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

Capala, Karol; Padash, Amin; Chechkin, Aleksei V.; Shokri, Babak; Metzler, Ralf; Dybiec, Bartlomiej

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.