I have posted this question on there, but got no answer.
The following theorem is Theorem 13.3.3 of Meyn and Tweedie's Markov Chains and Stochastic Stability on page 328:
Theorem 13.3.3. If $\Phi$ is positive Harris and aperiodic, then for every initial distribution $\lambda$ $$||\int\lambda(dx)P^n(x,\cdot)-\pi||\to 0,n\to\infty$$ where $\pi$ is invariant for $P$.
For the continuous-time positive Harris Markov process, does the theorem also hold true? Or is there any similar property? As the book restricts to the discrete time case, could someone provide me some reference for the continuous-time case? Thanks very much!