Exponentially quasi-mixing limits for killed symmetric Lévy processes

Quasi-mixing limits of the killed symmetric Lévy process are studied. It is proved that (intrinsic) ultracontractivity of the underlying process implies the existence of its (uniformly) exponentially quasi-mixing limits. As a by-product, this implication ensures that the process has (uniformly) exponential quasi-ergodicity and (uniformly) exponentially fractional quasi-ergodicity on ${L^{p}}$ ($p\ge 1$). It is noteworthy that precise rates of convergence and precise limiting equalities are provided, which are determined by spectral gaps and eigenfunction ratios of the underlying process. Finally, three examples are provided to demonstrate the theoretical results.