The Branching Annihilating Process: An Interacting Particle System

Abstract

The branching annihilating process (BAP) is a special case of the branching annihilating walk of Bramson and Gray. In the BAP each particle places offspring on neighbouring sites at unit rate, but when two particles occupy the same site they annihilate each other. We show that the product measure with density 1/2 is the limit starting from any $A \neq \varnothing$. Also considered will be the DBAP (double BAP) in one dimension. In this model a particle always places offspring on both neighbouring sites. The limiting measure here depends critically on whether the initial number of particles is odd or even.