On the surviving probability of an annihilating branching process and application to a nonlinear voter model

We study a discrete time interacting particle system which can be considered as an annihilating branching process on Z where at each time every particle either performs a jump as a nearest neighbor random walk, or splits (with probability ε) into two particles which will occupy the nearest neighbor sites. Furthermore, if two particles come to the same site, then they are removed from the system. We show that if the branching probability ε>0 is small enough, and the number of particles at initial time is finite, then the surviving probability, i.e. the probability p(t) that there is at least one particle at time t decays to zero exponentially fast. This result is applied to a nonlinear discrete time voter model (in a random and nonrandom environment) obtained as a small perturbation with parameter ε of the classical voter model. For this class of models, we show that if ε>0 is small enough, then the process converges to a unique invariant probability measure independently on the initial distribution. It is known that the classical one-dimensional voter model (in a random environment as well as without environment) is not ergodic, that is there exist at least two extremal invariant probability measures. Our results prove therefore the phase transition in ε=0.