Table of Contents
International Journal of Statistical Mechanics
Volume 2013, Article ID 831390, 9 pages
http://dx.doi.org/10.1155/2013/831390
Research Article

Solution and Analysis of a One-Dimensional First-Passage Problem with a Nonzero Halting Probability

Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo, Tokyo 112-8551, Japan

Received 24 April 2013; Accepted 9 September 2013

Academic Editor: Michel Droz

Copyright © 2013 Ken Yamamoto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper treats a kind of a one-dimensional first-passage problem, which seeks the probability that a random walker first hits the origin at a specified time. In addition to a usual random walk which hops either rightwards or leftwards, the present paper introduces the “halt” that the walker does not hop with a nonzero probability. The solution to the problem is expressed using a Gauss hypergeometric function. The moment generating function of the hitting time is also calculated, and a calculation technique of the moments is developed. The author derives the long-time behavior of the hitting-time distribution, which exhibits power-law behavior if the walker hops to the right and left with equal probability.