# L\'evy flights versus L\'evy walks in bounded domains

## TL;DR

This paper compares Le9vy flights and Le9vy walks in bounded domains, analyzing their differences and similarities in terms of statistical properties like survival probability and mean first passage time, influenced by boundary conditions and jump distribution.

## Contribution

It provides a detailed analytical and numerical comparison of Le9vy flights and Le9vy walks within bounded domains, highlighting conditions for their similarity.

## Key findings

- Similarity depends on boundary conditions and stability index.
- Le9vy flights exhibit pathological properties addressed by Le9vy walks.
- Models show comparable results under certain conditions.

## Abstract

L\'evy flights and L\'evy walks serve as two paradigms of random walks resembling common features but also bearing fundamental differences. One of the main dissimilarities are discontinuity versus continuity of their trajectories and infinite versus finite propagation velocity. In consequence, well developed theory of L\'evy flights is associated with their pathological physical properties, which in turn are resolved by the concept of L\'evy walks. Here, we explore L\'evy flights and L\'evy walks models on bounded domains examining their differences and analogies. We investigate analytically and numerically whether and under which conditions both approaches yield similar results in terms of selected statistical observables characterizing the motion: the survival probability, mean first passage time and stationary PDFs. It is demonstrated that similarity of models is affected by the type of boundary conditions and value of the stability index defining asymptotics of the jump length distribution.