The Basic Dichotomy

A natural counterpart to the finite interval is the first-passage properties of the semi-infinite interval [0,∞] with absorption at x = 0. Once again, this is a classical geometry for studying first-passage processes, and many of the references mentioned at the outset of Chap. 2 are again pertinent. In particular, the text by Karlin and Taylor (1975) gives a particularly comprehensive discussion about first-passage times in the semi-infinite interval with arbitrary hopping rates between neighboring sites. Once again, however, our focus is on simple diffusion or the nearest-neighbor randomwalk. For these processes, the possibility of a diffusing particle making arbitrarily large excursions before certain trapping takes place leads to an infinite mean lifetime. On the other hand, the recurrence of diffusion in one dimension means that the particle must eventually return to its starting point. This dichotomy between infinite lifetime and certain trapping leads to a variety of extremely surprising first-passage-related properties both for the semi-infinite interval and the infinite system.

Perhaps the most amazing such property is the arcsine law for the probability of long leads in a symmetric nearest-neighbor random walk in an unbounded domain. Although this law applies to the unrestricted random walk, it is intimately based on the statistics of returns to the origin and thus fits naturally in our discussion of first-passage on the semi-infinite interval. Our natural expectation is that, for a random walk which starts at x = 0, approximately one-half of the total time would be spent on the positive axis and the remaining one-half of the time on the negative axis. Surprisingly, this is the least probable outcome.