Feller Fractional Diiusion and L Evy Stable Motions

{ Fractional calculus allows to generalize the standard (linear and one dimensional) diiusion equation by replacing the second-order space derivative by a derivative of fractional order. If this is taken as the pseudo-diierential operator introduced by Feller in 1952 the fundamental solution of the resulting diiusion equation is a probability density evolving in time and stable in the sense of L evy. Like the standard diiusion equation through its fundamental solution, the Gaussian density, yields the well-known Brownian motion, so the Feller diiusion equation yields the so called L evy stable motions, whose increments are independent and stably distributed. We show how to approximate each of these motions by a discrete-time, discrete-space random walk model, which is based on an integer-valued random variable lying in the domain of attraction of the corresponding L evy probability distribution.