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, f (η), characterized by a Lévy index µ ∈ (0, 2], which includes standard random walks (µ = 2) and Lévy 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

Noncrossing run-and-tumble particles on a line

“…This effective 'extrapolation' length or 'shifting of the wall' also happens in a class of neutron scattering problems where the shift is known as the Milne extrapolation length-hence we have denoted it by ξ Milne . Similar Milne-like extrapolation lengths also emerge in certain trapping problems of discretetime random walks [23][24][25]. Thus our main conclusion from this section is that while the exponent 1/2 characterizing the power-law decay of S(x 0 , t) is the same for both the passive Brownian and the active RTP, the fingerprint of the 'activeness' actually is manifest in the amplitude of this power-law decay (and not in the exponent).…”

“…Only q − (0, s) remains unknown and yet to be fixed. The pair of linear equations (24) and (25) can be easily solved by inverting the (2 × 2) matrix…”

Survival probability of stochastic processes beyond persistence exponents

“…However, if one aims to evaluate the time t to wait for observing a first-passage event with a given likelihood, or to determine the dependence of the survival probability on the initial distance to the target, one needs to know the prefactor S 0 , which turns out to be much less characterized than the persistence exponent θ. Even for Markovian random walks this problem is not trivial [15], as exemplified by recent studies for one dimensional Levy flights [16], while only scaling relations for S 0 (with the initial distance to the target) are known [17] in fractal domains. However, if the dynamics of the random walker results from interactions with other degrees of freedom, the process becomes non-Markovian and the determination of S 0 becomes much more involved [18].…”

Splitting Probabilities of Jump Processes

“…is the survival probability, ie the probability that the process never crosses 0 during its n first steps, and F 0,∞ (k|x 0 ) is the probability of crossing 0 after exactly steps. We next make use of the asymptotic behavior of q(x 0 , n) obtained in [29], which yields for 1 (x 0 /a µ ) µ n :…”

“…In order to determine the dependence on x 0 of the splitting probability, we use next the large n behavior of the survival probability given by [29]:…”