We derive general properties of anomalous diffusion and nonexponential relaxation from the theory of tempered $\alpha$-stable processes. The tempering results in the existence of all moments of operational time. The subordination by the inverse tempered $\alpha$-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 and the Cole-Davidson relaxation function. This model includes subdiffusion as a particular case.

• Received 17 June 2008

DOI:https://doi.org/10.1103/PhysRevE.78.051106