Annihilating branching processes

We consider Markov processes [eta]t [subset of] d in which (i) particles die at rate [delta] [greater-or-equal, slanted] 0, (ii) births from x to a neighboring y occur at rate 1, and (iii) when a new particle lands on an occupied site the particles annihilate each other and a vacant site results. When [delta] = 0 product measure with density is a stationary distribution; we show it is the limit whenever P([eta]0[not equal to] ø) = 1. We also show that if [delta] is small there is a nontrivial stationary distribution, and that for any [delta] there are most two extremal translation invariant stationary distributions.