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

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. 1. Introduction The first-passage problem is a useful and fundamental problem in statistical physics [1]. In this problem we seek the probability that a diffusion particle or a random walker first reaches a specific set of sites at a specified time. The first-passage problem has many applications in statistical physics, such as reaction-rate theory [2], neuron dynamics [3], and economic analysis [4]. The simplest form of the first-passage problem is a random walker hopping on a one-dimensional lattice until it hits the origin. This problem is called the gambler's ruin problem in probability theory [5]—the motion of the random walker is regarded as the dynamics of a gambler's bankroll. Consider a random walker on a one-dimensional lattice hopping to the right with probability and left with in a single time step. The problem seeks statistical properties of duration , the time at which the walker from the position first hits the origin (of course, is a random variable.) The central quantity is the probability that the walker at first hits the origin after time . According to [5], the solution is obtained as by solving the equation with initial and boundary conditions and ( ). The coefficient before in (1) is the number of different paths from hitting the origin first at time , and it is connected with the reflection principle of a random walk [6]. Obviously, holds when , because . The present paper analyzes an extended form of the above one-dimensional first-passage problem; a random walker hops to the right with probability , to the left with , and it does not hop with probability . Figure 1 schematically shows the problem. The only difference from the original ruin problem is that the halting probability is introduced, but the results change greatly. In fact, the solution (1) of nonhalting case ( ) is superseded by a formula involving a Gauss