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.Lévy flights (LFs) and Lévy walks (LWs) are the prime example in the investigation of non-standard transport processes whose stationary solution does not converge towards the Boltzmann form [1,2,4,3]. Being subject to the generalised central limit theorem [5,6], LFs correspond to a Markov process in which extremely long excursions can occur with appreciable probability, whereas in LWs long excursions are penalised through a time cost introduced via a spatiotemporal coupling [7,8]. Applications of LFs and LWs range from the famed flight of an albatross [9], the spreading of spider-monkeys [10], or the grazing patterns of bacteria [11], over economical data [12] to molecular collisions [13] and plasmas [14]. Despite their broad usage, the detailed behaviour of even the simpler, uncoupled LF processes, on which we concentrate in the following, in external potentials and under nontrivial boundary conditions is still not fully explored. Thus, there have recently been discovered bifurcations between multimodal states of the probability density function (PDF) of LFs in steeper than harmonic external fields, in whose presence also the variance becomes finite [15,16], and rich band structures have been reported for LFs in periodic potentials [17].

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Lévy flights in steeper than harmonic potentials have been shown to exhibit finite variance and a critical time at which a bifurcation from an initial mono-modal to a terminal bimodal distribution occurs (Chechkin et al., Phys. Rev. E 67, 010102(R) (2003)). In this paper, we present a detailed study of Lévy flights in potentials of the type U (x) ∝ |x| c with c > 2. Apart from the bifurcation into bimodality, we find the interesting result that for c > 4 a trimodal transient exists due to the temporal overlap between the decay of the central peak around the initial δ-condition and the building up of the two emerging side-peaks, which are characteristic for the stationary state. Thus, for certain system parameters there exists a transient tri-modal distribution of the Lévy flight. These properties of LFs in external potentials of the power-law type can be represented by certain phase diagrams. We also present details about the proof of multi-modality and the numerical procedures to establish the probability distribution of the process.

We investigate the statistical behavior of Lévy flights confined in a symmetric, quartic potential well U(x) proportional, variant x(4). At stationarity, the probability density function features a distinct bimodal shape and decays with power-law tails which are steep enough to give rise to a finite variance, in contrast to free Lévy flights. From a delta-initial condition, a bifurcation of the unimodal state is observed at t(c)>0. The nonlinear oscillator with potential U(x)=ax(2)/2+bx(4)/4, a,b>0, shows a crossover from unimodal to bimodal behavior at stationarity, depending on the ratio a/b.

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