Path Integral Formulation for Lévy Flights - Evaluation of the Propagator for Free, Linear and Harmonic Potentials in the Over- and Underdamped Limits

Abstract

Lévy 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.

21 pages, 5 figures

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• Unusual eigenvalue spectrum and relaxation in the Lévy Ornstein-Uhlenbeck process
• Characterization of the Most Probable Transition Paths of Stochastic Dynamical Systems with Stable Lévy Noise