Does a drifting walk ever come home?

Question

A walker starts at 0 and each step moves +1 with probability p and -1 with probability q = 1-p, with p 
eq q (a biased walk). What is the probability that the walk ever returns to 0 after it leaves? Express your answer in terms of p and q.

Accepted Answer

First we need a building block: for a biased walk, what is the probability of ever moving one net step against the drift? Suppose p q (upward drift). The probability of ever reaching -1 starting from 0, i.e. ever dropping one level, call it ho{\downarrow}. By the same self-similar logic used for these walks, ho\downarrow satisfies ho\downarrow = q + p\,ho\downarrow^2 (drop directly, or first rise and then make two independent drops). The relevant root is ho\downarrow = \frac{q}{p} \quad (	ext{when } p q), while moving one step with the drift is certain, ho\uparrow = 1. (Symmetric…

Does a drifting walk ever come home?

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