Average Time Spent by Levy Flights and Walks on an Interval with Absorbing Boundaries

Abstract

We consider a Levy flyer of order alpha that starts from a point x0 on an interval [O,L] with absorbing boundaries. We find a closed-form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed. These two quantities are equivalent to the mean first passage times for Levy flights and Levy walks, respectively. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for both quantities in the continuous limit. We show that numerical solutions for the discrete Levy processes converge to the continuous approximations in all cases except the case of alpha approaching 2 and the cases of x0 near absorbing boundaries. For alpha larger than 2 when the second moment of the flight length distribution exists, our result is replaced by known results of classical diffusion. We show that if x0 is placed in the vicinity of absorbing boundaries, the average total length has a minimum at alpha=1, corresponding to the Cauchy distribution. We discuss the relevance of this result to the problem of foraging, which has received recent attention in the statistical physics literature.