Radiative Transfer with Long-Range Interactions: Regularity and Asymptotics

This work is devoted to radiative transfer equations with long-range interactions. Such equations arise in the modeling of high frequency wave propagation in random media with long-range dependence. In the regime we consider, the singular collision operator modeling the interaction between the wave and the medium is conservative, and as a consequence wavenumbers take values on the unit sphere. Our goal is to investigate the regularizing effects of grazing collisions, the diffusion limit, and the peaked-forward limit. As in the case where wavenumbers take values in Rd+1, we show that the transport operator is hypoelliptic, which implies in particular that the solutions are infinitely differentiable in all variables. Using probabilistic techniques, we show as well that the diffusion limit can be carried out as in the case of a regular collision operator and that as a consequence, the diffusion coefficient is nonzero and finite. Finally, we consider the regime where grazing collisions are dominant.