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

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

2003-09-19

arxiv.org/abs/cond-mat/0309449v1

Physics

Condensed Matter

Statistical Mechanics

8 pages, 3 figures, iopart.cls style, accepted to J. Phys. A (Lett)

Scientific paper

10.1088/0305-4470/36/41/L01

We discuss the first passage time problem in the semi-infinite interval, for homogeneous stochastic Markov processes with L{\'e}vy stable jump length distributions $\lambda(x)\sim\ell^{\alpha}/|x|^{1+\alpha}$ ($|x|\gg\ell$), namely, L{\'e}vy 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 $\sim t^{-3/2}$. 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.

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