The problem of analytical calculation of barrier crossing characteristics for Levy flights

TL;DR

This paper analytically investigates barrier crossing times for Levy flights using fractional Fokker-Planck equations, deriving explicit formulas for free flights and cubic potentials, advancing understanding of Levy noise-driven escape processes.

Contribution

It introduces a general differential equation for nonlinear relaxation time and provides analytical solutions for Levy flights in specific potentials, enhancing theoretical modeling of such stochastic processes.

Findings

Analytical expression for nonlinear relaxation time of free Levy flights.

Closed-form quadrature expression for barrier crossing in cubic potential.

Advancement in theoretical understanding of Levy noise-induced barrier crossing.

Abstract

By using the backward fractional Fokker-Planck equation we investigate the barrier crossing event in the presence of Levy noise. After shortly review recent results obtained with different approaches on the time characteristics of the barrier crossing, we derive a general differential equation useful to calculate the nonlinear relaxation time. We obtain analytically the nonlinear relaxation time for free Levy flights and a closed expression in quadrature of the same characteristics for cubic potential.