Crossover between Levy and Gaussian regimes in first passage processes

Abstract

We propose a new approach to the problem of the first passage time. Our method is applicable not only to the Wiener process but also to the non--Gaussian L$\acute{\rm e}$vy flights or to more complicated stochastic processes whose distributions are stable. To show the usefulness of the method, we particularly focus on the first passage time problems in the truncated L$\acute{\rm e}$vy flights (the so-called KoBoL processes), in which the arbitrarily large tail of the L$\acute{\rm e}$vy distribution is cut off. We find that the asymptotic scaling law of the first passage time $t$ distribution changes from $t^{-(\alpha +1)/\alpha}$-law (non-Gaussian L$\acute{\rm e}$vy regime) to $t^{-3/2}$-law (Gaussian regime) at the crossover point. This result means that an ultra-slow convergence from the non-Gaussian L$\acute{\rm e}$vy regime to the Gaussian regime is observed not only in the distribution of the real time step for the truncated L$\acute{\rm e}$vy flight but also in the first passage time distribution of the flight. The nature of the crossover in the scaling laws and the scaling relation on the crossover point with respect to the effective cut-off length of the L$\acute{\rm e}$vy distribution are discussed.