Survival probability of random walks and Lévy flights on a semi-infinite line

Abstract

We consider a one-dimensional random walk (RW) with a continuous and symmetric jump distribution, , characterized by a Lévy index , which includes standard random walks () and Lévy flights (). We study the survival probability, , representing the probability that the RW stays non-negative up to step n, starting initially at . Our main focus is on the x0-dependence of for large n. We show that displays two distinct regimes as x0 varies: (i) for (‘quantum’ regime), the discreteness of the jump process significantly alters the standard scaling behavior of and (ii) for (‘classical’ regime) the discrete-time nature of the process is irrelevant and one recovers the standard scaling behavior (for this corresponds to the standard Brownian scaling limit). The purpose of this paper is to study how precisely the crossover in occurs between the quantum and the classical regime as one increases x0.