Average Time Spent by a Lévy Flyer on an Interval with Absorbing Boundaries

We consider a Lévy flyer that starts from a point x0 on an interval [O,L] with absorbing boundaries. Using fractional differential equations, we show that the average number of flights the flyer takes before it is absorbed is 2(πα)−1 sin(πα/2) [ x0(L− x0)l −2 0 ]α/2 , where 0 < α < 2 and l0 > 0 are the parameters of the distribution of the flight length l: P (|l| > r) = (l0/r) α, r > l0, and l0 is a small number with respect to x0 and L− x0. For α > 2, when the second moment of the flight length distribution exists, our result is replaced by known results of classical diffusion. We also find a closed form expression for the total length of the flights before absorption, and show that it can be well approximated by the product of the average number of flights and the average length of the flight started at x0. We show that when x0 is small comparatively to L, the average total length has a minimum at α = 1. We discuss the relevance of this result to the problem of foraging, which has received recent attention in the statistical physics literature.