Lévy flights are paradigmatic generalised random walk processes, in which the independent stationary increments—the ‘jump lengths’—are drawn from an -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 first-passage 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 and the skewness (asymmetry) parameter . The other approach is based on the stochastic Langevin equation with -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.

We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Lévy flights by leveraging the Fokker–Planck equation with a $\delta$-sink and an asymmetric space-fractional derivative operator with stable index $\alpha$ and asymmetry (skewness) parameter $\beta$. We find exact analytical results for the probability density of first-arrival times and the search efficiency, and we analyse their behaviour within the limits of short and long times. We find that when the starting point of the searcher is to the right of the target, random search by Brownian motion is more efficient than Lévy flights with $\beta \le 0$ (with a rightward bias) for short initial distances, while for $\beta >0$ (with a leftward bias) Lévy flights with $\alpha \to 1$ are more efficient. When increasing the initial distance of the searcher to the target, Lévy flight search (except for $\alpha =1$ with $\beta =0$) is more efficient than the Brownian search. Moreover, the asymmetry in jumps leads to essentially higher efficiency of the Lévy search compared to symmetric Lévy flights at both short and long distances, and the effect is more pronounced for stable indices $\alpha$ close to unity.

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.

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 perform numerical studies of a thermally driven, overdamped particle in a random quenched force field, known as the Sinai model. We compare the unbounded motion on an infinite 1-dimensional domain to the motion in bounded domains with reflecting boundaries and show that the unbounded motion is at every time close to the equilibrium state of a finite system of growing size. This is due to time scale separation: inside wells of the random potential, there is relatively fast equilibration, while the motion across major potential barriers is ultraslow. Quantities studied by us are the time dependent mean squared displacement, the time dependent mean energy of an ensemble of particles, and the time dependent entropy of the probability distribution. Using a very fast numerical algorithm, we can explore times up top 10¹⁷ steps and thereby also study finite-time crossover phenomena.

We provide the first-passage statistics of Lévy flights in a one-dimensional semi-infinite domain, considering the effects of a constant drift µ while the jump length distribution is symmetric. By solving the space-fractional Fokker–Planck equation governing the evolution of the probability density function (PDF), we derive expressions for both the survival probability and the first-passage time for given values of the stability index α and drift parameter µ. Our findings are further validated by comparing them with simulations from the stochastic Langevin equation driven by α-stable noise. Additionally, we make use of the Skorokhod theorem for processes with stationary independent increments and show that our numerical results are in good agreement with the analytical expressions for the PDF of first-passage times and survival probabilities. Specifically, in the asymptotic long-time limit, we show that for α = 2 (Brownian motion), the first-passage time distribution follows the classical Lévy–Smirnov form and decays exponentially, irrespective of the drift direction. The survival probability exhibits a distinct asymptotic behaviour: for positive drift, it decreases exponentially, while for negative drift, it saturates to a finite value, indicating a nonzero probability for the particle to escape to − ∞ . For α = 1 (Cauchy process), the survival probability and the first-passage time density exhibit a power-law decay, with µ-dependent exponents. For 0 < α < 1 , we find that the long-time asymptotic remains consistent with the drift-free case, but with a µ-dependent prefactor. Finally, for α-stable processes with 1 < α < 2 , the first-passage time density follows a power-law decay, whose exponent depends on both α and the drift direction; concurrently, the survival probability decays in power-law form for positive drift, while in the case of negative drift, it saturates to a finite value.