Average Time Spent by Levy Flights and Walks on an Interval with Absorbing Boundaries (original) (raw)

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...

Should i stay or should i go? zero-size jumps in random walks for lévy flights

Fractional Calculus and Applied Analysis

We study Markovian continuos-time random walk models for Lévy flights and we show an example in which the convergence to stable densities is not guaranteed when jumps follow a bi-modal power-law distribution that is equal to zero in zero. The significance of this result is two-fold: i) with regard to the probabilistic derivation of the fractional diffusion equation and also ii) with regard to the concept of site fidelity in the framework of Lévy-like motion for wild animals.

Diffusion regimes in Lévy flights with trapping

Physica A: Statistical Mechanics and its Applications, 1999

The diffusion of a walk in the presence of traps is investigated. Different diffusion regimes are obtained considering the magnitude of the fluctuations in waiting times and jump distances. A constant velocity during the jump motion is assumed to avoid the divergence of the mean squared displacement. Using the limit theorems of the theory of Lévy stable distributions we have provided a characterization of the different diffusion regimes. 02.90 +p, 61.40 +b, 81.90 +c

Introduction to the Theory of Lévy Flights

Anomalous Transport, 2008

Lévy flights, also referred to as Lévy motion, stand for a class of non-Gaussian random processes whose stationary increments are distributed according to a Lévy stable distribution originally studied by French mathematician Paul Pierre Lévy. Lévy stable laws are important for three fundamental properties: (i) similar to the Gaussian law, Lévy stable laws form the basin of attraction for sums of random variables.

First passage time moments of asymmetric Lévy flights

Journal of Physics A: Mathematical and Theoretical

We investigate the first-passage dynamics of symmetric and asymmetric Lévy flights in semi-infinite and bounded intervals. By solving the space-fractional diffusion equation, we analyse the fractional-order moments of the first-passage time probability density function for different values of the index of stability and the skewness parameter. A comparison with results using the Langevin approach to Lévy flights is presented. For the semi-infinite domain, in certain special cases analytic results are derived explicitly, and in bounded intervals a general analytical expression for the mean first-passage time of Lévy flights with arbitrary skewness is presented. These results are complemented with extensive numerical analyses.

Lévy flights versus Lévy walks in bounded domains

Physical Review E, 2017

Lévy flights and Lévy walks serve as two paradigms of random walks resembling common features but also bearing fundamental differences. One of the main dissimilarities are discontinuity versus continuity of their trajectories and infinite versus finite propagation velocity. In consequence, well developed theory of Lévy flights is associated with their pathological physical properties, which in turn are resolved by the concept of Lévy walks. Here, we explore Lévy flights and Lévy walks models on bounded domains examining their differences and analogies. We investigate analytically and numerically whether and under which conditions both approaches yield similar results in terms of selected statistical observables characterizing the motion: the survival probability, mean first passage time and stationary PDFs. It is demonstrated that similarity of models is affected by the type of boundary conditions and value of the stability index defining asymptotics of the jump length distribution.

First passage times of Lévy flights coexisting with subdiffusion

Physical Review E, 2007

We investigate both analytically and numerically the first passage time ͑FPT͒ problem in one dimension for anomalous diffusion processes in which Lévy flights and subdiffusion coexist. We analyze the FPT for three subclasses of Lévy stable motions: ͑i͒ symmetric Lévy motions characterized by Lévy index , 0Ͻ Ͻ 2, and skewness parameter ␤ =0, ͑ii͒ one-sided Lévy motions with , 0Ͻ Ͻ 1, and skewness ␤ = 1, and ͑iii͒ twosided skewed Lévy motions, the extreme case, 1 Ͻ Ͻ 2, and skewness ␤ = −1. In all three cases the waiting times between successive jumps are heavy tailed with index ␣. We show that in all three cases the FPT distributions are power laws. Our findings extend earlier studies on FPTs of Lévy flights by considering the interplay between long rests and the Lévy long jumps.

Towards deterministic equations for Levy walks: The fractional material derivative

Physical Review E, 2003

Lévy walks are random processes with an underlying spatiotemporal coupling. This coupling penalizes long jumps, and therefore Lévy walks give a proper stochastic description for a particle's motion with broad jump length distribution. We derive a generalized dynamical formulation for Lévy walks in which the fractional equivalent of the material derivative occurs. Our approach will be useful for the dynamical formulation of Lévy walks in an external force field or in phase space for which the description in terms of the continuous time random walk or its corresponding generalized master equation are less well suited.