Transport and diffusion of high frequency waves in random media: speckle formation and the Gaussian conjecture

A well-known conjecture in physical literature states that high frequency waves propagating over long distances through turbulence eventually follows Gaussian statistics, with the second moment following a transport equation. 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. 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 Gaussian distribution is fully characterized by its first and second moments.