Lévy walks with rests: Long-time analysis

Area of Science:

• Physics
• Mathematics
• Statistical Mechanics

Background:

• Lévy walks are fundamental models for anomalous transport.
• Understanding the asymptotic behavior of random walks with resting periods is crucial for modeling complex systems.

Purpose of the Study:

• To analyze the asymptotic behavior of Lévy walks incorporating resting periods.
• To identify the limiting processes governing these random walks under various parameter regimes.

Main Methods:

• Application of functional convergence theorems for continuous-time random walks.
• Mathematical analysis of asymptotic properties.
• Numerical simulations to validate theoretical findings.

Main Results:

• Demonstration of standard Lévy walk as a possible limit.
• Identification of a novel limiting process balancing subdiffusion and Lévy flights.
• Discovery of other complex limiting behaviors based on model parameters.

Conclusions:

• The asymptotic behavior of Lévy walks with rests is diverse and parameter-dependent.
• The study provides a comprehensive framework for understanding anomalous diffusion phenomena.