  We perform the asymptotic analysis of parabolic equations with stiff
transport terms. This kind of problem occurs, for example, in collisional
gyrokinetic theory for tokamak plasmas, where the velocity diffusion of the
collision mechanism is dominated by the velocity advection along the Laplace
force corresponding to a strong magnetic field. This work appeal to the
filtering techniques. Removing the fast oscillations associated to the singular
transport operator, leads to a stable family of profiles. The limit profile
comes by averaging with respect to the fast time variable, and still satisfies
a parabolic model, whose diffusion matrix is completely characterized in terms
of the original diffusion matrix and the stiff transport operator. Introducing
first order correctors allows us to obtain strong convergence results, for
general initial conditions (not necessarily well prepared).
