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

We discuss the first passage time problem in the semi-infinite interval, for homogeneous stochastic Markov processes with Levy stable jump length distributions lambda(x) similar to l(alpha)/\x\(1+alpha) (\x\ much greater than f), namely, Levy 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 similar tot(-3/2). Conversely, we show that for LF's 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.