A theorem exhibiting the duality between certain infinite systems of interacting stochastic processes and a type of branching process is proved. This duality is then used to study the ergodic properties of the infinite system. In the case of the vector model a complete understanding of the ergodic behavior is obtained.
"Ergodic Theorems for Weakly Interacting Infinite Systems and the Voter Model." Ann. Probab. 3 (4) 643 - 663, August, 1975. https://doi.org/10.1214/aop/1176996306