  We develop an analytical tool which is adept for detecting shapes of
oscillatory functions, is useful in decomposing homogenization problems into
limit-problems for kinetic equations, and provides an efficient framework for
the validation of multi-scale asymptotic expansions. We apply it first to a
hyperbolic homogenization problem and transform it to a hyperbolic limit
problem for a kinetic equation. We establish conditions determining an
effective equation and counterexamples for the case that such conditions fail.
Second, when the kinetic decomposition is applied to the problem of enhanced
diffusion, it leads to a diffusive limit problem for a kinetic equation that in
turn yields the effective equation of enhanced diffusion.
