[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Consider a random walker who moves on the integers 0, 1,

\ldots, [math]N[/math], moving one step to the right with probability [math]p[/math] and one step to the left with probability [math]q = 1 - p[/math]. If the walker ever reaches 0 or [math]N[/math] he stays there. (This is the Gambler's Ruin problem of Exercise Exercise.) If [math]p = q[/math] show that the function

[[math]] f(i) = i [[/math]]

is a harmonic function (see Exercise Exercise), and if [math]p \ne q[/math] then

[[math]] f(i) = \biggl(\frac {q}{p}\biggr)^i [[/math]]

is a harmonic function. Use this and the result of Exercise Exercise to show that the probability [math]b_{iN}[/math] of being absorbed in state [math]N[/math] starting in state [math]i[/math] is

[[math]] b_{iN} = \left \{ \matrix{ \frac iN, &\mbox{if}\,\, p = q, \cr \frac{({q \over p})^i - 1}{({q \over p})^{N} - 1}, & \mbox{if}\,\,p \ne q.\cr}\right. [[/math]]

For an alternative derivation of these results see Exercise Exercise.