On quasi-stationaries for symmetric Markov processes

Abstract

This paper intends to give some mild sufficient conditions for the existence and uniqueness of quasi-stationary distributions (QSDs) for general symmetric Markov processes. Under the same conditions, it is also proved that: the unique QSD attracts exponentially all initial distributions supported in the allowed states; the considered process admits a quasi-ergodic stationary distribution (QESD); the Lp-spectral bounds of its associated semigroup are independent of 1≤p≤∞. Finally, we present three typical examples to illustrate these results.