Realization of Lévy walks as Markovian stochastic processes

Area of Science:

• Statistical Physics
• Stochastic Processes
• Mathematical Physics

Background:

• Lévy flights are non-Brownian, long-tailed random walks crucial in various physical phenomena.
• Existing models often struggle to represent Lévy flights as continuous processes, limiting their applicability.

Purpose of the Study:

• To present a continuous process realization of Lévy flights.
• To derive generalized Langevin and Fokker-Planck equations for Lévy flights under specific conditions.

Main Methods:

• Utilized multivariate Langevin processes as the theoretical framework.
• Derived generalized Langevin equation and Fokker-Planck equation for a particle with friction and velocity-dependent stochastic force.
• Employed a procedure analogous to the Smoluchowski limit treatment of the Kramers-Fokker-Planck equation.

Main Results:

• Successfully realized Lévy flights as a continuous process.
• Explicitly derived the generalized Langevin equation and the corresponding generalized Fokker-Planck equation describing Lévy flights.
• Established a method applicable to systems with friction and velocity-dependent stochastic forces.

Conclusions:

• The presented approach offers a continuous framework for Lévy flights.
• This method provides a foundation for future investigations in inhomogeneous media or systems with boundaries.
• The derived equations are fundamental for understanding complex particle dynamics.