Deterministically driven random walks in a random environment on $\mathbb{Z}$

Abstract

We introduce the concept of a deterministic walk in a deterministic environment on a state space $S$ (DWDE), focusing on the case where $S$ is countable. For the deterministic walk in a fixed environment we establish properties analogous to those found in Markov chain theory, but for systems that do not in general have the Markov property (in the stochastic process sense). In particular, we establish hypotheses ensuring that a DWDE on $\mathbb{Z}$ is either recurrent or transient. An immediate consequence of this result is that a symmetric DWDE on $\mathbb{Z}$ is recurrent. Moreover, in the transient case, we show that the probability that the DWDE diverges to $+ \infty$ is either 0 or 1. In certain cases we compute the direction of divergence in the transient case.