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Zbl 0946.60039
Gorenflo, R.; Mainardi, F.
Random walk models for space-fractional diffusion processes. (English)
[J] Fract. Calc. Appl. Anal. 1, No.2, 167-191 (1998). ISSN 1311-0454; ISSN 1314-2224/e

This survey paper provides a generalization of the classical random walk model of the standard diffusion equation to the so-called Feller (space-)fractional diffusion equation $$ {\frac {\partial u} {\partial t}} = D_{\theta}^{\alpha} u, \quad u=u(x,t;\alpha,\theta),\ x \in R, t \in R_0^{+}, \quad 0 < \alpha \leq 2, \tag 1 $$ obtained by replacing the second (space-)derivative with a special pseudo-differential operator $$ D_{\theta}^{\alpha} = - \left[c_{+} (\alpha,\theta) I_{+}^{-\alpha} + c_{-} (\alpha,\theta) I_{-}^{-\alpha} \right], \tag 2 $$ arising by inversion of linear combinations of left- and right-sided Riemann-Liouville-Weyl operators (in case of both coefficients $c_{\pm} (\alpha,0)$ being equal, this leads to inversion of the Riesz potential operators). \par In a pioneering paper, {\it W. Feller} [Meddel. Lunds Univ. Mat. Sem. Suppl.-Band M. Riesz, 73-81 (1952; Zbl 0048.08503)] considered the problem of generating all the stable probability distributions through the Green function of the Cauchy problem for the linear evolution equation of fractional order (1), in view of semigroup theory. From a probabilistic point of view, these semigroups represent stable distributions in the sense of {\it P. Lévy} [``Calcul des probabilités'' (1925; JFM 51.0380.02)]. To honour both, Lévy and Feller, the authors call the considered stohastic processes (which are of Markov type), characterized by the two parameters $\alpha$ (for the order of differentiation) and $\theta$ (for the ``skewness'' of the spatial pseudo-differential operator), as ``Lévy-Feller processes''. They give a completely revised version of Feller's work and complement his analysis by an original finite-difference approach to approximate and to find the Green function of the Cauchy problem for (1), excluding the singular case $\alpha=1$. Their essential idea is to approximate the inverse operators in the expression (2) for the Feller derivative, by the Grünwald-Letnikov scheme, see, for example, {\it R. Gorenflo} [in: Fractals and fractional calculus in continuum mechanics. CISM Courses and Lectures 378, 277-290 (1997)]. The authors provide also numerical computations of all the stable, non-Gaussian densities, with exception to those of Cauchy type.
[Virginia S.Kiryakova (Sofia)]

MSC 2000:: *60G50 Sums of independent random variables
33E20 Functions defined by series and integrals
44A20 Integral transforms of special functions
45K05 Integro-partial differential equations
60E07 Infinitely divisible distributions
26A33 Fractional derivatives and integrals (real functions)
60J65 Brownian motion

Keywords: stochastic processes; diffusion equations; fractional calculus; Riesz-Feller potentials; stable probability distributions; random walks; JFM 51.0380.02

Citations: Zbl 0048.08503; JFM 51.0380.02

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