Controlled random walk with a target site

Abstract

We consider a symmetric simple random walk $\{W_i\}$ on $\mathbb{Z}^d, d=1,2$, 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 $\mathbb{P}(W_n=0)$.  We show that for $d=1$, if $m_n \gg (\log n)^{2+\gamma}$ for some $\gamma>0$, there is a strategy for each $n$ yielding $\mathbb{P}(W_n = 0) \to 1$.  For $d=2$, if $m_n \gg n^\epsilon$ for some $\epsilon>0$ then there are strategies yielding $\liminf_n \mathbb{P}(W_n=0)>0$.