Revisiting the derivation of the fractional diffusion equation
  The fractional diffusion equation is derived from the master equation of continuous-time random walks (CTRWs) via a straightforward application of the Gnedenko-Kolmogorov limit theorem. The Cauchy problem for the fractional diffusion equation is solved in various important and general cases. The meaning of the proper diffusion limit for CTRWs is discussed. 
Enrico Scalas, Rudolf Gorenflo, Francesco Mainardi and Marco Raberto