Crossover between Levy and Gaussian regimes in first-passage processes

We propose an 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 Levy 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 Levy flights (the so-called KoBoL processes from Koponen, Boyarchenko, and Levendorskii), in which the arbitrarily large tail of the Levy 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 Levy regime) to t(-3/2)-law (Gaussian regime) at the crossover point. This result means that an ultraslow convergence from the non-Gaussian Levy regime to the Gaussian regime is observed not only in the distribution of the real time step for the truncated Levy 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 cutoff length of the Levy distribution are discussed.