Question 11.2.33: Consider a random walker who moves on the integers 0, 1, .........

Consider a random walker who moves on the integers 0, 1, …, N, moving one step to the right with probability p and one step to the left with probability q =1 − p. If the walker ever reaches 0 or N he stays there. (This is the Gambler’s Ruin problem of Exercise 23.) If p = q show that the function

f(i)= i

is a harmonic function (see Exercise 27), and if p ≠ q then

f(i)= \left(\frac{q}{p} \right)^i

is a harmonic function. Use this and the result of Exercise 27 to show that the probability b_iN of being absorbed in state N starting in state i is

b_{iN}=\begin{cases} \frac{i}{N}, & if ~p = q,\\ \frac{(\frac{q}{p} )^i-1}{(\frac{q}{p} )^N-1}, & if ~p ≠ q.\end{cases}

For an alternative derivation of these results see Exercise 24.