Marian Smoluchowski Institute of Physics, and Mark Kac Center for Complex Systems Research, Jagiellonian University, ul. St. Łojasiewicza 11, 30-348 Kraków, Poland.
Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel.
Phys Rev E. 2017 May;95(5-1):052102. doi: 10.1103/PhysRevE.95.052102. Epub 2017 May 3.
Lévy flights and Lévy walks serve as two paradigms of random walks resembling common features but also bearing fundamental differences. One of the main dissimilarities is the discontinuity versus continuity of their trajectories and infinite versus finite propagation velocity. As a consequence, a well-developed theory of Lévy flights is associated with their pathological physical properties, which in turn are resolved by the concept of Lévy walks. Here, we explore Lévy flight and Lévy walk models on bounded domains, examining their differences and analogies. We investigate analytically and numerically whether and under which conditions both approaches yield similar results in terms of selected statistical observables characterizing the motion: the survival probability, mean first passage time, and stationary probability density functions. It is demonstrated that the similarity of the models is affected by the type of boundary conditions and the value of the stability index defining the asymptotics of the jump length distribution.
Lévy 飞行和 Lévy 漫步是两种随机漫步模式，它们具有相似的特征，但也存在根本的区别。其中一个主要的区别是它们的轨迹是不连续的还是连续的，以及传播速度是无限的还是有限的。因此， Lévy 飞行理论的发展与它们的病理物理特性有关，而这些特性又通过 Lévy 漫步的概念得到了解决。在这里，我们研究了有界域上的 Lévy 飞行和 Lévy 漫步模型，考察了它们的差异和相似之处。我们通过分析和数值方法研究了这两种方法在选择描述运动的统计量方面是否以及在何种条件下会产生相似的结果：生存概率、平均首次通过时间和稳定概率密度函数。结果表明，模型的相似性受到边界条件的类型和定义跳跃长度分布渐近的稳定性指数值的影响。
