Lévy Flights are paradigmatic generalised random walk processes, in which the independent stationary increments-the "jump lengths"-are drawn from an α-stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Lévy Flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Lévy Flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Lévy Flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of firstpassage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index α and the skewness (asymmetry) parameter β. The other approach is based on the stochastic Langevin equation with α-stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.∞ 0 t℘(t)dt can capture some aspects of this dynamics, ‡ the full information encoded in ℘(t) provides significant additional insight [114,115,116]. Here we study the first-passage properties for a general class of α-stable Lévy laws. We go beyond previous approaches [113,117,118,119,120,121,122,123,124,125,126,127,128] focusing on symmetric and one-sided α-stable relocation distributions and consider α-stable laws with arbitrary asymmetry in semi-infinite and bounded domains. Our approach is based on the convenient formulation of LFs in terms of the space-fractional diffusion equation. We derive these integro-differential equations for LFs based on general asymmetric α-stable distributions of relocation lengths in finite domains, and thus go beyond studies of the exit time and escape probability in bounded domain for symmetric LFs [129,130,131]. An important aspect of this study is that we complement our results with numerical analyses of the (stochastic) Langevin equation for LFs and show how both approaches complement each other.The paper is organised as follows. In section 2 we define Lévy stable laws and the associated fractional diffusion equation. In section 3 we set up our numerical model for the fractional diffusion equation and the associated Langevin equation. Moreover, a comparison between the numerical method and α-stable distributions for symmetric and asymmetric den...

The escape from a potential well is an archetypal problem in the study of stochastic dynamical systems, representing real-world situations from chemical reactions to leaving an established home range in movement ecology. Concurrently, Lévy noise is a well-established approach to model systems characterized by statistical outliers and diverging higher order moments, ranging from gene expression control to the movement patterns of animals and humans. Here, we study the problem of Lévy noise-driven escape from an almost rectangular, arctangent potential well restricted by two absorbing boundaries, mostly under the action of the Cauchy noise. We unveil analogies of the observed transient dynamics to the general properties of stationary states of Lévy processes in single-well potentials. The first-escape dynamics is shown to exhibit exponential tails. We examine the dependence of the escape on the shape parameters, steepness, and height of the arctangent potential. Finally, we explore in detail the behavior of the probability densities of the first-escape time and the last-hitting point.

We investigate the first-passage dynamics of symmetric and asymmetric Lévy flights in a 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.