  We study specific properties of particles transport by exploring an exact
solvable model, a so-called comb structure, where diffusive transport of
particles leads to subdiffusion. A performance of L\'evy -- like process
enriches this transport phenomenon. It is shown that an inhomogeneous
convection flow, as a realization of the L\'evy--like process, leads to
superdiffusion of particles on the comb structure. A frontier case of
superdiffusion that corresponds to the exponentially fast transport is studied
and the log--normal distribution is obtained for this case.
