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

We consider a one-dimensional random walk (RW) with a continuous and symmetric jump distribution, f (η), characterized by a Levy index μϵ (0, 2], which includes standard random walks (μ = 2) and Lvy flights (0 < μ <2). We study the survival probability, q(x₀, n), representing the probability that the RW stays non-negative up to step n, starting initially at x₀≤ 0. Our main focus is on the x₀-dependence of q(x₀, n) for large n. We show that q(x₀, n) displays two distinct regimes as x0 varies: (i) for x₀ = O(1) (quantum regime), the discreteness of the jump process significantly alters the standard scaling behavior of q(x₀, n) and (ii) for x0 = O(n^1/μ) (classical regime) the discrete-time nature of the process is irrelevant and one recovers the standard scaling behavior (for μ = 2 this corresponds to the standard Brownian scaling limit). The purpose of this paper is to study how precisely the crossover in q(x₀, n) occurs between the quantum and the classical regime as one increases x0.