Probability Question

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Problem 3. Consider the following Markov chain on the state space Q = {2k, k = - 2, -1,-0,1,2, - } . If your current position is x = 2k, then the next step you jump to 2.c = 2k+1 or x/2 = 2k-1 with equal probabilities. (a) Let D = {2-N,2-N+1 - , 1,2, , 2N }, for some N > 1. Suppose you absorb the chain when it hits Dc. Show that f(x) = log(x) is a harmonic function at any x in D.

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