Stochastic wave-kinetic theory in the Liouville approximation

The behavior of scalar wave propagation in a wide class of asymptotically conservative, dispersive, weakly inhomogeneous and weakly nonstationary, anisotropic,random media is investigated on the basis of a stochastic, collisionless, Liouville_type equation governing the temporal evolution of a phase_space Wigner distribution density function. Within the framework of the first_order smoothing approximation, a general diffusion-convolution_type kinetic or transport equation is derived for the mean phase_space distribution function containing generalized (nonloral, with memory) diffusion,friction, and absorption operators in phase space. Various levels of simplification are achieved by introducing additional constraints. In the long_time, Markovian, diffusion approximation, a general set of Fokker-Planck equations is derived. Finally, special cases of these equations are examined for spatially homogeneous systems and isotropic media.