To hit or to pass it over - remarkable transient behavior of first arrivals and passages for Lévy flights in finite domains

The term 'Lévy flights' was coined by Benoit Mandelbrot, who thus poeticized α-stable Lévy random motion, a Markovian process with stationary independent increments distributed according to the α-stable Lévy probability law. Contrary to the Brownian motion, the trajectories of the α-stable Lévy motion are discontinous, that is exhibit jumps. This feature implies that the process of first passage through the boundary of a given space domain, or the first escape, is different from the process of first arrival (hit) at the boundary. Here we investigate the properties of first escapes and first arrivals for Lévy flights and explore how the asymptotic behavior of the corresponding (passage and hit) probabilities is sensitive to the size of the domain. In particular, we find that the survival probability to stay in a large enough, finite domain has a universal Sparre Andersen temporal scaling ${t}^{-1/2}$, which is transient and changes to an exponential non-universal decay at longer times. Also, the probability to arrive at a finite domain possesses a similar transient Sparre Andersen universality that turns into a non-universal and slower power-law decay in course of time. Finally, we demonstrate that the probability density of the leapover length ℓ over the boundary, related to overshooting events, has an intermediate asymptotics ${{\ell }}^{-(1+\alpha /2)}$ ($0\lt \alpha \lt 2$) which is inherent for the escape from a semi-infinite domain. However, for larger leapovers the probability density decays faster according to the ${{\ell }}^{-(1+\alpha )}$ law. Thus, we find that the laws derived for the α-stable processes on the semi-infinite domain, manifest themselves as transients for Lévy flights on the finite domain