Probabilistic Theory of Transport Processes with Polarization

We derive a probabilistic representation for solutions of matrix-valued transport equations that account for polarization effects. Such equations arise in radiative transport for the Stokes parameters that model the propagation of light through turbulent atmospheres. They also arise in radiative transport for seismic wave propagation in the earth's crust. The probabilistic representation involves an augmented scalar transport equation in which the polarization parameters become independent variables. Our main result is that the linear moments of the augmented transport equation with respect to the polarization variables are the solution of the matrix-valued transport equation. The augmented scalar transport equation is well suited to analyzing the hydrodynamic regime of small mean free paths. It is also well suited to getting approximate solutions by Monte Carlo simulation.