Absorbing boundary in one-dimensional anomalous transport.

Abstract

In this paper we study the space-time probability distribution Q(x,t) of a random walk subject to an absorbing boundary at the origin x=0 for motion controlled by L\'evy flights and L\'evy walks characterized by the exponent \ensuremath{\gamma}. We find that the method of images, usually applicable to Brownian motion, may break down for L\'evy processes. We calculate the distribution Q(x,t) to be at x>0 after time t>0 having started at the origin assuming that the boundary is effective at time t>0. We show that Q(x,t) depends on the details of the underlying process, Q(x,t)\ensuremath{\sim}${\mathit{x}}^{\ensuremath{\gamma}/2}$/${\mathit{t}}^{1+1/\ensuremath{\gamma}}$, 1\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}2 for small x, while total survival is independent of the spatial realization of motion and displays a universal behavior. We also discuss the related Smoluchowski boundary condition problem.