Abstract
We investigate the first-passage dynamics of symmetric and asymmetric Lévy flights in semi-infinite and bounded intervals. By solving the space-fractional diffusion equation, we analyse the fractional-order moments of the first-passage time probability density function for different values of the index of stability and the skewness parameter. A comparison with results using the Langevin approach to Lévy flights is presented. For the semi-infinite domain, in certain special cases analytic results are derived explicitly, and in bounded intervals a general analytical expression for the mean first-passage time of Lévy flights with arbitrary skewness is presented. These results are complemented with extensive numerical analyses.
Highlights
Lévy flights (LFs) correspond to a class of Markovian random walk processes that are characterised by an asymptotic power-law form for the distribution of jump lengths with a diverging variance [1,2,3,4,5]
LFs appear as traces of light beams in disordered media [24], and in optical lattices the divergence of the kinetic energy of single ions under gradient cooling are related to Lévytype fluctuations [25]
We show that the mean first-passage time (MFPT) of LFs in a finite interval is representative for the first-passage time probability density function (PDF) by analysing the associated coefficient of variation
Summary
Lévy flights (LFs) correspond to a class of Markovian random walk processes that are characterised by an asymptotic power-law form for the distribution of jump lengths with a diverging variance [1,2,3,4,5]. The first part of this paper, based on our previous results in [60], is devoted to the study of fractional order moments of the first-passage time PDF of LFs in a semiinfinite domain for symmetric (0 < α < 2 with β = 0), one-sided (0 < α < 1 with β = 1), extremal two-sided (1 < α < 2 with β = ±1), and a general form (α ∈ (0, 2] with β ∈ [−1, 1], excluding α = 1 with β = 0) α-stable laws.
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