Non-uniqueness of the first passage time density of Lévy random processes (original) (raw)

On the first passage time and leapover properties of Lévy motions

Physica A: Statistical Mechanics and its Applications, 2007

We investigate two coupled properties of Le´vy stable random motions: the first passage times (FPTs) and the first passage leapovers (FPLs). While, in general, the FPT problem has been studied quite extensively, the FPL problem has hardly attracted any attention. Considering a particle that starts at the origin and performs random jumps with independent increments chosen from a Le´vy stable probability law l a,b (x), the FPT measures how long it takes the particle to arrive at or cross a target. The FPL addresses a different question: given that the first passage jump crosses the target, then how far does it get beyond the target? These two properties are investigated for three subclasses of Le´vy stable motions: (i) symmetric Le´vy motions characterized by Le´vy index a(0oao2) and skewness parameter b ¼ 0, (ii) one-sided Le´vy motions with 0oao1, b ¼ 1, and (iii) two-sided skewed Le´vy motions, the extreme case, 1oao2, b ¼ À1.

Asymptotic behaviour of first passage time distributions for Lévy processes

Probability Theory and Related Fields, 2013

In this paper we establish local estimates for the first passage time of a subordinator under the assumption that it belongs to the Feller class, either at zero or infinity, having as a particular case the subordinators which are in the domain of attraction of a stable distribution, either at zero or infinity. To derive these results we first obtain uniform local estimates for the one dimensional distribution of such a subordinator, which sharpen those obtained by Jain and Pruitt . In the particular case of a subordinator in the domain of attraction of a stable distribution our results are the analogue of the results obtained by the authors in [6] for non-monotone Lévy processes. For subordinators an approach different to that in [6] is necessary because the excursion techniques are not available and also because typically in the non-monotone case the tail distribution of the first passage time has polynomial decrease, while in the subordinator case it is exponential. ‡ Research funded by the CONACYT Project Teoría y aplicaciones de procesos de Lévy where b denotes the drift and Π the Lévy measure of X. We will write ψ * for the exponent of {Xt − bt, t ≥ 0}, so that ψ * (λ) := ψ(λ) − bλ, λ ≥ 0.

Lévy-Brownian motion on finite intervals: Mean first passage time analysis

Physical Review E, 2006

We present the analysis of the first passage time problem on a finite interval for the generalized Wiener process that is driven by Lévy stable noises. The complexity of the first passage time statistics (mean first passage time, cumulative first passage time distribution) is elucidated together with a discussion of the proper setup of corresponding boundary conditions that correctly yield the statistics of first passages for these non-Gaussian noises. The validity of the method is tested numerically and compared against analytical formulae when the stability index α approaches 2, recovering in this limit the standard results for the Fokker-Planck dynamics driven by Gaussian white noise.

Dynamical approach to Lévy processes

We derive the diffusion process generated by a correlated dichotomous fluctuating variable y starting from a Liouville-like equation by means of a projection procedure. This approach makes it possible to derive all statistical properties of the diffusion process from the correlation function of the dichotomous fluctuating variable ⌽ y (t). Of special interest is that the distribution of the times of sojourn in the two states of the fluctuating process is proportional to d 2 ⌽ y (t)/dt 2 . Furthermore, in the special case where ⌽ y (t) has an inverse power law, with the index ␤ ranging from 0 to 1, thus making it nonintegrable, we show analytically that the statistics of the diffusing variable approximate in the long-time limit the ␣-stable Lévy distributions. The departure of the diffusion process of dynamical origin from the ideal condition of the Lévy statistics is established by means of a simple analytical expression. We note, first of all, that the characteristic function of a genuine Lévy process should be an exponential in time. We evaluate the correction to this exponential and show it to be expressed by a harmonic time oscillation modulated by the correlation function ⌽ y (t). Since the characteristic function can be given a spectroscopic significance, we also discuss the relevance of our results within this context.

First passage and arrival time densities for Lévy flights and the failure of the method of images

Journal of Physics A: Mathematical and General, 2003

We discuss the first passage time problem in the semi-infinite interval, for homogeneous stochastic Markov processes with Lévy stable jump length distributions λ(x) ∼ ℓ α /|x| 1+α (|x| ≫ ℓ), namely, Lévy flights (LFs). In particular, we demonstrate that the method of images leads to a result, which violates a theorem due to Sparre Andersen, according to which an arbitrary continuous and symmetric jump length distribution produces a first passage time density (FPTD) governed by the universal long-time decay ∼ t −3/2 . Conversely, we show that for LFs the direct definition known from Gaussian processes in fact defines the probability density of first arrival, which for LFs differs from the FPTD. Our findings are corroborated by numerical results.

Random time averaged diffusivities for Lévy walks

The European Physical Journal B, 2013

We investigate a Lévy-Walk alternating between velocities ±v0 with opposite sign. The sojourn time probability distribution at large times is a power law lacking its mean or second moment. The first case corresponds to a ballistic regime where the ensemble averaged mean squared displacement (MSD) at large times is x 2 ∝ t 2 , the latter to enhanced diffusion with x 2 ∝ t ν , 1 < ν < 2. The correlation function and the time averaged MSD are calculated. In the ballistic case, the deviations of the time averaged MSD from a purely ballistic behavior are shown to be distributed according to a Mittag-Leffler density function. In the enhanced diffusion regime, the fluctuations of the time averages MSD vanish at large times, yet very slowly. In both cases we quantify the discrepancy between the time averaged and ensemble averaged MSDs.

First passage and first hitting times of Lévy flights and Lévy walks

New Journal of Physics

For both Lévy flight and Lévy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For Lévy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it (‘leapovers’), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the Lévy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer a...

A note on first passage probabilities of a Lévy process reflected at a general barrier

Probability and Mathematical Statistics

A NOTE ON FIRST PASSAGE PROBABILITIES OF A LÉVY PROCESS REFLECTED AT A GENERAL BARRIERIn this paper we analyze a Lévy process reflected at a general possibly random barrier. For this process we prove the Central Limit Theorem for the first passage time. We also give the finite-time first passage probability asymptotics.

One-dimensional stochastic Lévy-Lorentz gas

Physical Review E, 2000

We introduce a Lévy-Lorentz gas in which a light particle is scattered by static point scatterers arranged on a line. We investigate the case where the intervals between scatterers ͕ i ͖ are independent random variables identically distributed according to the probability density function ()ϳ Ϫ(1ϩ␥) . We show that under certain conditions the mean square displacement of the particle obeys ͗x 2 (t)͘уCt 3Ϫ␥ for 1Ͻ␥Ͻ2. This behavior is compatible with a renewal Lévy walk scheme. We discuss the importance of rare events in the proper characterization of the diffusion process.