  We investigate the role of the form of the spatial diffusion coefficient in
shock acceleration of fast particles. Referring to non-classical diffusion and
using the results of numerical (hybrid) simulations tailored for the downstream
shock population in quasi-perpendicualr high-Mach number collisionless shocks
to which we apply the theory, we demonstrate that the inferred diffusion
coefficients are in excellent agreement with the requirements of the theory and
its predictions. Diffusion in the collisionless regime turns out to be
non-classical Gibbsian (L\'evy flight), time-dependent though weak.
