convergence of these distributions over spaces of Hölder-continuous functions. Numerical simulations illustrate theoretical results.

This is joint work with Guillaume Bal.