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Random walk models for space-fractional diffusion processes. (English) Zbl 0946.60039

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

$\frac{\partial u}{\partial t}={D}_{\theta }^{\alpha }u,\phantom{\rule{1.em}{0ex}}u=u\left(x,t;\alpha ,\theta \right),\phantom{\rule{4pt}{0ex}}x\in R,t\in {R}_{0}^{+},\phantom{\rule{1.em}{0ex}}0<\alpha \le 2,\phantom{\rule{2.em}{0ex}}\left(1\right)$

obtained by replacing the second (space-)derivative with a special pseudo-differential operator

${D}_{\theta }^{\alpha }=-\left[{c}_{+}\left(\alpha ,\theta \right){I}_{+}^{-\alpha }+{c}_{-}\left(\alpha ,\theta \right){I}_{-}^{-\alpha }\right],\phantom{\rule{2.em}{0ex}}\left(2\right)$

arising by inversion of linear combinations of left- and right-sided Riemann-Liouville-Weyl operators (in case of both coefficients ${c}_{±}\left(\alpha ,0\right)$ being equal, this leads to inversion of the Riesz potential operators).

In a pioneering paper, 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 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, 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.

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