Fractional master equations and fractal time random walks

Fractional master equations containing fractional time derivatives of order 0<ω≤1 are introduced on the basis of a recent classification of time generators in ergodic theory. It is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks. The corresponding waiting time density ψ(t) is obtained exactly as ψ(t)=(${\mathit{t}}^{\mathrm{\omega }\mathrm{-}1}$/C)${\mathit{E}}_{\mathrm{\omega },\mathrm{\omega }}$(-${\mathit{t}}^{\mathrm{\omega }}$/C), where ${\mathit{E}}_{\mathrm{\omega },\mathrm{\omega }}$(x) is the generalized Mittag-Leffler function. This waiting time distribution is singular both in the long time as well as in the short time limit.