Explicit densities of multidimensional ballistic Lévy walks

Area of Science:

• Stochastic dynamics
• Mathematical physics
• Complex systems modeling

Background:

• Lévy walks are crucial for modeling real-world phenomena.
• Understanding ballistic Lévy walks is vital for various applications.
• Previous models lacked explicit density formulas for ballistic Lévy walks.

Purpose of the Study:

• Derive explicit density formulas for 2D and 3D ballistic Lévy walks.
• Provide efficient methods for evaluating these densities.
• Validate theoretical results with numerical simulations.

Main Methods:

• Derivation of explicit formulas for 2D and 3D ballistic Lévy walks.
• Utilizing hypergeometric functions and fractional derivatives for 2D case.
• Employing elementary functions for 3D case.
• Comparison with Monte Carlo simulations.

Main Results:

• Explicit formulas for 3D ballistic Lévy walk densities are elementary functions.
• 2D ballistic Lévy walk densities are expressed using hypergeometric functions and fractional derivatives.
• Theoretical derivations show perfect agreement with Monte Carlo simulations.

Conclusions:

• The derived formulas offer efficient analytical and numerical tools for studying ballistic Lévy walks.
• This work advances the application of Lévy walk models in diverse scientific fields.
• The findings facilitate a deeper understanding of complex stochastic processes.