Average time spent by Levy flights and walks on an interval with absorbingboundaries - art. no. 041108

We consider a Levy flyer of order alpha that starts from a point x(o) 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 equi valent to the mean first passage times for Levy flights and Levy walks, res pectively. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for both quantities in the continuous lim it. We show that numerical solutions for the discrete Levy processes conver ge to the continuous approximations in all cases except the case of alpha-- >2, and the cases of x(o)-->0 and x(o)-L. For alpha >2, when the second mom ent of the flight length distribution exists, our result is replaced by kno wn results of classical diffusion. We show that if x(o) is placed in the vi cinity of absorbing boundaries, the average total length has a minimum at a lpha =1, corresponding to the Cauchy distribution. We discuss the relevance of this result to the problem of foraging, which has received recent atten tion in the statistical physics literature.