Zeitschrift für Analysis und ihre Anwendungen Vol. 18, No. 2, pp. 231  246 (1999) 

Approximation of LévyFeller Diffusion by Random WalkR. Gorenflo and F. MainardiR. Gorenflo: Free Univ. of Berlin, Dept. Math. & Comp. Sci., Arnimallee 26, D14195 Berlin, gorenflo@math.fuberlin.deF. Mainardi: Univ. of Bologna, Dept. Phys., Via Irnerio 46, I40126 Bologna, Italy, mainardi@bo.infn.it Abstract: After an outline of W. Feller's inversion of the (later so called) Feller potential operators and the presentation of the semigroups thus generated, we interpret the twolevel difference scheme resulting from the GrünwaldLetnikov discretization of fractional derivatives as a random walk model discrete in space and time. We show that by properly scaled transition to vanishing space and time steps this model converges to the continuous Markov process that we view as a generalized diffusion process. By reinterpretation of the proof we get a discrete probability distribution that lies in the domain of attraction of the corresponding stable Lévy distribution. By letting only the timestep tend to zero we get a random walk model discrete in space but continuous in time. Finally, we present a random walk model for the timeparametrized Cauchy probability density. Keywords: stable probability distributions, RieszFeller potentials, pseudodifferential equations, Markov processes, random walks Full text of the article:
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