School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, People's Republic of China.
Phys Rev E. 2019 Jan;99(1-1):012135. doi: 10.1103/PhysRevE.99.012135.
Memory effects, sometimes, cannot be neglected. In the framework of continuous-time random walk, memory effect is modeled by the correlated waiting times. In this paper, we derive the two-point probability distribution of the correlated waiting time process, as well as the one of its inverse process, and present the Langevin description of Lévy walk with memory. We call this model a Lévy-walk-type model with correlated waiting times. Based on the built Langevin picture, the properties of aging and nonstationary are discussed. This Langevin system exhibits sub-ballistic superdiffusion 〈x^{2}(t)〉∝t^{2-α^{2}β/αβ+1} if the friction force is involved, while it displays superballistic diffusion or hyperdiffusion 〈x^{2}(t)〉∝t^{2+α/αβ+1} if there is no friction. The parameter 0<α<1 is for the white α-stable Lévy noise, while 0≤β≤1 is to characterize the strength of the correlation of waiting times; β=0 corresponds to uncorrelated case and β=1 the strongest correlation. It is discovered that the correlation of waiting times suppresses the diffusion behavior whether a friction is involved or not. The stronger the correlation of waiting times becomes, the slower the diffusion is. In particular, the correlation function, correlation coefficient, ergodicity, and scaling property of the corresponding stochastic process are also investigated.
记忆效应有时不容忽视。在连续时间随机游走的框架下，通过相关等待时间来模拟记忆效应。在本文中，我们推导出相关等待时间过程的两点概率分布，以及其逆过程的概率分布，并给出了具有记忆的 Lévy 游走的 Langevin 描述。我们将此模型称为具有相关等待时间的 Lévy 游走型模型。基于构建的 Langevin 图，讨论了老化和非平稳性的特性。此 Langevin 系统在涉及摩擦力时表现出亚弹道扩散〈x^{2}(t)〉∝t^{2-α^{2}β/αβ+1}，而在没有摩擦力时则表现出超弹道扩散或超扩散〈x^{2}(t)〉∝t^{2+α/αβ+1}。参数 0<α<1 表示白色 α-稳定 Lévy 噪声，而 0≤β≤1 则用于描述等待时间相关的强度；β=0 对应于无相关的情况，β=1 则对应于最强的相关性。研究发现，等待时间的相关性无论是否存在摩擦力都会抑制扩散行为。等待时间相关性越强，扩散越慢。特别地，还研究了相应随机过程的相关函数、相关系数、遍历性和标度性质。
