Inoue Jun-ichi, Sazuka Naoya
Complex Systems Engineering, Graduate School of Information Science and Technology, Hokkaido University, N14-W9, Kita-ku, Sapporo 060-0814, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2007 Aug;76(2 Pt 1):021111. doi: 10.1103/PhysRevE.76.021111. Epub 2007 Aug 17.
We propose an approach to the problem of the first-passage time. Our method is applicable not only to the Wiener process but also to the non-Gaussian Lévy flights or to more complicated stochastic processes whose distributions are stable. To show the usefulness of the method, we particularly focus on the first-passage time problems in the truncated Lévy flights (the so-called KoBoL processes from Koponen, Boyarchenko, and Levendorskii), in which the arbitrarily large tail of the Lévy distribution is cut off. We find that the asymptotic scaling law of the first-passage time t distribution changes from t(-(alpha+1)/alpha)-law (non-Gaussian Lévy regime) to t(-32)-law (Gaussian regime) at the crossover point. This result means that an ultraslow convergence from the non-Gaussian Lévy regime to the Gaussian regime is observed not only in the distribution of the real time step for the truncated Lévy flight but also in the first-passage time distribution of the flight. The nature of the crossover in the scaling laws and the scaling relation on the crossover point with respect to the effective cutoff length of the Lévy distribution are discussed.
我们提出了一种解决首次通过时间问题的方法。我们的方法不仅适用于维纳过程，也适用于非高斯 Lévy 飞行或分布稳定的更复杂随机过程。为了证明该方法的实用性，我们特别关注截断 Lévy 飞行中的首次通过时间问题（即来自 Koponen、Boyarchenko 和 Levendorskii 的所谓 KoBoL 过程），其中 Lévy 分布的任意大尾部被截断。我们发现，在交叉点处，首次通过时间 t 分布的渐近标度律从 t^(-(α + 1)/α) 律（非高斯 Lévy regime）变为 t^(-3/2) 律（高斯 regime）。这一结果意味着，不仅在截断 Lévy 飞行的实时步长分布中，而且在飞行的首次通过时间分布中，都观察到了从非高斯 Lévy regime 到高斯 regime 的超慢收敛。讨论了标度律中的交叉性质以及关于 Lévy 分布有效截止长度的交叉点处的标度关系。
