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

الملخص EN

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.