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Diffusion and Relaxation Controlled by Tempered α-stable Processes

We derive general properties of anomalous diffusion and nonexponential relaxation from the theory of tempered α-stable processes. Its most important application is to overcome the infinite-moment difficulty for the α-stable random operational time τ. The tempering results in the existence of all moments of τ. The subordination by the inverse tempered α-stable process provides diffusion(relaxation) that occupies an intermediate place between subdiffusion (Cole-Cole law) and normal diffusion (exponential law). Here we obtain explicitly the Fokker-Planck equation, the mean square displacement and the relaxation function. This model includes subdiffusion as a particular case.