LEVY FLIGHT SUPERDIFFUSION: AN INTRODUCTION

After a short excursion from the discovery of Brownian motion to the Richardson "law of four thirds" in turbulent diffusion, the article introduces the Levy flight superdiffusion as a self-similar Levy process. The condition of self-similarity converts the infinitely divisible characteristic function of the Levy process into a stable characteristic function of the Levy motion. The Levy motion generalizes the Brownian motion on the base of the alpha-stable distributions theory and fractional order derivatives. Further development on this idea lies on the generalization of the Langevin equation with a non-Gaussian white noise source and the use of functional approach. This leads to the Kolmogorov's equation for arbitrary Markovian processes. As a particular case we obtain the fractional Fokker-Planck equation for Levy flights. Some results concerning stationary probability distributions of Levy motion in symmetric smooth monostable potentials, and a general expression to calculate the nonlinear relaxation time in barrier crossing problems are derived. Finally, we discuss the results on the same characteristics and barrier crossing problems with Levy flights, recently obtained by different approaches.