Existence of quasi-stationary measures for asymmetric attractive particle systems on \ZZd

We show the existence of nontrivial quasi-stationary measures for conservative attractive particle systems on \ZZd conditioned on avoiding an increasing local set \A. Moreover, we exhibit a sequence of measures {.n}, whose .-limit set consists of quasi-stationary measures. For zero-range processes, with stationary measure \nur, we prove the existence of an L2(\nur) nonnegative eigenvector for the generator with Dirichlet boundary on \A, after establishing a priori bounds on the {.n}.