# Multi-phase long-term autocorrelated diffusion: Stationary continuous-time Weierstrass walk vs. flight

## TL;DR

This paper analyzes the diffusion properties of stationary continuous-time Weierstrass walks and flights, revealing complex multi-phase behaviors including anomalous, Brownian non-Gaussian, and hierarchical diffusion phases.

## Contribution

It introduces a detailed diffusion phase diagram for CTWW and compares it with CTWF, highlighting the hierarchical structure and multi-phase nature of these processes.

## Key findings

- CTWW exhibits multiple diffusion phases, including anomalous and Brownian non-Gaussian.
- The diffusion phase diagram of CTWW is more complex than that of CTWF.
- Hierarchical coupling defines the unique diffusion behaviors of these processes.

## Abstract

In this paper we are examining diffusion properties of stationary continuous-time Weierstrass walk (CTWW). We are showing it is a multi-phase representation of the L\'evy walk. The hierarchical spatial-temporal coupling, combined with coupling between dynamic variables define the CTWW process. The walker moves here with a piecewise constant velocity between trajectory turning points. We have found the diffusion phase diagram of the CTWW consisting not only of anomalous non-Gaussian or non-fBm phases but also Brownian yet non-Gaussian ones. We compare the diffusion phase diagram of the stationary CTWW with the corresponding hierarchical continuous-time Weierstress flight (CTWF). The instantaneous jumps between trajectory turning points preceded by waiting define the CTWF process. It is a hierarchical representation of the L\'evy flight. We have found the diffusion phase diagram of the CTWF to be a small part of the corresponding CTWW one.