Conservative Random Walk

TL;DR

This paper extends the coin-turning walk to higher dimensions, analyzing its fundamental properties like transience, recurrence, and scaling limits, thereby enriching the understanding of correlated and persistent random walks.

Contribution

Introduces higher-dimensional analogues of the coin-turning walk and investigates their fundamental properties, including transience, recurrence, and scaling limits.

Findings

Analysis of transience and recurrence in higher dimensions

Identification of scaling limits for the new processes

Comparison with correlated and persistent random walks

Abstract

Recently, in ["The coin-turning walk and its scaling limit", Electronic Journal of Probability, 25 (2020)], 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.