Stretched-Gaussian asymptotics of the truncated Lévy flights for the diffusion in nonhomogeneous media

Abstract

The Lévy, jumping process, defined in terms of the jumping size distribution and the waiting time distribution, is considered. The jumping rate depends on the process value. The fractional diffusion equation, which contains the variable diffusion coefficient, is solved in the diffusion limit. That solution resolves itself to the stretched Gaussian when the order parameter $\mu \to 2$. The truncation of the Lévy flights, in the exponential and power-law form, is introduced and the corresponding random walk process is simulated by the Monte Carlo method. The stretched Gaussian tails are found in both cases. The time which is needed to reach the limiting distribution strongly depends on the jumping rate parameter. When the cutoff function falls slowly, the tail of the distribution appears to be algebraic.

19 pages, 5 figures

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Cited by in corpus (10)

• Paradoxical diffusion: Discriminating between normal and anomalous random walks
• Fractional Fokker-Planck equation for Lévy flights in nonhomogeneous environments
• Multiplicative Lévy processes: Itô versus Stratonovich interpretation
• Subordinated diffusion and CTRW asymptotics
• Nonlinear stochastic equations with multiplicative Lévy noise
• Anomalous diffusion in systems driven by the stable Levy noise with a finite noise relaxation time and inertia
• Cumulant Approach of Arbitrary Truncated Levy Flight
• Fluctuations in multiplicative systems with jumps
• Asymmetric Levy flights in nonhomogeneous environments
• Anomalous diffusion for a correlated process with long jumps