Path Integral Formulation for L\'{e}vy Flights - Evaluation of the Propagator for Free, Linear and Harmonic Potentials in the Over- and Underdamped Limits

TL;DR

This paper develops a path integral approach to analyze Lévy flights, enabling the evaluation of propagators for various potentials in different damping regimes, advancing the mathematical tools for anomalous diffusion.

Contribution

It introduces a novel method to evaluate path integrals involving fractional derivatives, specifically applied to Lévy flights in different potential and damping scenarios.

Findings

Derived propagators for free Lévy flights in various damping regimes

Evaluated propagators for linear and harmonic potentials

Provided a practical method for path integral calculations with fractional operators

Abstract

L\'{e}vy flights can be described using a Fokker-Planck equation which involves a fractional derivative operator in the position co-ordinate. Such an operator has its natural expression in the Fourier domain. Starting with this, we show that the solution of the equation can be written as a Hamiltonian path integral. Though this has been realized in the literature, the method has not found applications as the path integral appears difficult to evaluate. We show that a method in which one integrates over the position co-ordinates first, after which integration is performed over the momentum co-ordinates, can be used to evaluate several path integrals that are of interest. Using this, we evaluate the propagators for (a) free particle (b) particle subjected to a linear potential and (c) harmonic potential. In all the three cases, we have obtained results for both overdamped and underdamped cases.