Análise de caminhadas de Lévy em trajetórias curvas 2D

Abstract

A crucial problem in the study of anomalous diffusion and transport refers to adequate analysis of trajectory data. The analysis and inference of Lévy walk model from empirical or simulated trajectories of particles in two and three-dimensions (2D and 3D) is much more hard than in 1D because path curvature is nonexistent in 1D but pretty common in higher dimensions. Lately, a new method to detect Lévy walks, which considers 1D projections of 2D or 3D trajectory data, has been proposed by Humphries et al. The main idea of this method is to explore the fact that a 1D projection of a high-dimensional Lévy walk is itself a Lévy walk. In this work, we ask whether or not this projection method is capable enough to clearly distinguish a 2D Lévy walk with curvature from a simple Markovian correlated random walk. We focus this work in challenging case in which both 2D walks have the same probability density functions (pdf) of step sizes as well as of turning angles between succesive steps. Our approach extends the original projection the original projection method by introducing a rescaling of the projected data. After a projection and coarse graining, the renormalized pdf for the travel distances between successive turnings is seen to possess a fat tail when there is an underlying Lévy process. We exploit this effect to infer a Lévy walk process in the original high-dimensional curved trajectory. In contrast, there is no fat tail when a (Markovian) is analyzed. We show that this procedure works very well in clearly identifying a Lévy walk even when there is noise from curvature. The present protocol may be useful in realistic contexts involving ongoing debates on the presence (or not) of Lévy walks related to animal movement on land (2D) and air and oceans (3D).