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 µ → 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.

Nonlinear stochastic equations with multiplicative Lévy noise

“…Since in the physical phenomena process values are usually finite, introducing truncated distributions is realistic. In the random walk theory, the truncated Lévy flights are often considered [5,6,[14][15][16]. They agree with the Lévy flights for an arbitrarily large jump value; deviations appear only at very far tails [14].…”

Multiplicative Lévy processes: Itô versus Stratonovich interpretation

“…Details of the derivation are presented in Refs. [31,38]. The Fox functions are well suited for problems which involve Lévy processes since any Lévy distribution, both symmetric and asymmetric, can be expressed as the function H 1,1 2,2 (x) [39].…”

Asymmetric Lévy flights in nonhomogeneous environments

“…Usually, one observes the power-law distributions with the tails falling faster than for the stable Lévy distributions. This is the case for the financial market [46][47][48] and the minority game [49]; fast falling power-law tails result from a multifractal analysis of the extreme events [50] and emerge when one introduces a power-law truncation to the distribution [37,51]. We shall demonstrate that Eq.…”