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Previous work has shown that a primary gravity wave of sufficient amplitude will always decay through resonant wave-wave interactions with two secondary gravity waves. The interaction among the resonant trio can be reasonably rapid and may be an important process responsible for energy exchange among gravity waves of different wavelengths in the atmosphere. By taking statistical ensemble averages to the interaction equations we obtain an hierarchy of moment equations, the closure condition of which can be effected by making a random phase approximation. The obtained kinetic equation is Boltzmann-like and describes the time evolution of wave action. The nonlinear kinetic equation is impossible to solve in general except numerically, but the equation is drastically simplified in three limiting cases identified as elastic scattering, parametric subharmonic instability, and induced diffusion processes. Through elastic scattering, an upgoing wave is scattered into a downgoing wave by interacting resonantly with a vertical shear. This process is thus responsible for making the atmospheric spectrum vertically symmetric if it is not so initially. The parametric subharmonic instability process is responsible for transferring energetic large-scale waves to small-scale waves at one-half the frequency. The induced diffusion process is responsible for time evolution of small-scale waves by a three-wave process involving two nearly identical waves of small scales interacting resonantly with a large-scale vertical shear. The rapidity with which these three processes take place in the atmosphere and their implication in the atmospheric spectrum are investigated and discussed.