Properties of Lévy flights on an interval with absorbing boundaries

Abstract

We consider a Lévy flyer of order α that starts from a point x 0 on an interval [0, L] with absorbing boundaries. We find a closed-form expression for an arbitrary average quantity, characterizing the trajectory of the flyer, such as mean first passage time, average total path length, probability to be absorbed by one of the boundaries. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for these quantities in the continuous limit. We find numerically the eigenfunctions and the eigenvalues of these equations. We study how the results of Monte-Carlo simulations of the Lévy flights with different flight length distributions converge to the continuous approximations. We show that if x 0 is placed in the vicinity of absorbing boundaries, the average total path length has a minimum near α=1, corresponding to the Cauchy distribution. We discuss the relevance of these results to the problem of biological foraging and transmission of light through cloudy atmosphere.