Approximation of Lévy-Feller Diffusion by Random Walk

R. Gorenflo and F. Mainardi

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 two-level difference scheme resulting from the Grünwald-Letnikov 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 re-interpretation 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 time-step 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 time-parametrized Cauchy probability density.

Keywords: stable probability distributions, Riesz-Feller potentials, pseudo-differential equations, Markov processes, random walks

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