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 $m_n\gg n^{ϵ}$ for some $ϵ>0$ then there are strategies yielding $\underset{n}{lim inf}\mathbb{P}(W_n=0)>0$.