  In this paper we consider a class of one-dimensional interacting particle
systems in equilibrium, constituting a dynamic random environment, together
with a nearest-neighbor random walk that on occupied/vacant sites has a local
drift to the right/left. We adapt a regeneration-time argument originally
developed by Comets and Zeitouni for static random environments to prove that,
under a space-time mixing property for the dynamic random environment called
cone-mixing, the random walk has an a.s. constant global speed. In addition, we
show that if the dynamic random environment is exponentially mixing in
space-time and the local drifts are small, then the global speed can be written
as a power series in the size of the local drifts. From the first term in this
series the sign of the global speed can be read off. The results can be easily
extended to higher dimensions.
