2008/09/01 by A. A. Dubkov, Bernardo Spagnolo, B. Spagnolo +1 · 5 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Applied mathematics #Brownian motion #Differential equation #Diffusion and Search Dynamics #Fokker–Planck equation #Fractional Brownian motion #Fractional Differential Equations Solutions #Langevin equation #Lévy flight #Lévy process #Mathematical analysis #Mathematical physics #Mathematics #Physics #Random walk #Statistical physics #White noise #cond-mat.stat-mech #stochastic dynamics and bifurcation
paper · pdf · doi:10.1142/s0218127408021877
published as Intern. Journ. of Bifurcation and Chaos, Vol. 18, No. 9, 2649 - 2672 (2008) · 32 pages, 9 figures, to appear in Int. J. of Bifurcation and Chaos
openalex publication_date 2008/09/01 · arxiv created 2008/10/08 · arxiv updated 2015/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
After a short excursion from the discovery of Brownian motion to the Richardson "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 α-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 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.