We survey our recent articles dealing with one dimensional attractive zero range processes moving under site disorder. We suppose that the underlying random walks are biased to the right and so hyperbolic scaling is expected. Under the conditions of
our model the process admits a maximal invariant measure. The initial focus of the project was to find conditions on the initial law to entail convergence in distribution to this maximal distribution, when it has a finite density. Somewhat surprisingly, necessary and sufficient conditions were found. In this part hydrody-namic results were employed chiefly as a tool to show distributional convergence but subsequently we developed a theory for hydrodynamic limits treating profiles possessing densities that did not admit corresponding equilibria. Finally we derived strong local equilibrium results.
We consider dynamic random walks where the nearest neighbour jump rates are determined by an underlying supercritical contact process in equilibrium. This has previously been studied by den Hollander and dos Santos and den Hollander, dos Santos, Sido
ravicius. We show the CLT for such a random walk, valid for all supercritical infection rates for the contact process environment.
