A Wigner-based volumetric transport framework for paraxial waves in random media

We develop a Wigner-based phase-space framework for mean paraxial wave propagation in random media. Starting from the random parabolic wave equation, we derive the exact evolution of the realization-dependent Wigner distribution and identify the ensemble-averaged Wigner function as the natural second-order state variable. The averaged equation contains a closure defect, given by a mixed field--medium correlation, which prevents a closed transport equation from being obtained without additional assumptions. We therefore organize the modelling as a hierarchy from the random wave equation to an exact Wigner formulation, then to a nonlocal kinetic closure, and finally to a local Fokker--Planck reduction in the small-angle regime. For the minimal homogeneous isotropic Fokker--Planck model, we derive closed evolution laws for the quadratic moments, exhibit the cubic-in-distance contribution to beam spreading, and obtain explicit Gaussian and Gauss--Schell propagation formulas. These analytical results are used to validate a phase-space splitting solver in one-dimensional transverse benchmarks. Comparisons with nonlocal kinetic models show that the diffusive approximation is accurate for narrow momentum-transfer kernels and loses validity in a controlled way as finite-jump effects become significant. Finally, we introduce a first atmospheric specialization based on a regularized turbulence spectrum, yielding an effective diffusion coefficient expressed in terms of standard atmospheric parameters.