Quasi-mixing limits, mean-ratio quasi-ergodicity and Q-processes of Markov processes

Under mild assumptions, we establish that the (uniform) exponential quasi-mixing limits of the underlying Markov process can be guaranteed by (intrinsic) ultracontractivity of its transition semigroup and dual semigroup. Other interesting properties are also considered, including (uniform) mean-ratio quasi-ergodicity proportional to time, and the conditional process converging (uniformly) exponentially to a (uniform) exponential ergodic process. It is worth emphasizing that the reference measure in this paper is permitted to be infinite. Finally, two examples are provided.