Conservative random walk

Recently, in [11], the "coin-turning walk" was introduced on 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 Z(d), d >= 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.