SIAM Journal on Applied Mathematics, Volume 80, Issue 3, Page 1175-1196, January 2020.
Many physical and biological processes are modeled by “particles" undergoing Lévy random walks. A feature of significant interest in these systems is the mean square displacement (MSD) of the particles. Long-time asymptotic approximations of the MSD have been established, via the Tauberian theorem, for systems in which the distribution of the step durations is asymptotically a power law of infinite variance. We extend these results, using elementary analysis, and obtain closed-form expressions as well as power law bounds for the MSD in equilibrium, and representations of the MSD as sums of superlinear, linear, and sublinear terms. We show that the superlinear components are determined by the mean and asymptotics of the step durations, but also that the linear and sublinear components (whose size has implications for the accuracy of the asymptotic approximation) depend on the entire distribution function.

中文翻译：

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LévyWalks中的均方位移

SIAM应用数学杂志，第80卷，第3期，第1175-1196页，2020年1月。
许多物理和生物过程都是通过经历Lévy随机游走的“粒子”来建模的，这些系统中最引人关注的一个特征是粒子的均方位移（MSD），通过这种方法已经建立了MSD的长期渐近近似。陶伯定理，对于步长分布是渐近无穷大的幂定律的系统，我们使用基本分析扩展了这些结果，并获得了平衡态MSD的闭式表达式以及幂定律边界， MSD表示为超线性，线性和亚线性项的和。我们表明，超线性分量由步长的均值和渐近性决定，但是线性和亚线性分量（其大小对渐近逼近的精度有影响）也取决于整个分布函数。