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We consider a one-dimensional totally asymmetric nearest-neighbor zero-range process with site-dependent jump-rates -an environment. For each environment p we prove that the set of all invariant measures is the convex hull of a set of product measures with geometric marginals. As a consequence we show that for environments p satisfying certain asymptotic property, there are no invariant measures concentrating on configurations with density bigger than ρ * (p), a critical value. If ρ * (p) is finite we say that there is phase-transition on the density. In this case we prove that if the initial configuration has asymptotic density strictly above ρ * (p), then the process converges to the maximal invariant measure.

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E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. AbstractThe subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.