Non-Markovian process with variable memory functions

We present a treatment of the non-Markovian character of memory by incorporating different forms of Mittag-Leffler (ML) function, which generally arises in the solution of a fractional master equation, as different memory functions in the Generalized Kolmogorov-Feller Equation (GKFE). The cross-over from the short time (stretched exponential) to long time (inverse power law) approximations of the ML function incorporated in the GKFE is proven. We have found that the GKFE solutions are the same for negative exponential and upto first order expansion of the stretched exponential function for very small tau -> 0. A generalized integro-differential equation form of the GKFE along with an asymptotic case is provided.