Controlled random walk with a target site

We consider a simple random walk W_i in 1 or 2 dimensions, in which the walker may choose to stand still for a limited time. The time horizon is n, the maximum consecutive time steps which can be spent standing still is m_n and the goal is to maximize P(W_n=0). We show that for dimension 1, if m_n grows faster than (\log n)^{2+γ} for some γ>0, there is a strategy for each n such that P(W_n = 0) approaches 1. For dimension 2, if m_n grows faster than a positive power of n then there are strategies keeping P(W_n=0) bounded away from 0.

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