Conservative random walk

Abstract

Recently, in [11], the “coin-turning walk” was introduced on $\mathbb{Z}$. It is a non-Markovian process where the steps form a (possibly) time-inhomogeneous Markov chain. In this article, we follow up the investigation by introducing analogous processes in ${\mathbb{Z}}^d,d\ge 2$: at time n the direction of the process is “updated” with probability $p_n$; otherwise the next step repeats the previous one. We study some of the fundamental properties of these walks, such as transience/recurrence and scaling limits.

Our results complement previous ones in the literature about “correlated” (or “Newtonian”) and “persistent” random walks.