We consider a Markov chain on a locally compact metric space with an absorbing set. Necessary and sufficient conditions are provided for the existence of a quasistationary probability distribution.
"On the existence of a quasistationary measure for a Markov chain." Ann. Probab. 29 (1) 437 - 446, February 2001. https://doi.org/10.1214/aop/1008956338