TY - JOUR

T1 - First passage and arrival time densities for Lévy flights and the failure of the method of images

AU - Chechkin, Aleksei V.

AU - Metzler, Ralf

AU - Gonchar, Vsevolod Y.

AU - Klafter, Joseph

AU - Tanatarov, Leonid V.

PY - 2003/10/17

Y1 - 2003/10/17

N2 - 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) ∼ ℓα/|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.

AB - 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) ∼ ℓα/|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.

UR - http://www.scopus.com/inward/record.url?scp=0142184745&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/36/41/L01

DO - 10.1088/0305-4470/36/41/L01

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AN - SCOPUS:0142184745

SN - 1751-8113

VL - 36

SP - L537-L544

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 41

ER -