Abstract

The term Levy motion, or “Levy flights” (LFs) was coined for a family of non-Gaussian random processes whose stationary increments are distributed with the Levy stable probability laws discovered by French mathematician Paul Pierre Levy. These laws are of importance due to three remarkable properties: (i) as the Gaussian law, stable laws are attractive to the distributions of sums of random variables, thus naturally appear when evolution of the physical (chemical, biological, ...) system or the result of an experiment is determined by the sum of random factors (Generalized Central Limit Theorem); (ii) in contrast to the Gaussian law, the probability density functions (PDFs) of the stable laws possess slowly decaying power-law asymptotics; thus they naturally serve for the description of fluctuation processes with bursts or large outliers; and (iii) as the Brownian motion, which increments are distributed with the Gaussian law, the Levy motions are statistically self-similar, or self-affine, thus naturally suited for the description of random fractal processes. In my talk I will review at tutorial level some recent results in the LFs theory. The properties of LFs are discussed with the use of analytical and numerical solutions of space and time fractional kinetic equations as well as numerical simulation based on the Langevin approach.