The biased annihilating branching process

In the biased annihilating branching process, particles place offspring on empty neighboring sites at rate A and destroy neighbors at rate 1. It is conjectured that for any . . 0 the population will spread to ., and this is shown in one dimension for The process on a finite graph when starting with a non-empty configuration has limiting distribution v. /(. +1), the product measure with density ./(1 +.). It is shown that v. /(. +1) and . Ø are the only stationary distributions on Moreover, if and the initial configuration is non-empty, then the limiting measure is v. /(. +1) provided the initial measure converges.