Minimal quasi-stationary distributions under nullR-recurrence

The existence of quasi-stationary distributions (QSD) in an absorbing Markov chain entails a stationary behaviour before absorption. In general, depending on the initial distribution, several QSDs may exist. Under some conditions upon the transition matrix between non-absorbing states, we prove that the QSD associated with any Dirac initial distribution, when it exists, is unique, and is the minimal QSD. In other words, if we take this QSD as an initial distribution, the process has the smallest probability of not being absorbed in the first jump.[PUBLICATION ABSTRACT]