ANNIHILATING BRANCHING-PROCESSES

We consider Markov processes eta-t-subset-of Z(d) in which (i) particles die at rate sigma greater-than-or-equal-to 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 sigma = 0 product measure with density 1/2 is a stationary distribution; we show it is the limit whenever P(eta-o not-equal phi) = 1. We also show that if sigma is small there is a nontrivial stationary distribution, and that for any sigma there are most two extremal translation invariant stationary distributions.