Absorbing boundary in one-dimensional anomalous transport

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 Lévy flights and Lévy walks characterized by the exponent γ. We find that the method of images, usually applicable to Brownian motion, may break down for Lévy 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)∼xγ/2/ t1+1/γ, 1≤γ≤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.