The present evidence about the origin of high-energy cosmic rays is that two ranges exist: one below 1018 eV related with galactic sources, and one up to 1020 eV, corresponding to extragalactic processes (Kiràly et al. 1979). However the continuity of the spectrum indicates that the physical mechanisms must be correlated. In the global energetics the spectral range above 1012 eV is irrelevant and the bulk of CR energy is actually provided by supernovae, pulsars, X-ray binaries, etc. in the Galaxy. Nevertheless none of these objects seems capable of producing CR above 1015 eV/nucleon. We have investigated the possibility that the acceleration at higher energies is statistical, taking place over a hierarchy of scales. A model has been developed in terms of the quasi-linear theory of particle acceleration by MHD turbulence. Cosmic rays with ≲1015 eV/nucleon injected by single sources into interstellar space undergo momentum diffusion by stockastic interaction with long wavelength MHD perturbations; small wavelength modes provide pitchangle scatterings. These MHD perturbations, Alfvèn and last magnetosonic waves, are generated by the dynamic interaction of supernova remnants with the interstellar medium. From observations (Jokipii 1977), the range of possible wavelengths extends from the proton gyroradius to the size of the so-called “superbubbles”, up to 100 pc, with a power-law spectrum. Correspondingly acceleration is efficient up to 1018 eV/nucleon; in fact for B 10−3 + 10−5 G, n = 0.01 ÷ 1 cm−3 and L = 0.1 ÷ 100 pc, we find that the acceleration timescale
(ε = eBL is the maximum allowed energy) is always shorter than the time scales of losses and turbulent structure lifetimes; α = 3.5 ÷ 4 is the speciral index of turbulent modes. Contrary to the original Ferm i mechanism, in this theory the time scale for acceleration up to any energy ε is fixed by the final phase, previous steps being negligible.