Consider a CT Markov Process $X=(X\_t)\_{t\geq0}$ with state space $E\in\mathbb{R}^N$. Are there any general conditions under which a stationary distribution $\pi$ for $X$ is also a limiting distribution and viceversa?


Any useful reference will be much appreciated.



Stationary and limiting distributions
For Markov chains, a very useful condition is Harris recurrence,
see <https://en.wikipedia.org/wiki/Harris_chain>.
This has been generalized to continuous time, see  

<https://www.jstor.org/stable/3690386?seq=1#metadata_info_tab_contents>


