Should I stay or should I go? Zero-size jumps in random walks for L\'evy flights

TL;DR

This paper investigates how zero-size jumps in Le9vy flights affect convergence to stable densities, impacting fractional diffusion models and animal movement theories.

Contribution

It presents a novel example showing non-convergence in Le9vy flights with bi-modal power-law jump distributions that are zero at zero.

Findings

Non-guaranteed convergence to stable densities with zero-included jumps

Implications for fractional diffusion equations

Relevance to animal movement modeling

Abstract

We study Markovian continuous-time random walk models for L\'evy flights and we show an example in which the convergence to stable densities is not guaranteed when jumps follow a bi-modal power-law distribution that is equal to zero in zero. The significance of this result is two-fold: i) with regard to the probabilistic derivation of the fractional diffusion equation and also ii) with regard to the concept of site fidelity in the framework of L\'evy-like motion for wild animals.