First passage and arrival time densities for Lévy flights and the failure of the method of images

Abstract

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].

Lévy flights do not always optimize random blind search for sparse targets

“…In contrast to Brownian search with frequent returns to previously visited points in space, as originally pointed out by Shlesinger and Klafter (5) these long jumps improve the efficiency of the search process by leading the random walker to uncorrelated areas of the search space. We recall that for LFs long leap-overs with length distribution pðℓÞ ' ℓ −1−α=2 across a point may frequently occur, and thus the probability of actually arriving at a point is significantly smaller than the passage of the walker across this point (32)(33)(34). Leap-overs are crucial to the understanding of the first arrival of LFs.…”

“…1 generalizes the first arrival dynamics in absence of a drift of ref. 32. Owing to the sink term, f ðx; tÞ is not normalized, that is, the cumulative survival S ðtÞ = R ∞ −∞ f ðx; tÞdx is a decreasing function of time.…”

Lévy flights in the presence of a point sink of finite strength

“…It is well known that encounters in biology such as foraging, predator-prey dynamics and pollination have significant advantages by adopting Lévy strategies for their search [22][23][24], and they can be studied using a fractional reaction-diffusion equation [15]. With a finite sink as given in Eq.…”

Space-fractional Fokker–Planck equation and optimization of random search processes in the presence of an external bias

“…The length of these leapovers is distributed as ℘ l (ℓ) ≃ ℓ −1−α/2 , and is thus wider than the original jump length distribution (1) [48]. This property renders the first arrival of an LF different from the process of first passage, and the first arrival efficiency worsens with decreasing α [49].…”