Deterministically driven random walks in a random environment on Z

TL;DR

This paper introduces deterministic walks in deterministic environments on countable spaces, establishing conditions for recurrence or transience, and analyzing divergence behavior, extending classical Markov chain concepts to non-Markovian systems.

Contribution

It develops a framework for deterministic walks in deterministic environments, providing criteria for recurrence, transience, and divergence direction, which are analogous to Markov chain properties.

Findings

Symmetric DWDE on Z is recurrent.

Transient DWDE diverges with probability 0 or 1.

Conditions for recurrence and transience are established.

Abstract

We introduce the concept of a deterministic walk in a deterministic environment on a countable state space (DWDE). 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 particular, we establish hypotheses ensuring that a DWDE on $\Z$ is either recurrent or transient. An immediate consequence of this result is that a symmetric DWDE on $\Z$ 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.