A uniform diffusion limit for random wave propagation with turning point

A random wave propagation problem with turning point is considered for a refractive, layered random medium. The variations of the medium structure are assumed to have two spatial scales; microscopic random fluctuations are superposed upon slowly varying macroscopic variations. An extension of a limit theorem for stochastic differential equations with multiple spatial scales is derived and proved to obtain a uniformly valid diffusion limit for random multiple scattering up to the turning point region. The scale dependence of the infinitesimal generator of the backward Kolmogorov equation provides an insight into the interplay of internal refraction and random scattering as one approaches the turning point.