The survival of nonattractive interacting particle systems on Z

Abstract

We consider interacting particle systems on $Z$ which allow five types of pairwise interaction: Annihilation, Birth, Coalescence, Death and Exclusion with corresponding rates a, b,c, d, e . We show that whatever the values of a, c, d, e, if the birthrate is high enough there is a positive probability the particle system will survive starting from any finite occupied set. In particular: an IPS with rates a b c d e has a positive probability of survival if

$$ b > 4d + 6a, \quad c + a \geq d + e$$

or

$$ b > 7d + 3a - 3c + 3e, \quad c + a < d + e.$$

We create a suitable supermartingale by extending the method used by Holley and Liggett in their treatment of the contact process.