The theory of diffusive shock acceleration is extended to the case of superdiffusive transport, i.e., when the mean square deviation grows proportionally to t α, with α 1. Superdiffusion can be described by a statistical process called Lévy random walk, in which the propagator is not a Gaussian but it exhibits power-law tails. By using the propagator appropriate for Lévy random walk, it is found that the indices of energy spectra of particles are harder than those obtained where a normal diffusion is envisaged, with the spectral index decreasing with the increase of α. A new scaling for the acceleration time is also found, allowing substantially shorter times than in the case of normal diffusion. Within this framework we can explain a number of observations of flat spectra in various astrophysical and heliospheric contexts, for instance, for the Crab Nebula and the termination shock of the solar wind.
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