ar X iv : p hy si cs / 0 60 60 38 v 1 5 J un 2 00 6 Crossover between L é vy and Gaussian regimes in first passage processes

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 Lévy flights or more complicated stochastic processes whose distributions are stable. To show the usefulness of the method, we especially focus on the first passage time problems in the truncated Lévy flights, in which arbitrarily large tail of the Lévy distribution is cut off. We find that the asymptotic scaling law of the first passage time t distribution changes from t-law (non-Gaussian Lévy regime) to t-law (Gaussian regime) at the crossover point. This result means that an ultra-slow convergence from the non-Gaussian Lévy regime to the Gaussian regime is observed not only in the distribution of the real time step for the truncated Lévy 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 Lévy distribution are discussed.