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

In many areas of science and applied mathematics, important events often get initiated when a stochastic variable crosses a threshold value. Financial assets may be sold at a particular price, or a chemical reaction initiated when two particles come close enough. In such processes, the time of triggering is known as the first-hitting or first-passage time. For one of the simplest cases – standard Brownian motion – important new results have been achieved within the last decade. For non-Brownian stochastic processes, things are more difficult, as such processes typically exhibit anomalous diffusion: for long times, the mean squared displacement of the random variable grows not linearly with time, but in proportion to time raised to an exponent ν. Subdiffusion or dispersive transport occurs for 0 < ν < 1, and superdiffusion or enhanced transport for 1 < ν < 2. Varieties of anomalous diffusion arise in turbulent flows, charge carrier transport in amorphous semiconductors, as well as in human travel and biological cell migration.
Unusual first-passage and first-hitting properties also arise for Lévy flight (LF) and Lévy walk (LW) processes, among the most prominent models for the description of superdiffusive processes. These random processes have scale-free probability distributions for individual movements, and are non-ergodic in the sense that long time and ensemble averages of physical observables are different. In a new paper, LML External Fellow Rainer Klages and colleagues explore the first-passage and first-hitting properties of LFs and LWs in one dimension as functions of the Lévy stable index α, which determines the statistical likelihood of increasingly large jumps.
As they show, for α > 1, these two models give qualitatively identical results at long times due to the finite average length of a relocation. The situation changes markedly for α < 1, in which case the scaling of the probability distribution functions depends strongly on the exact model. The authors suggest that the quantitative data they have derived for the associated first-passage and first-hitting time probability distribution functions should be valuable for practical applications of Lévy flights and Lévy walks, especially in areas such as random search processes and the analysis of computer algorithms.
The paper is available at

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