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Zbl 1157.82300
Dubkov, A.A.; Spagnolo, B.; Uchaikin, V.V.
Lévy flight superdiffusion: an introduction. (English)
[J] Int. J. Bifurcation Chaos Appl. Sci. Eng. 18, No. 9, 2649-2672 (2008). ISSN 0218-1274

Summary: After a short excursion from the discovery of Brownian motion to the Richardson \lq\lq law of four thirds'' in turbulent diffusion, the article introduces the Lévy flight superdiffusion as a self-similar Lévy process. The condition of self-similarity converts the infinitely divisible characteristic function of the Lévy process into a stable characteristic function of the Lévy motion. The Lévy motion generalizes the Brownian motion on the base of the $\alpha$-stable distributions theory and fractional order derivatives.\par 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 Lévy flights. Some results concerning stationary probability distributions of Lévy 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 Lévy flights, recently obtained by different approaches.

MSC 2000:: *82-01 Textbooks (statistical mechanics)
82C32 Neural nets
82B41 Random walks, etc. (statistical mechanics)

Keywords: Lévy process; Lévy motion; Lévy flights; stable distributions; fractional differential equation; barrier crossing

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Zentralblatt MATH Berlin [Germany]