Introduction to the Theory of Lévy Flights
Lévy flights, also referred to as Lévy motion, stand for a class of non-Gaussian random processes whose stationary increments are distributed according to a Lévy stable distribution originally studied by French mathematician Paul Pierre Lévy. Lévy stable laws are important for three fundamental properties: (i) similar to the Gaussian law, Lévy stable laws form the basin of attraction for sums of random variables. This follows from the theory of stable laws, according to which a generalized
... a generalized central limit theorem exists for random variables with diverging variance. The Gaussian distribution is located at the boundary of the basin of attraction of stable laws; (ii) the probability density functions of Lévy stable laws decay in asymptotic power-law form with diverging variance and thus appear naturally in the description of many fluctuation processes with largely scattering statistics characterized by bursts or large outliers; (iii) Lévy flights are statistically self-affine, a property catering for the description of random fractal processes. Lévy stable laws appear as statistical description for a broad class of processes in physical, chemical, biological, geophysical, or financial contexts, among others. We here review the fundamental properties of Lévy flights and their underlying stable laws. Particular emphasis lies on recent developments such as the first passage time and leapover properties of Lévy flights, as well as the behavior of Lévy flights in external fields. These properties are discussed on the basis of analytical and numerical solutions of fractional kinetic equations as well as numerical solution of the Langevin equation with white Lévy noise.