Dependent double branching annihilating random walk

Double (or parity conserving) branching annihilating random walk, introduced in [19], is a one-dimensional non-attractive particle system in which positive and negative particles perform nearest neighbor hopping, produce two offsprings to neighboring lattice points and annihilate when they meet. Given an odd number of initial particles, positive recurrence as seen from the leftmost particle position was first proved in [2] and, subsequently in a much more general setup, in [16]. These results assume that jump rates of the various moves do not depend on the configuration of the particles not involved in these moves. The present article deals with the case when the jump rates are affected by the locations of several particles in the system. Motivation for such models comes from non-attractive interacting particle systems with particle conservation. Under suitable assumptions we establish the existence of the process, and prove that the one-particle state is positive recurrent. We achieve this by arguments similar to those appeared in [16]. We also extend our results to some cases of long range jumps, when branching can also occur to non-neighboring sites. We outline and discuss several particular examples of models where our results apply.