Limit properties of Lévy walks

Abstract

In this paper we study properties of the diffusion limits of three different models of Lévy walks (LW). Exact asymptotic behavior of their trajectories is found using LePage series representation. We also prove an existing conjecture about total variation of LW sample paths. Based on this conjecture we verify martingale properties of the limit processes for LW. We also calculate their probability density functions and apply this result to determine the potential density of the associated non-symmetric α-stable processes. The obtained theoretical results for continuous LW can be used to recognize and verify this type of processes from anomalous diffusion experimental data. In particular they can be used to estimate parameters from experimental data using maximum likelihood methods.

Machine-Learning Solutions for the Analysis of Single-Particle Diffusion Trajectories

“…There exist plenty of other models to explain the occurrence of anomalous diffusion, − apart from the mentioned SBM and FBM. We here also consider continuous-time random walk (CTRW), with random waiting times between successive jumps, ,, Lévy walks (LW), − and annealed transient time motion (ATTM) . We provide short descriptions of each of these models in the Supporting Information.…”

Preface: characterisation of physical processes from anomalous diffusion data

“…Maraj et al [34] introduces the empirical anomaly measure as a means to measure the distance between the anomalous diffusion process and normal diffusion. Limit properties of Lévy walks are shown to be useful in the recognition and verification of Lévy walk-type motion, as well as the parameter estimation in maximum likelihood methods [35]. Wang et al [36] studies the emerging residual nonergodicity in fractional Brownian motion with random diffusivity that may help distinguish and categorise certain nonergodic and non-Gaussian features of particle displacements.…”

Preface: Characterisation of Physical Processes from Anomalous Diffusion Data

“…[33] introduces the empirical anomaly measure as a means to measure the distance between the anomalous diffusion process and normal diffusion. Limit properties of Lévy walks are shown to be useful in the recognition and verification of Lévy walk-type motion, as well as the parameter estimation in maximum likelihood methods [34]. Ref.…”