In recent years in physics and in other fields, e.g. stochastic economical processes, pseudo-differential evolutionary equations modelling processes of anomalous diffusion are becoming more and more popular. Interesting mathematical models can be achieved by replacing in the common diffusion equation the second-order space derivative or the first-order time derivative by suitable pseudo-differential operators which can be interpreted respectively as a fractional space derivative (in the sense of Riesz-Feller) and a fractional time derivative (in the sense of Caputo). In the former case the resulting process is of Markov-type, mathematically a semigroup, which generates all Levy stable probability distributions. In the latter case the resulting process exhibits memory effects, which generates probability distributions with variance proportional to a power law of time. Discrete models can be obtained by discretization in space and in time. By taking care in constructing these, they can be interpreted not only as difference schemes for approximating the solution of initial value problems, but also as random walk models for simulating particle paths by the Monte Carlo technique. A report is given on possible accesses to find consistent models, which generalize the classical Brownian motion related to the common diffusion equation. The poster resumes recent work (published and in progress) by the author with the fruitful collaboration of Professor Rudolf GORENFLO (Dept. of Mathematics, Free University of Berlin). Other contributors include Prof. Graziano SERVIZI (Dept. of Physics, University of Bologna) and Mainardi's former students: Dr. Gianni DE FABRITIIS (Queen Mary College, London) and Dr. Paolo PARADISI (DIENCA, Engineering, University of Bologna and ISA0-CNR, Bologna)

1. R. Gorenflo, G. De Fabritiis and F. Mainardi
   "Discrete random walk models for symmetric Levy-Feller diffusion processes" Physica A 269,84-94 (1999) [http://xxx.lanl.gov/cond-mat/9903264]
2. F. Mainardi, P. Paradisi and R. Gorenflo
   "Probability distributions generated by fractional diffusion equations",
   J. Kertesz and I. Kondor "Econophysics: an Emerging Science", Kluwer, Dordrecht , pp. 39 (1999)
   [http://www.ge.infm.it/econophysics/ : papers -> mainardi]
3. R. Gorenflo, and F. Mainardi
   "Fractional calculus and stable probability distributions", Archives of Mechanics 50,377-388 (1998)
   [http://www.ge.infm.it/econophysics/ : papers -> mainardi]
4. R. Gorenflo, and F. Mainardi
   "Random walk models for space-fractional diffusion processes",
   Fractional Calculus and Applied Analysis 1,167-190 (1998)
   [http://www.ge.infm.it/econophysics/ : papers -> mainardi]

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