Enhancement of stability of metastable states in the presence of L\'{e}vy noise

TL;DR

This paper investigates how symmetric Lévy noise influences the stability of metastable states, deriving equations for mean residence time and showing that Lévy noise can enhance the stability of these states.

Contribution

It derives a fractional Fokker-Planck framework for Lévy flights in metastable potentials and provides a closed-form expression for mean residence time in a specific case.

Findings

Lévy noise can increase the mean residence time in metastable states.

Analytical expressions for residence time are obtained for specific Lévy indices.

The results suggest noise-induced stabilization of metastable states.

Abstract

The barrier-crossing event for superdiffusion characterized by symmetric L\'{e}vy flights is analyzed. Starting from the fractional Fokker-Planck equation, we derive an integro-differential equation along with the necessary conditions to calculate the mean residence time of a particle within a fixed interval. We consider an arbitrary smooth potential profile, particularly metastable, with a sink and L\'{e}vy noise characterized by both an arbitrary index $\alpha$ and arbitrary noise intensity parameter. For the specific case of L\'{e}vy flights with an index $\alpha = 1$ and a cubic metastable potential, a closed expression for the mean residence time is obtained in quadratures. The analytical results reveal an enhancement of the mean residence time in the metastable state due to the influence of L\'{e}vy noise.