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Einstein's explanation of Brownian motion provided one of the cornerstoneswhich underlie the modern approaches to stochastic processes. His approach isbased on a random walk picture and is valid for Markovian processes lackinglong-term memory. The coarse-grained behavior of such processes is described bythe diffusion equation. However, many natural processes do not possess theMarkovian property and exhibit to anomalous diffusion. We consider here thecase of subdiffusive processes, which are semi-Markovian and correspond tocontinuous-time random walks in which the waiting time for a step is given by aprobability distribution with a diverging mean value. Such a process can beconsidered as a process subordinated to normal diffusion under operational timewhich depends on this pathological waiting-time distribution. We derive twodifferent but equivalent forms of kinetic equations, which reduce to knowfractional diffusion or Fokker-Planck equations for waiting-time distributionsfollowing a power-law. For waiting time distributions which are not pure powerlaws one or the other form of the kinetic equation is advantageous, dependingon whether the process slows down or accelerates in the course of time

From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion