L\'evy noise-driven escape from arctan potential wells

TL;DR

This paper investigates how L{\'e}vy noise influences the escape dynamics from an arctan potential well, revealing exponential tail behavior and dependence on potential parameters, with implications for stochastic systems modeling.

Contribution

It introduces a detailed analysis of L{\'e}vy noise-driven escape in an arctan potential, highlighting transient dynamics and probability density behaviors not previously explored.

Findings

Escape times exhibit exponential tails.

Escape probability depends on potential shape parameters.

Probability densities of escape times and last-hitting points are characterized.

Abstract

The escape from a potential well is an archetypal problem in the study of stochastic dynamical systems, representing real-world situations from chemical reactions to leaving an established home range in movement ecology. Concurrently, L{\'e}vy noise is a well-established approach to model systems characterized by statistical outliers and diverging higher-order moments, ranging from gene expression control to the movement patterns of animals and humans. Here, we study the problem of L\'evy noise-driven escape from an almost rectangular, arctan potential well restricted by two absorbing boundaries. We unveil analogies of the observed transient dynamics to the general properties of stationary states of L{\'e}vy processes in single-well potentials. The first escape dynamics is shown to exhibit exponential tails. We examine the dependence of the escape on the shape parameters, steepness and height, of the arctan potential. Finally, we explore in detail the behavior of the probability densities of the first-escape time and the last-hitting point.