  This work concerns the asymptotic analysis of high-frequency wave propagation
in randomly layered media with fast variations and long-range correlations. The
analysis takes place in the 3D physical space and weak-coupling regime. The
role played by the slow decay of the correlations on a propagating pulse is two
fold. First we observe a random travel time characterized by a fractional
Brownian motion that appears to have a standard deviation larger than the pulse
width, which is in contrast with the standard O'Doherty-Anstey theory for
random propagation media with mixing properties. Second, a deterministic pulse
deformation is described as the solution of a paraxial wave equation involving
a pseudo-differential operator. This operator is characterized by the
autocorrelation function of the medium fluctuations. In case of fluctuations
with long-range correlations this operator is close to a fractional Weyl
derivative whose order, between 1 and 2, depends on the power decay of the
autocorrelation function. In the frequency domain, the pseudo-differential
operator exhibits a frequency-dependent power law attenuation with exponent
corresponding to the order of the fractional derivative, and a
frequency-dependent phase modulation, both ensuring the causality of the
limiting paraxial wave equation as well as the Kramers-Kronig relations. The
mathematical analysis is based on an approximation-diffusion theorem for random
ordinary differential equation with long-range correlations.
