Conservative random walk

Recently, in [11], the “coin-turning walk” was introduced on Z. It is a non-Markovian process where steps form (possibly) time-inhomogeneous Markov chain. In this article, we follow up investigation by introducing analogous processes Zd,d≥2: at time n direction of “updated” with probability pn; otherwise next step repeats previous one. We study some fundamental properties these walks, such as transience/recurrence and scaling limits. Our results complement ones literature about “correlated” (or “Newtonian”) “persistent” random walks.