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 minus p If the walker ever reaches 0 or N he stays there (This is the Gamblers Ruin problem of Exercise 23 ) If p q show that the function f(i)...

In each case Exercise 27 sho ...

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

Question:

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 6= q then
f(i) = (q/ p)i
is a harmonic function. Use this and the result of Exercise 27 to show that the probability biN of being absorbed in state N starting in state i is
For an alternative derivation of these results see Exercise 24.