Let $S$ be a countable set and let $\xi_t$ be a Markov process on the subsets of $S$. Harris has given criteria for the existence of a dual process $\xi_t^\ast$ on the finite subsets of $S$. By extending Harris's notion of duality the class of $\xi_t$ which have dual processes is enlarged. The dual processes are then used to study the ergodic behavior of $\xi_t$. Also treated is a class of $\xi_t$ which have growing dual processes.
"Applications of Duality to a Class of Markov Processes." Ann. Probab. 5 (4) 522 - 532, August, 1977. https://doi.org/10.1214/aop/1176995758