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In this paper, a wide view on the theory of the continuous time random walk ( CTRW ) and its relations to the space–time fractional diffusion process is given. We begin from the basic model of  CTRW  (Montroll and Weiss [ 19 ], 1965) that also can be considered as a compound renewal process. We are interested in studying the random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. We prove the relation between the integral equation of the  CTRW  having the above fat tails waiting and jump width distributions and the space–time fractional diffusion equations in the Laplace–Fourier domain. The space–time fractional Fokker–Planck equation could also be driven from the discrete Ehren–Fest model and is represented by the theory of  CTRW . These space–time fractional diffusion processes are getting increasing popularity in applications in physics, chemistry, finance, biology, medicine and many other fields. The asymptotic behavior of the Mittag–Leffler function plays a significant rule on simulating these models. The behaviors of the studied  CTRW  models are well approximated and visualized by simulating various types of random walks by using the Monte Carlo method.

From power laws to fractional diffusion processes with and without external forces, the non direct way