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 \(\tau \rightarrow 0\). A generalized integro-differential equation form of the GKFE along with an asymptotic case is provided.

我们通过结合不同形式的Mittag-Leffler（ML）函数（通常出现在分数主方程的解决方案中）作为通用Kolmogorov-Feller方程（GKFE）中的不同记忆函数，来介绍记忆的非马尔可夫特性）。证明了GKFE中包含的ML函数从短时间（拉伸指数级）到长时间（逆幂定律）的近似。我们发现，对于非常小的\（\ tau \ rightarrow 0 \），GKFE解对于负指数和拉伸指数函数的一阶展开都是相同的。提供了GKFE的广义积分微分方程形式以及渐近情况。