Deterministically Driven Random Walks in a Random Environment.

We introduce the concept of a deterministically driven random walk in a random environment on a state space S', focusing on the case where S is countable. Since our construction has a purely deterministic representation we refer to it as a deterministic walk in a deterministic environment (DWDE). For the deterministic walk in a fixed environment we establish properties analogous to those found in Markov chain theory, but for processes that do not in general have the Markov property. In the finite state space setting, we establish hypotheses for the recurrence or transience of a deterministic walk, and the existence of asymptotic occupation times. In the case of a DWDE on Z, we establish hypotheses that ensure that it 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 +00 is either 0 or 1. In certain cases we compute the direction of divergence in the transient case