The purpose of this note is to demonstrate survival for the long-range contact process. This process was introduced by Spitzer ; it possesses the same basic evolutionary rule as does the contact process on the integers, except that particles may appear at large distances from already extant particles instead of just as immediate neighbors. We consider here two variants of this model, and show that in both cases the system will survive if (i) it commences from a reasonably dense initial state and (ii) the birth rate for particles is moderately greater than the corresponding death rate. The methodology consists primarily of an energy argument, which provides a lower bound for the particle density of the system.
"A Note on the Survival of the Long-Range Contact Process." Ann. Probab. 9 (5) 885 - 890, October, 1981. https://doi.org/10.1214/aop/1176994316