PHYSICAL REVIEW E 105, 014114 (2022) Lévy walks with rests: Long-time analysis Marcin Magdziarz * Hugo Steinhaus Center, Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland Wladyslaw Szczotka Institute of Mathematics, University of Wroclaw, Plac Grunwaldzki 2/4, 50-384 Wroclaw, Poland (Received 2 July 2021; accepted 27 December 2021; published 18 January 2022) In this paper we analyze the asymptotic behavior of Lévy walks with rests. Applying recent results in the field of functional convergence of continuous-time random walks we find the corresponding limiting processes. Depending on the parameters of the model, we show that in the limit we can obtain standard Lévy walk or the process describing competition between subdiffusion and Lévy flights. Some other more complicated limit forms are also possible to obtain. Finally we present some numerical results, which confirm our findings. DOI: 10.1103/PhysRevE.105.014114 I. INTRODUCTION In recent years Lévy walks (LWs) proved very useful models for anomalous diffusion [1]. They have become an alternative to Brownian diffusive random walk as a process, which underlies random movement with constant velocity. The particle that performs Lévy walks moves with velocity v for a time period, which follows a power law ψ (t ) ∝ t −(1+α) with α> 0. Then it chooses randomly a new direction of the motion [1]. Here we focus on the case α ∈ (0, 1), which leads to a ballistic regime [2]. Lévy walks were used to describe the dynamics of blinking nanocrystals [3–6]. Other striking and sometimes very beautiful examples of applications include: migration of swarming bacteria [7], light transport in special optical materials (Lévy glass) [8], and foraging patterns of animals [9–12]. More examples are described in a review paper devoted to this model [1], see also Ref. [13]. Lévy walks can be analyzed in a framework of coupled continuous-time random walks (CTRWs). Except from the standard Lévy walk, we can distinguish two other important cases: the so-called wait-first or jump-first models [1]. These are examples of CTRWs, which are closely related to the standard LW. The particles that perform wait-first LW, instead of moving with the constant velocity v for certain random time T , remain motionless for time T and then executes a jump with length equal to v · T . As a result trajectories are discon- tinuous, contrary to the standard LW, see Fig. 1 (left panel). If we linearly interpolate the trajectory of wait-first model, we obtain the trajectory of standard LW, see Fig. 1 (middle panel, solid line). In this case the walker moves with constant veloc- ity (in this paper we will assume for simplicity that v = 1) and after a certain random period of time it changes its direction. The CTRW approach to LWs was analyzed in Refs. [14,15]. Although the jump models and the standard LW appear to be * Corresponding author: marcin.magdziarz@pwr.edu.pl very similar, they have very different statistical properties. In Ref. [2] a method to find probability density functions p(x, t ) (PDFs) for all these models in ballistic regime was proposed by Froemberg et al. For another approach to this problem for the jump models, see also Ref. [16]. It is also worth to mention that PDFs of multidimensional isotropic Lévy walks were found in Refs. [17,18]. Important results related to the ergodic properties of Lévy walks can be found in Refs. [19–21] Here we analyze the so-called LW with rests. Its trajec- tories are obtained by adding a random waiting period after each period of ballistic motion, see Fig. 1 (right panel). We underline that the ballistic regime is observed for α ∈ (0, 1). One usually assumes that the resting periods are chosen from a power-law distribution with some exponent γ . The formal definition of LW with rests will be given in the next sec- tion. LW with rests were first introduced and analyzed in Refs. [22,23], where the authors used CTRW to define the corresponding equation for the propagator. Next, this equation was used to analyze the asymptotic behavior of the PDF of the model as well as the asymptotics of the mean-square displacement. One of the most stimulating and important ap- plications of CTRWs with power-law waiting times and jumps can be found in the paper [24], where the authors showed that the dispersal of bank notes and human travel behavior can be described by LW-type of dynamics. In Ref. [25] the authors analyzed the ratio of times a particle spends in fly- ing and resting phases. For a general review on Lévy walks and their generalizations we refer the interested reader to Refs. [1,13]. We also note that recently it has been demon- strated in Ref. [26] that neuronal mRNP transport follows aging LW with truncated power-law run times. That way the authors confirmed that mRNP particles in the analyzed experiment display aging. In this paper we analyze the asymptotic, long-time behav- ior of LW with rests. Applying recent results in the field of functional convergence of continuous-time random walks and related models, we derive the corresponding limiting process. 2470-0045/2022/105(1)/014114(8) 014114-1 ©2022 American Physical Society