Existence of quasi-stationary measures for asymmetric attractive particle systems on ${\text{ZZ}}^d$

We show the existence of non-trivial quasi-stationary measures for conservative attractive particle systems on ${\text{ZZ}}^d$ conditioned on avoiding an increasing local set $\text{A}$. Moreover, we exhibit a sequence of measures $\{{\nu }_n\}$, whose $\omega$-limit set consists of quasi-stationary measures. For zero range processes, with stationary measure $\text{nur}$, we prove the existence of an $L^2(\text{nur})$ nonnegative eigenvector for the generator with Dirichlet boundary on $\text{A}$, after establishing a priori bounds on the $\{{\nu }_n\}$.